Research ArticleSocial Interaction and the City The Effect of Space on theReduction of Entropy

Vinicius M Netto1 Joao Meirelles2 and Fabiano L Ribeiro34

1Universidade Federal Fluminense (UFF) Niteroi RJ Brazil2Laboratory on Human-Environment Relations in Urban Systems (HERUS) Ecole Polytechnique Federale de Lausanne (EPFL)Lausanne Switzerland3Universidade Federal de Lavras (UFLA) Lavras MG Brazil4City University of London London UK

Correspondence should be addressed to Vinicius M Netto vmnettoiduffbr

Received 13 March 2017 Accepted 21 June 2017 Published 23 August 2017

Academic Editor Dimitri Volchenkov

Copyright copy 2017 Vinicius M Netto et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

How can individual acts amount to coherent systems of interaction In this paper we attempt to answer this key question bysuggesting that there is a place for cities in the way we coordinate seemingly chaotic decisions We look into the elementaryprocesses of social interaction exploring a particular concept ldquosocial entropyrdquo or how social systems deal with uncertainty andunpredictability in the transition from individual actions to systems of interaction Examining possibilities that (i) actions relyon informational differences latent in their environments and that (ii) the city itself is an information environment to actions wepropose that (iii) space becomes a form of creating differences in the probabilities of interactionWe investigate this process throughsimulations of distinct material scenarios to find that space is a necessary but not sufficient condition for the reduction of entropyFinally we suggest that states and fluctuations of entropy are a vital part of social reproduction and reveal a deep connection betweensocial informational and spatial systems

1 Introduction Challenges forSocial Reproduction

Among a number of questions that might linger in thesociological imagination [1] one refers to social systems in aparticularly acute way how do we put our actions togetherin a way to create a society How can individual actionsdevelop into something like a working coherent system ofinteractions These questions feel more crucial if we thinkof the seemingly increasing challenges faced by contempo-rary societies Part of these challenges has to do with thegrowing profusion of information and messages circulatingwithin social systems (see [2]) For instance computationalprocessing power doubles every eighteen months while thevolume of data doubles every twelve months [3 4] makingthe processing of available information impossible in the longrun Interestingly Luhmann [5] sees this problem bringingincreasing difficulties for dealing with possibilities of actionand how we put them together as interaction systems a

challenge of selection that seems more and more imposed onour daily experiences According to Luhmann a key questionfaced by social systems is as follows ldquoamong the possibilitiesof information and interaction which ones will be actual-izedrdquo Considering that we have more possibilities than wecan know about the very way we build our interactionsdepends on this selection Crucially having more optionsends up bringing more difficulty in anticipating what willhappen next For all intents and purposes more channels ofinformation and possibilities of action mean less certaintyand predictability

This apparent difficulty has a name entropy a measureof the probability of events and recognizing order in the faceof unpredictability and uncertainty In fact social systemsface entropy all the time Our daily actions are riddled withuncertainty from daily choices we have to make to theway they will play out once they merge into the actions ofother people But seeing that requires a somewhat unusualperspective one aware of the conditions through which

HindawiComplexityVolume 2017 Article ID 6182503 16 pageshttpsdoiorg10115520176182503

2 Complexity

choices are made conditions that are prior to our actionsand have to be there so these possibilities may be known anddecided upon

We suggest that among all systems of communicationcreated to give us information about what is available outthere so that we can have counterparts in interaction thereis a crucial rather old one the city We will propose thatpossibilities of interaction are presented to us and to someextent preselected for us by our own environment and thatthis environment is deeply spatial shaped in the form ofcities Cities have historically been a key part of the processof information selection and actualization of actions Wediscuss in this article how this is the case how cities playa key role in how seemingly chaotic decisions amount tocoherent systems of interaction In short we are interestedin knowing how social organization emerges once its materialand informational conditions are taken simultaneously intoaccount namely in the form of an urban environment

Essentially we propose to track the passage from possibil-ities of action surrounding our decisions to the performanceof action in our daily lives and the variation of entropy thatthese passages entail We suggest that looking carefully intothe place of cities in those passages will clarify the constantorderingdisordering of action helping to explain socialreproduction more clearly We do so arguing that city spaceshave the effect of reducing uncertainty and unpredictabilityin the transition from individual actions to ensembles ofinteraction As social agents we would unconsciously engageinto processes of increasing and decreasing entropywheneverwe perform socially in the city that is whenever our actionshave effects on the world Since the organization of actions inour daily lives transcends local contexts and is largely beyondobservation we designed simulations in silico to try andclarify the effect of cities These computational experimentsare intended to explore how every time we go about thecity we participate in the fragile but recursive emergence ofinteraction systems

Once the problem is set Section 2 discusses entropyconcentrating on its developments in information and socialtheories in order to reach a little explored instance ofsocial entropy active in the coordination of action Section 3establishes a presence for cities as information environmentwhile Section 4 advances the city as a frame of reference toactions and how they coevolve into systems In turn Section 5hypothesizes how the reduction of entropy mediated bythe city becomes an essential part of the self-organizationof action We examine this process in Section 6 proposingan agent based model (ABM) able to assess the role ofdifferent social and spatial factors in entropy such as personalorientations and an extensive space creating friction formobile agents able to create and retrieve information fromtheir environment In the final section we discuss whatour theory and model say about the role of cities in socialinteraction

2 What Is Social Entropy

The concept of entropy a term derived from the Greek wordtropos ldquotransformationrdquo and introduced by the German

physicist Rudolf Clausius in 1865 originated in the physicalsystems context specifically the second law of thermodynam-ics which describes the irreversible process of energy dissipa-tion It was advanced by Ludwig Boltzmannrsquos connection ofentropy and probability in 1877 addressing nonequilibriumprocesses and linking macroscopic properties of a system toits microscopic disorder There are more possible disorderedstates than ordered ones so systems are likely to movetoward disorder From the infinite variety of dynamic statesentropy is a measure of the propensity for certain states Itcharacterizes each macroscopic state in terms of the numberof ways of achieving this state [6]

However entropy is not an exclusive property of physicalsystems The property was translated into ldquohuman affairsrdquoby Shannon [7] in the context of information transmissionShannon sees entropy related to the probability of observingcertain events over time and the tendency to increase thevariability of entities leading to uncertainty and unpre-dictability aswe shall see below In turn a number of theoristshave seen entropy in societies as well They have exploredShannonrsquos measure of information since the 1960s in a fieldthat came to be known as Social Entropy Theory (SET) (eg[8 9]) Charvat et al [10 11] proposed notions like ldquosemanticentropyrdquo in decision making and ldquoentropy of behaviourrdquo tomeasure the homogeneity of needs and dependence betweensystems while Horan [12] developed a measure of propor-tional reduction of uncertainty Threats to the social systemcome not only from the accumulation of internally producedentropy but also from the external environment Entropywasessentially seen as the opposite of information something tobe controlled by system boundaries (see [13])

Certain arguments are particularly interesting to ourapproach Galtung [14] analyzed entropy at micro- andmacroscopic levels using two basic types actor entropy andinteraction entropy Strong forces would push a social systeminto a pendulum oscillating between low- and high-entropystates The overall outcome of actions is not random sinceindividual agents would act within generalized roles followgeneralized norms and pursue generalized goals Socialnorms rules culture language and other ldquosteering mediardquoof a social system [15] would be constraints on behaviourand keep it from reaching maximum entropy The degreeof entropy would fluctuate cyclically or noncyclically ratherthan maintain a constant level Bailey [16 17] advanced thisidea by posing a question how can social systems increasetheir organizational complexity or decrease their entropyover time He argues that ldquosociety does face a number ofrecurring needs and thus has recurring goals which are metthrough recurring actions Inasmuch as these actions arereplicated they are said to be orderly [ ] It is this degreeof order which results from the patterned replication ofsocial action over time that results in a degree of entropy lessthan maximum If the constraints on action were removedit is reasonable to expect that society would not decreaseits entropy levels but might in fact move in the direction ofincreased entropy levelsrdquo [16 page 127] We shall see throughour model that the reduction of entropy does not have todepend on routine or constraints over action as Galtung andBailey argue but may emerge out of the very coordination of

Complexity 3

actions between agents pursuing changing goals mediated byan informational space

Other approaches take societies as collections of agentsinteracting in space-time within geographic boundaries [18]Bailey [19 20] also related entropy to measures of city sizepopulation size and territory Features from social cybernet-ics were also introduced such as context-dependency andthe self-reflexivity of agents and applied to issues like thedivision of labour problematic communication adaptationin societies of growing complexity the role of informationin decision making and the reduction of entropy [21 22]However such conceptualizations have a thin spatiality Inthem space is mostly a background not a part of the problemof entropy or its resolution in time Cities are virtually absentas part of the environment of social systems and as complexsystems in their own right In turn we will not find muchsupport in spatial disciplines either they havemostly ignoredthe conditions of production of actions and thereforethe place of cities in it Beyond other means to stimulatesocial organization usually seen in social theory (see [23])our aim is to deal with space as an active environmentalcondition and a steering medium in the coordination ofaction In order to include cities systemically we suggestgetting back to our initial question how can seemingly unpre-dictable individual acts amount to interconnected systems ofaction

We find a way to answer this question in Luhmann [5]a social system faces a number of possibilities of actionlarger than it can convert into actual actions which inturn imposes the need of selection Luhmann understandssocieties as a network of interconnected subsystems whereevents are communicatively formed The structural elementsthat enable these subsystems are fragile the fleeting butsuccessive moments of selection and connection betweenactionsThe reproduction of a social system requires the abil-ity to produce these connections In societies with growingamounts of information and agency the number of possibleinteractions grows exponentially Accordingly we have agrowing difficulty of knowing these possibilities and choosingamong them We have an increase in a type of complexitythat Luhmann calls ldquounstructuredrdquo an entropic informationalcomplexity that may at any moment lead into semantic andorganizational loss Rather than describing the actual courseof events this is an approach designed to throw light oncounterfactual possibilities and risks that social systems faceall the time Luhmann uncovers the effort we do every daychoosing agents and activities to interact with in order toreduce risks of not performing at all [5 page 287]

In order to fully appreciate this scenario of social sys-tems prone to entropy let us get back to the pioneeringdefinition of Shannon [7] Shannon defined entropy in thecontext of communication The fundamental problem ofcommunication is that of reproducing at one point a messageselected at another point either exactly or approximatelyMessages have meanings correlated with certain physicalor conceptual entities according to some system If thenumber of messages in a set is finite entropy is a measureof the distribution of probabilities that certain messages willoccur

An example will help clarify what this means as shown inthe following

Reduction of Entropy in the Construction of Language [7]

(1) Zero-order approximation (symbols independentand equiprobable)

XFOML RXKHRJFFJUJ ZLPWCFWKCYJ FFJ-EYVKCQSGHYD QPAAMKBZAACIBZL-HJ-QD

(2) First-order approximation (symbols independent butwith frequencies of English text)

OCRO HLI RGWR NMIELWIS EU LL NBNE-SEBYA TH EEI ALHENHTTPA OOBTTVANAH BRL

(3) Second-order approximation (diagram structure as inEnglish)

ON IE ANTSOUTINYS ARE T INCTORE STBE S DEAMY ACHIN D ILONASIVE TUCO-OWE AT TEASONARE FUSO TIZIN ANDYTOBE SEACE CTISBE

(4) Third-order approximation (trigram structure as inEnglish)

IN NO IST LAT WHEY CRATICT FROUREBIRS GROCID PONDENOME OF DEMONS-TURES OF THE REPTAGIN IS REGOAC-TIONA OF CRE

(5) First-order word approximation Rather than con-tinue with tetragram 119899-gram structure it is easierand better to jump at this point to word unitsHere words are chosen independently but with theirappropriate frequencies

REPRESENTING AND SPEEDILY IS ANGOOD APT OR COME CAN DIFFERENTNAT-URAL HERE HE THE A IN CAMETHE TO OF TO EXPERT GRAY COME TOFURNISHES THE LINE MESSAGE HAD BETHESE

(6) Second-order word approximation The word transi-tion probabilities are correct but no further structureis included

THE HEADAND IN FRONTAL ATTACKONAN ENGLISH WRITER THAT THE CHAR-ACTER OF THIS POINT IS THEREFOREANOTHER METHOD FOR THE LETTERSTHATTHETIMEOFWHOEVERTOLDTHEPROBLEM FOR AN UNEXPECTED

In Shannonrsquos diagram line (1) shows independent letterswith the same probability of occurrence On a ldquofirst-orderrdquoapproximation (line (2)) letters are independent of eachother and have the same probability of appearing in sentences

4 Complexity

that they have in English In line (3) we have the additionof the frequency with which letters follow other letters inpairs in English On the third-order approximation (line (4))letters are randomly ordered according to the likelihood ofassociation with others in trios Line (5) shows words chosenindependent to each other but with the same probability ofappearing in sentences that they have in English Finallyline (6) comprises the probability that a word is followed byanother specific word Even without decoding the meaningsof words a nearly intelligible message is emerging Differ-ences in frequency and sequence in letters andwords producedifferences in the probability of occurrence and an increasedprobability of certain combinations In practice this meansthat the message components become more predictable andintelligible The reduction of entropy allows communication

One could argue that Shannon entropy is anything butsemantic This is a crucial point Shannon indeed definesinformation irrespective of meaning However Shannonrsquoscoauthor in their 1949 book The Mathematical Theory ofCommunication Warren Weaver had already attempted toincorporate semantic information into Shannonrsquos theoryTheargument recently expanded by Haken and Portugali [24]shows that Shannon information participates in semanticinformation and vice versa as themind relates to the environ-ment deflatinginflating information variations in Shannoninformation entail differentmeanings Furthermore differentmeanings affect the quantity of information as they areproduced and retrieved through differences in events orentities however subject to ambiguities Aswe shall see belowfor the sake of assessing entropy pragmatically differencesin semantic information can be sufficiently captured throughShannon information This allows one to estimate semanticentropy through Shannon entropy

But what does this have to do with social entropy Soci-eties are interaction systems Peoplersquos actions are mediated byinformation they have meanings from personal orientation[25] to informational [5] and practical contents [26] Mean-ings produce informational difference We recognize actionsthrough such informational differences say in the contentsof utterances or in the tasks we perform with someone Inprinciple the greater the diversity of orientations and actionsthe harder it is to predict what someonersquos next action willbe But there is more to this temporal condition A state ofhigh entropy surrounds actions in a potential state before theycome into being when a number of possibilities of action lieahead Bringing Shannonrsquos and Luhmannrsquos insights togetherthe informational problem that societies face on a daily basislies in dealing with the selection of actions

We experience entropy whenever we deal with uncer-tainty options decisions unpredictable situations or toomuch information but we rarely realize we do so Entropy is aphenomenon ldquobeyond observationrdquo we cannot see or touchentropy even though we experience it on a daily basis Weare not used to think about the challenges we face in orderto make choices or put our actions together in any workablesense The key question here is how such a heightened fieldof choices and the taken-for-granted process of selectioninvolve our environment We need to understand how citiesare part of the semantic exchanges that constitute collective

action And the first step here is to understand city space asinformation in its own right

3 The City as Information Environment

Luhmann shows that social systems are immersed in seman-tic production meaning becomes their environment Wewish to further develop this idea exploring urban space aspart of such an environment In fact the idea of space as anenvironment able to contain social information has gainedgreat support recently FromVygotsky [27] toWilson [28] andHaken and Portugali [24] a number of cognitive properties ofspace and spatial properties of cognition have been identified

(i) Cognition Is Situated and Extended Our cognitive activityoccurs in the context of a real environment and inherentlyinvolves perception Internal cognitive processes are shapedby their coordination with external resources [27] Whilecognitive processes occur perceptual information continuesto be captured so as to affect our actions The ways in whichwe dive in cognitive activity are linked to our continuousinteraction with the environment [28] Theories of extendedmind assert a causal flow as the mind uses resources inthe environment and vice versa a two-way interaction in acoupled cognitive system [29] Order and systematicity inhuman cognition and action derive in part from the stabilityof our environment (Michaelian and Sutton 2013)

(ii) We Load the Built Environment with Information Cer-tain cognitive processes trigger associations with elementsof the environment through the incorporation of sociallyacquired information [30] In turn information is classifiedinto potentially shared categories [31] Clustering processesconnect pieces of information through similarities of physicalaspects of the environment or through relations between theirmeanings [32] Environmental elements (say buildings orplaces) are interpreted through such socially sharedmeanings[33] Visible elements of a city convey different amounts ofinformation [24] Small details (like a building entrance)can be related to higher orders of information and refer tolarger spatial formations (such as a main street or a citycentre) evolving to rich multilevel hierarchical structuresin which the interactions between information units in thelower levels generate patterns of environmental elements inhigher levels and vice versa The probability of a building orplace evoking a mental representation shared by people isenhanced by its physical appearance and visual identity alongwith its visibility and location in the environment and thesocial information associated with activities performed there([34] cf [35])

(iii) Spatial Information Relieves Our Cognitive Work Wedeal with the environment also through memory Howeverour short-term memory is constrained in its capability toprocess information [24] Luckily our knowledge of spatialproperties and patterns can integrate semantic visual andconfigurational aspects projected into an urban environmentas symbolic off-loading The spatial environment becomesan external memory carrying information about activities

Complexity 5

and agencies found in it for future retrieval [28] Insteadof trying to keep all the relevant details about activities inour short-term memory we retrieve these details from theenvironment itself as an extension memory and informationsource [36] Environmental information is a means to reducethe effort of memorization Other mnemonic features such askinesthetic images have been shown to functionally preservespatial and semantic properties of the external world [37]This external semantic resource relieves the cognitive loadand what Wilson [28] calls ldquorepresentational bottleneckrdquo thelimits faced by our internal memory which lead us to extendour mnemonics to the environment in the first place Theroles of long-term and short-term memories for agents ableto retrieve social information from space will be of particularinterest to our approach

(iv) Cognition Is Pressed for Time Daily life requires responsecapacity More cognitive capacity can be built up fromsuccessive layers of real-time interaction with our spatialcontext We create mental models of the environment fromwhich we create action plans Sophisticated forms of situatedcognition happen in any activity that involves the continuousupdating of plans in response to changing conditions ofaction Time pressure demands spatial decisions [38ndash40]

(v) Spatial Information Serves Action Agents build instruc-tions about events in the environment in an indexical form[41] The practical aims of agents can thus be directlyconnected to their situation [42 43] We manipulate theenvironment as a way of dealing with our practical problemsEmbodied cognition includes mechanisms serving adaptiveactivities [44] Our cognitive memory evolved in the serviceof perception and action in a three-dimensional environ-ment [28] Cognition serves action through a flexible andsophisticated strategy where information is stored for futureuse without firm commitments on what the future use maybe Spatial information can be absorbed through a varietyof uses for which it was not originally encoded This meansthat new uses may be derived from a stored representationof space They need not be triggered by direct observationof the environment and its affordances [45] Environmentalrepresentations either schematic or detailed appear to belargely free of a particular purpose or at least contain infor-mation beyond what is necessary for a specific action This iscertainly an adaptive cognitive strategyThe fact that humansencode the physical world using spatially and semanticallystructured mental models offer a huge advantage in solvingproblems

This nondiscursive knowledge includes spatial propertiesand heterogeneities and enables us to build inferences forexample when we try to imagine a likely street where tofind certain activity In turn Lakoff and Johnson [46] arguethat mental models are based on a modelling of the physicalworld and bolster in analogies between abstract and concretedomains Other works tie semantics with imagetic schemesable to incorporate knowledge of the physical world as away to encode relationships between events In perceivingsomething we perceive not only its observed form butalso the potential information enfolded in it [24] This very

property is crucial to the presence of urban space in theselection of actions to be performed

So our actions encode space with social informationand such informationally differentiated space plays a partin easing the representational bottleneck in our memoriesWe are able to retrieve information from this semanticarrangement that takes the form of the city before and duringour actions and experiences Such semantic layer is storedin activity places and in the spatial relations between themandmay be related to physical and functional heterogeneitiessuch as accessibility and centrality patterns [34] Such layersof spatial information can be evoked in our memories andused when we need to make inferences such as where wecould find a particular social situation or place Howevermost theories of situated cognition still do not recognize howthis informational space is part of the way we create actionsystems Let us see how this could be the case

4 The City as a Frame of Reference

Our aim has strong parallels with Hutchinsrsquos [47 48] viewon cultural ecosystems operating cognitively at larger spatialand temporal scales than an individual person According toHutchins all instances of cognition can be seen as emergingfrom distributed processes that is from interactions betweenelements in a system Cognitive properties are not predictablefrom individual cognitive capacities Humans create theircognitive powers by creating the environment in which theyexercise those powers This socially distributed cognition is acollective operation produced by interactions among agentsin active relation to their environment in contexts of ongoingactivity Possibilities for individual learning depend on thestructure of the environment to the point that the distributionof cognitive skills is determined by the distribution of prac-tices engaged in by people As Hutchins we are concernedwith the tasks people confront in their everyday world takingpart in processes of coordination of action as they movethrough physical spaces Ultimately our aim relates well tohis proposition that ldquocultural practices decrease entropy andincrease the predictability of experiencerdquo [48 page 46] Ourwork follows parallel lines through a systems approach ableto encompass large-scale systems of action and spaces

We propose to explore the informational role of city spacein how agents build connections between actions We allknow that urban space is produced to express human activityBut theories of cognition suggest much more space wouldbe semanticized by our actions That means that activityplaces are informationally differentiated and acquire a levelof semantic definition similar to types of action [49 50]This differentiation is enacted in and associated with the verystructure of urban space Space is differentiated in at leastthree levels (i) its physical properties of extensity and config-uration such as accessibility and centrality patterns in a city(ii) recognizable shapes in its visual and tridimensional form[24] and (iii) the semantic contents of activities performed inbuildings and places (cf [34 35]) (Figure 1)

How can this informational space become a part ofactionThere is a remarkable lack of theoretical and empiricalwork on this problem First we seem to retrieve meanings

6 Complexity

(a) (b) (c)

Figure 1 Differential space like a code of instructions for collective action urban space finds differentiation both physically as hierarchiesof accessibility and distribution of built form (a) and semantically through actions performed in buildings and places ((b) (c))

associated with our actions in architectural and urban spacesWe arguably grasp useful information about the activityperformed in a place not only by visual features but alsoinferring what people do there Space not only representsactivity it is actuated and thus loaded with meanings relatedto performance This is a semantic dimension of space spaceldquomeansrdquo as much as our actions precisely because it issemanticized by our actionsThis enacted information is veryimportant Action is temporally related to spatial locations

Second these locations allow us to access potential coun-terparts in interaction As we shall see below knowing acity means that we can find places where certain activitiesare performed Recognizing spatial differences and patternsallows us to infer such locations Apart from other sources ofinformation such as linguistic exchanges we do so becausewecan retrieve environmental information about activities andlocation patterns

Third places are means of connection we can actualizeconnections with certain people or activities during a periodof time These spatial links allow us to create complexes ofinteraction In practice places ldquodrawrdquo courses of action withdifferent orientations and contents and tie them togethermomentarily Flows of convergent and divergent actionshappen all the time when people leave their homes to workto seek services or go about to socialize and do so oftenwithout prior knowledge of activities counterparts or placesthey wish to visit (Figure 2) Urban space becomes whatParsons [51] would call an ldquoaction frame of referencerdquo aconnective fabric which both enables and constrains choicesmaterially and socially

5 Entropy as a Way into Social Organization

In order to be a part of selection the urban environmentneeds to be informational enough and not just to suggest

t3

t2

t1

L1 L2 L3 L4

Figure 2 As a reference system urban space converges courses ofaction in time (119905

1 1199052 and 119905

3) and space (119871

1 1198712 1198713 and 119871

4)

particular options but allow unforeseen changes in the courseof our actions Luhmann suggests that a desired social activitymust be within reach ldquoBecause complete interdependence isunattainable however interdependencies only come aboutby selection [ ] Successfully established interdependenciesthen serve as perspectives for and constraints on the struc-tural selections that connect onto themrdquo [5 page 284] As asocial event this connection is produced through communi-cation But before communication comes into being certainconditions have to be there spatially there Places must bechosen as conjunctive pieces that materialize connections

Complexity 7

Could cities stimulate connections If they could andgiven the fact that they are formed by space and thatspace is not fully malleable is it reasonable to supposethat certain spatial formations could ease such momentaryconnections We know that proximity increases the intensityof interaction [52 53] But what form should urbanized spacetake in order to meet such systemic expectations We mayunderstand such spatial conditions through a counterfactualscenario Imagine a fully chaotic urban formation with norecognizable concentrations or paths Visual elements couldnot convey information as spatial differences would notdevelop into structures emerging out of absolute heterogene-ity References to activities of interest would have to relyonly on our memory Selection of actions would becomedependent on very detailed mental representations of such achaotic formation Inferences about more likely places to findparticular activities would be very hard tomakeWewould beimmersed in a spatial environment without patterns to guideour choices reduce our efforts and amplify our interactionsIn a world of unstructured spaces actions would face lackof information noise and relentless entropy The possibilitythat space may have an informational presence in interactionexpresses the possibility that the physical environment mayhave informational properties and that the world is rich ininformation

Now not only our daily experience denies that cities aresuch completely chaotic structureless places but also spatialeconomics and urban studies offer plenty of theories andevidences that as complex as they may be cities are (at leastpartially) structured places From location patterns describedsince Alonso [54] and accessibility patterns described fromHansen [55] to Hillier [56] we know that urban patternsmatter they imply that activities can be arranged in ways thatare recognizable and accessible And from the point of view ofselection the realization that activities are somehow arrangedalong a spatial structure is a form of preselection Activitiestend to be distributed according to levels of accessibilityland values and practical dependencies facilitating comple-mentary actions Intentionally or not and however subject tocontingencies urban space is also structured (physically andinformationally) which allows possibilities of action to bemore easily known selected and viable In short at first theurban structure affords a range of possibilities for selectionIn a second moment the structure itself suggests gradients ofattractive possibilities

In a view of social systems constituted by connectionsof actions this means that space can be part of the constanttransition from individual acts to complexes of actions (indi-vidual act rarr places as connections rarr action systems) Ourhypothesis here to be tested in computational experimentsin the following section is that cities become a crucial part ofthis connectivity They would reduce risks such as the lack ofinformation or courses of action too costly to emerge

What does this say about the effect of cities on entropyBeginning with Luhmann interactions have structural valuebecause they represent selections from combinatorial possi-bilities If we relate this to Shannonrsquos entropy the very emer-gence of urban structures eliminates spatial scenarios whereevery possible connection would have the same probability

Another counterfactual scenario might help here Entirelyhomogeneous cities (or entirely heterogeneous for thatmatter) free from structure would present even probabilitiesof choice between actions Of course this would be a problemfor social systems connections potentially interesting tomostpeople would be as difficult to find as any If urban spacewas purely homogeneous or purely random the probabilityof finding an activity would be distributed homogeneouslyin space That is a scenario of maximum entropy The spatialenvironment would be useless in stimulating certain sought-after connections which could otherwise be there say whenwe concentrate activities like final supply or input-outputexchanges in production Effort time spent and economiccosts involved in finding activities of interest would rise

On the other hand when urban space finds differentiatedcontents and structures say location patterns along acces-sible paths it naturally creates differences in the probabilityof interaction and decreases in entropy Location patterns arein fact material and informational expressions of interactionsystems The differences engendered by such patterns gener-ate differences in the chances of finding an agent a service orcommerce If a city did not have recognizable internal forma-tions such as a centralities or high streets agents would havemore difficulties in knowingwhere they aremost likely to findwhat they are after A partially structured environment allowsus to build inferences about actions and agencies available in asocial systemWhen these agencies are also related temporallythrough proximity and functional complementarity there isan even greater potential for differences in probability toemerge suggesting new connections and sequences Whatall this means is that cities internally differentiated yet farfrom absolute heterogeneity have the effect of reducing socialentropy

Before experimenting with these relationships in silicolet us attempt an introductory description of what thesedifferences in probability entail in theway social systems han-dle entropy (Figure 3) Linearizing for the sake of simplicityprocesses that in fact occur simultaneously the reductionof social entropy would take the following form (a) In aninitial state free of space think of the actions as lines movingin time Agents can do anything and we cannot foreseewhat they will do a potential state of high entropy Colorsof the lines in Figure 4 represent different informationalcontents in personal orientations (b) Then agents startselecting actions and converging into different positions in aspatial system arranged as places or buildings (representedby colors in the vertical strip) in order to connect withothers Action lines might have subtle differences in relationto the places they approach so they converge into placesby informational proximity Connections can only happenif all other possibilities of connection are excluded (c)As these convergences happen the initially chaotic mazebecomes a momentarily coordinated system where agentscooperate The overall entropy of actions is reduced (d)After each spatially held event actions may change againaccording to changing orientations Accordingly the colorsof lines change Entropy increases as actions go into anothermomentary state of unpredictability as new possibilities arepresented to agents and somemust be selected if actions are to

8 Complexity

Momentarilycoherent systems

State of unpredictabledecisions

Action systems

Reduction of entropy

Coordination of actions

Selectionof actions

High entropy state

Information growth(Hidalgo 2015)

Possibilities of action(Luhmann 1995)

Differences in theprobability ofinteractions

(Shannon 1948)

Agents interact Orientations align inactivity places

Actions mediated byinformation

(Luhmann 1995)

City as informationenvironment(Haken and Portugali 2015)

Figure 3 Recursive process that conducts interaction systems from chaotic states (high entropy) into coherence (low entropy)

(a) (b) (c)

(e) (d)

Figure 4 The cycle of social entropy and space clockwise from the high unpredictability of potential actions to a momentarily orderedsystem as actions converge into places only to dive in entropy again

Complexity 9

Actionagent x

Actionagent y

Activity place

Cooperation

Change insubjectiveorientation

Figure 5 Aligning different actions in the cooperation process

be actualized (e) And then actions ldquoreferrdquo again to meaningsand positions in space starting a new cycle of entropy(Figure 4)

These distinct moments are theoretical of course Inreality reminding Prigogine and Stengers [6 page 124] thisldquomultitude of eventsrdquo takes place simultaneously and mayrender any clearly emergent cycle invisible But that neithermeans that individual cycles do not happen or aggregate inbunches nor they compensate one another like peaks andtroughs in colliding sound waves In fact routinization indaily life leads to rhythms and cycles unfolding from orfolding over the seemingly chaotic maze of actions But evenif actions were far from synchronic the fabric of interactionstill faces the challenge of entropy that is the challenge ofproducing differences in the probability of connections as a wayto ease organization whenever agents feel compelled to createnew interactions and this process is remarkably recursiveWe converge to places to act with others and do so on adaily basis Once engaged in an activity place interactionrequires levels of cooperationWemay bring our own distincttendencies or latent orientations in the form of long-termmemories of how to act along with short-term memoriesof a previous action both shaping our next actions As weact socially we bring our orientations into contact in away to open them in the communication process regulatingcollective action A result of this coordination is to reduceinitial differences Our actions ldquoalignrdquowithin an activity place(Figure 5)

All this suggests that social life involves a continuousalthough unconscious handling of entropy a vital oscillationbetween unpredictability and predictability As this processengages an enormous number of agents and events andtranscends places and contexts and therefore cannot beobserved as a whole entropy can only be fully grasped intheoretical representations A useful way to examine space asan information environment for action is through a compu-tational model a means to examine a counterfactual worldwhere other steering media such as language or social ruleswere ldquoswitched offrdquo keeping only properties of space activealong with agentsrsquo cognitive abilities to align orientationswith activities Once we abstract from other media wemay assess whether space could have any presence in theself-organization of action Let us examine this possibilitythrough some experiments

LBuildingactivity type

Figure 6The theoretical one-dimensional citywith a set of differentactivity places Here there are 119871 buildings distributed in a ringconstituting the city space Colors represent activity types

6 Digital ExperimentsA Unidimensional Model

61 Design Concepts An agent based model (ABM) is pro-posed Agents are defined to perform actions at each time-step and places by corresponding activity types The decisionon which action to perform next may be influenced bythree different conditions namely (i) latent orientation120590in

119894(119905)

representing the tendency of a single agent to act arounda particular type of action initially randomly distributedthis condition remains over time (ii) current action 120590

119894(119905)

performed by an agent as heshe selects an activity placein order to perform a new action this means an influenceof the current action while allowing gradual changes oforientation in time influenced by other agents and activitiesand (iii) types of activity places 120590

1199091015840(119905) where agents perform

supporting specific types of action Agents follow simplerules selecting their activity places closer to their latentorientations and current actions while being influencedby those activities Activity places are also influenced byvisiting agents but they change at a slower rate In shortagents coevolve with their spatial and social environmentThe model simulates different situations under this basicstructure Differences in the weight of these factors overthe next action may lead to quite different levels of socialentropy suggesting new possibilities for understanding thephenomenonAlthough there is a tradition inmodelling dailyactivities including a growing literature using digital loca-tional data (eg [57]) our ABM is not designed to representempirical sequences of actions or actual routines Instead itfocuses on trends that might emerge from the interfaces ofsimplified systems of action information and space

62 Variables and Scale Consider a unidimensional cityformed by agents and places Places form a ring with length(or perimeter) 119871 as shown in Figure 6 Our choice forrepresenting the city by a ring is justified by the minimalsufficient representation of spatial distance as a factor whose

10 Complexity

role in social organization is to be assessed This is allowedby a unidimensional model (eg [58ndash60]) The ring formallows continuous movement across a linear sequence oflocations eliminating centrality factors while taking intoaccount periodic boundary conditions in order to reduceborder effects eliminating the role of topologywhile isolatingthe problem of distance (see [61ndash63]) This stylized cityhouses 119873 agents which in each time-step 119905 select and visita specific place located at the position 119909 within the cityConsider that the position of the 119894th agent at the time 119905 isrepresented by 119909

119894(119905) The position of an agent can assume the

integer values 1 2 119871 according to the place that shehechooses to visit This means that at every time-step eachagent will choose one activity place to perform her nextaction

We considered 120590119894(119905) as the current action of the 119894-th agent

and 120590119909(119905) the activity performed in a place located at the

position 119909 = (1 2 119871) of the city both in time 119905 Wequantify action orientations weighting the three variables(latent orientation current action and activity place) Firstvariables are translated into different numeric values Let usassume that an orientation is an integer number between 0and 1000 each number represents a particular type of actionand numbers in their vicinity mean similar actions Theresulting orientation in time 119905 for each agent is the weightedaverage of those three parametersThedifference |120590

119894(119905)minus120590119909(119905)|

gives us the difference in orientation between an agent andthe activity place At every time-step each agent assesses thedifference between her orientation and every activity placein the city An agent chooses a specific place based on herorientation affinity with this place minimum difference

As we hypothesized above the urban structure has a rolein this selection In this experiment we reduced ldquourban struc-turerdquo to distance In order to investigate how distance may ornot interfere in this selection we propose two scenarios Inthe first one the agent selects her next activity based only onthe similarity in orientations between herself and the activityplace Agents can move across a space free of friction Spaceis not a constraintThis scenario considers only differences inorientation The parameter 119864

119894(119909 119905) estimates the interaction

between agent 119894 and the place located at 119909 at the time 119905 Theagent selects activity locations that minimize this quantityWe propose

119864119894 (119909 119905) =

1003816100381610038161003816120590119894 (119905) minus 1205901199091003816100381610038161003816 (1)

In the second more realistic scenario space imposesfriction to movement The agent takes into account thephysical distance between her current location in the city andthe location of her next activity and opts to minimize thisdistance along with the difference between her orientationand that of the activity place We propose

119864119894 (119909 119905) =

1003816100381610038161003816(119909119894 (119905) minus 119909) (120590119894 (119905) minus 120590119909)1003816100381610038161003816 (2)

Summing up an agent 119894 selects at time 119905 a particularactivity place located at119909 thatminimizes the function119864

119894(119909 119905)

In the first scenario she will choose activity places closerto her orientation In the second she will choose activityplaces closer in both orientation and physical distance This

brings a form of ldquoenergy costrdquo into the factors consideredSpace takes on a function role because of the energy used inldquovisitingrdquo Agents move in this linear city willing to minimizethis energy while searching for social information latentin places close to their changing orientations Changes inentropy levels are derived from this energy function In itsspatial version the model assesses effects of the effort toreduce energy in action minimizing informational differ-ences (between orientations and activity places) and spatialdistances (between agents and activity places)

63TheEvolution of ActionOrientations Orientation evolvesin time according to the following rule When an agentchooses and joins a particular place both agent and placebecome a little closer in terms of orientation This meansthat activity places are not immune to what agents performin them and that the urban activity system also changes intime To illustrate how orientations are constantly updatedconsider that agent 119894 selects an activity place located at 1199091015840Orientations will be updated according the rule

120590119894 (119905 + 1) =

1120572 + 120573 + 120574

(120572120590in119894(119905) + 120573120590119894 (119905) + 1205741205901199091015840 (119905)) (3)

Selection also depends on three additional parameterslatent orientation 120590in

119894(119905) weighted by parameter 120572 current

orientation120590119894(119905)weighted by parameter120573 and type of activity

place 1205901199091015840(119905) weighted by parameter 120574 This means that agents

will consider the three factors previously described (120590in119894(119905)

120590119894(119905) 1205901199091015840(119905)) with different weights (120572 120573 120574) to estimate which

activity she wants to perform in the next time-stepThe latent orientation evolves from an initial randomly

distributed behaviour 1205900119894 We consider that 120590in

119894(119905) is normally

distributed around 1205900119894 meaning that every agent has an

ldquoaveragerdquo latent orientation whose value will be derived ateach time-step from a normal distribution peaking at theagentrsquos average orientation If 120572 is sufficiently small in relationto 120573 and 120574 then the latent orientation does not play a rolein the agentrsquos action However if 120572 is sufficiently large (oncein relation to 120573 and 120574) then the action becomes stronglydependent on the latent orientation This would lead to apretty conservative city where inhabitants are not open tochanges in behaviour

The current action of an agent may influence a newaction with an intensity that depends on parameter 120573 If thisparameter is small in comparison to 120572 and 120574 then the agenthas no memory of an ongoing orientation However if 120573 issufficiently large then the new action is strongly dependenton a previous one (a Markovian behaviour)

The intensity of the activity placersquos influence on the agentrsquosaction depends on the value of 120574 If this parameter is smallin comparison to 120572 and 120573 then the activity place does notaffect her next actionmeaning that the place does not changebehaviours However if this parameter is sufficiently largethen the place plays a strong influence on future actions Theactivity place will be updated according to the rule

1205901199091015840 (119905 + 1) = 1205901199091015840 (119905) + 120579sum

119894isin1199091015840

120590119894 (119905) (4)

Complexity 11

where 120579 is a parameter sufficiently small That is placesare less influenced by agents than the other way roundThis means that at every time-step a place would have itstype of activity closer to the average of orientations of itsvisitors Therefore the main activity of the place will changeslightly according to visiting agents while changing theirorientations

64 Frequency of Orientations and Entropy Levels We cal-culate entropy assessing the distribution of resulting orien-tations of all agents in the social system at each time-stepusing Shannon entropy Our quantitative interpretation ofinformation removes ambiguities from the relation betweenShannon information and semantic information as differentorientations actions and types of activity places are repre-sented by numerical values This allows us to estimate socialentropy through Shannon entropy The entropy of actions iscalculated as follows consider119873(120590 119905) as the number of agentswith orientation 120590 at the time 119905 (note that the total populationis 119873 = sum

120590119873(120590 119905)) We compute the frequency (or density)

120588(120590 119905) of this orientation within a population with

120588 (120590 119905) = 1119873119873 (120590 119905) (5)

Equation (5) calculates the probability of observing anorientation 120590 at time 119905 We compute the entropy level for anydistribution of orientations with

119878 (119905) = minussum120590

120588 (120590 119905) ln (120588 (120590 119905)) (6)

Equation (6) describes how uneven is the probability of find-ing different orientations Higher values mean that differentorientations have almost the same probability to happenwhile lower values indicate a system with clear orientationtrends The reduction of entropy implies that the probabilityof certain actions increases that is actions grow in similarityIn the limit as entropy falls to zero all agents in the systemwould reach the same orientation

65Model Procedure Themodel performs the following pro-cedures

(1) The city is created by generating the following

(a) The type of activity in places that is (120590119909=1(119905 =

0) 120590119909=2(119905 = 0) 120590

119909=119871(119905 = 0)) also generated

from a random distribution This step createsactivity places in the ring-city and assign themrandomly distributed activity values 120590

1199091015840(119905)

(b) The initial position of 119873 agents that is (1199091

1199092 119909

119873) and their respective orientations

(1205901(119905 = 0) 120590

2(119905 = 0) 120590

119873(119905 = 0)) generated

from a random distribution This step createsmoving agents in random locations and assignthem latent orientations based on the initialrandomly distributed orientation 120590in

119894(119905) = 120590

119894(0)

(2) At each time-step 119905 every agent chooses a place toact This place should minimize the function 119864

119894(119909 119905)

Suppose that the chosen place by 119894 is defined as 1199091015840Then agent and place update their orientations 120590

119894and

1205901199091015840 as described above

(3) Update the distribution of orientations 120588(120590 119905)(4) Compute Shannon entropy 119878(119905)

66 Results We have developed a number of observationson the behaviour of social systems working under differentparameters

(i) Entropy reduction is only found in scenarios wherespace imposes friction to movement that is wherethe spatial distance between agents and activity placesis considered as an active factor in selection Entropyrequires an extensive space in order to be reduced Butspace cannot do it alone Surprisingly space is a nec-essary but not sufficient condition Figure 7 comparestwo scenarios In the first one distance is an issue inthe selection of activities (red line)Thedistribution oforientations changes from a homogeneous one at thestart of simulation (a) to a nearly normal distributionat the end of simulation (b) where certain kindsof action are more likely to happen Informationalcontents in extensive space help aligning contents inactions In the second scenario agents move free ofspatial friction (blue line) Orientations are initiallyrandomly distributed (a) and continue to be so at theend of simulation (b) as a similar number of agentsare distributed along different orientationsThis resultsuggests that a materially active space becomes ameans for increasing the probability of certain inter-actions easing the collective coordination of action

Nowassessing the relative influence of social and personalfactors on the selection of a next action namely the socialinformation in activity places latent orientation and the cur-rent action under the influence of spatial friction (Figure 8)we can say the following

(i) There is structure in the relation between selectionfactors and the reduction of entropy Entropy beginsto fall and fluctuate around specific values accordingto different combinations of factors For some com-binations there is a great reduction of entropy Forothers the reduction is minimal almost negligibleFluctuations derive from the random component inthe decisionmaking of the agent (under the influenceof latent orientation)

(ii) A strong latent orientation (120572 sufficiently large) leadsto increasing entropy (blue lines in Figures 8 and9) since agents cannot align their actions with otheragents and activity places Like a long-term memorysystems whose actions are dictated by latent orienta-tions are likely to preserve initial orientations whichwere randomly distributed and therefore homoge-neous This condition is responsible for limiting thereduction of entropy

(iii) Current action alone does not shape new actions inany specific way (green lines in Figures 8 and 9)

12 Complexity

Start of simulation

00000

00005

00010

00015

00020

o

f tot

al ac

tions

250 500 750 10000Type of action

Materially inactive spaceMaterially active space

t = 0

(a)

End of simulation

00000

00005

00010

00015

00020

o

f tot

al ac

tions

250 500 750 10000Type of action

Materially inactive spaceMaterially active space

t = 50

(b)

Figure 7 Histogram of probability distribution of actors performing different types of action at the start (a) and at the end (b) of simulationsin scenarios where distance is not considered (blue) and distance is considered aligning orientations in action coordination (red) Resultsare averaged for 30 runs for 125 parameter combinations

activityplace

currentaction

latentorientation

225

250

275

300

325

Entr

opy

10 20 30 40 500Time

Figure 8 Evolution of entropy colors in a RGB scale refer toparameter weights in decisions on actions in a next time-step andtheir impact on entropy levels in different simulations (under thefriction of distance) In the palette triangle each color representsa specific combination of the three parameters In each vertex aparameter reaches maximum value while the others have minimumvalue Results are averaged for 30 different runs for 125 parametercombinations

It just keeps what is already happening workingas ldquoreinforcement feedbackrdquo for either direction theagent (and the system) is going Current actions aremeans to conserve tendencies in the system

(iv) Activity places play a key role in the reduction ofentropy (red lines in Figures 8 and 9)The social infor-mation in space ldquocontaminatesrdquo agents they aligntheir actions through the social contents of placesWe have seen that space matters as extension Nowwe see that space also has a very active informationalpresence

(v) Different combinations of factors have differenteffects over entropy Figure 8 shows the transitions

between the three parameters and how nuancedweights resulting from blending factors matter inthe reduction of entropy For instance blending theweight of activity places and latent orientation on thenext action displays a strong reduction of entropy(pink lines in Figure 8) That means that when agentsdo not keep information from previous selections(ie 120573 asymp 0) the action system reduces more entropythan in other parameter combinations A strongshort-term memory leads the system into conservingitself and into a poorer capability to coordinateactions Also when current actions 120573 and activityplaces 120574 share a similar weight over the next action(orange lines on top) the social system does notexperience a great reduction of entropy as agents tendto reproduce their actions

(vi) Finally the reduction of entropy implies that theprobability of certain actions and interactionsincreases In practical terms this means morealignments between agents and more connectionsbetween actions (interactions) However if all agentsin the system reached the same orientation andentropy dropped to zero the system would lose inter-nal differentiation Agents would behave in the sameway say in a world with no personal differentiationspecialization or division of labour New orientations(therefore entropy) are necessary if the social systemis to keep differentiated agencies On the otherhand a system with maximum entropy would createdifference in actions to a point where no action coor-dination or interaction would be possible Clearly thissituation cannot be the case A social system requiresbalances neither full entropy nor total predictability

These results also suggest that we are dealing with distinctldquosocial memoriesrdquo a long-term memory is active in what we

Complexity 13

20

25

30En

trop

y

04 0602-latent orientation

04 0602-current action

20

25

30

Entr

opy

20

25

30

Entr

opy

04 0602-activity place

Figure 9 Parameter exploration isolated importance of each parameter in entropy (under the friction of distance) Each vertical set of dotsrepresents one fixed value for the highlighted parameter with the two other parameters assuming every possible combination within thetested range Each dot corresponds to the entropy value when a sufficiently large time-frame is used

call latent orientations Short-term memory is active in theinfluence of a current orientation over a new action In turnthe social information latent in activity places is a result ofsocial arrangements (say as firms a local economy and soon) which find certain stability changing much slower thenactions It is an extension of the social system projected ontourban space stabilizing the system to some extent

7 Cities and Social Interaction Conclusions

What does this approach bring to the state of the art on therelations of social interaction information and space and onsocial entropy and ABM in particular We have seen thatprevious approaches overlooked the coordination of actionas a key empirical and analytical problem Furthermorethey tend to have the thin spatiality of a passive territorialbackground (eg [16 20]) In turn although the conceptof ldquocollective actionrdquo is finally getting attention in urbanstudies [64] the problem of social organization is still largelyunderestimated The problem of entropy has been dealt withsince Wilsonrsquos [65] work on spatial interaction but has notreached the mainstream of the discipline and deals mostlywith urban form (eg [24]) and spatial distributions [66]not with social information and action Regarding agentbased models we are not aware of an approach that dealswith the problem of how cities and space are part of socialorganization In a sense our model is closer to Axelrodrsquos[67] ABM model of dissemination of culture His agentsexchange ldquoculturerdquo through direct contact with neighboursin a cellular automata model whereas our agents are mobileand actively deal with distance and social information inspace recognizing types of activities in places The intensityof exchanges between agents in Axelrodrsquos model is a functionof similarities between them In our model exchanges aremediated by the selection and interaction with space In ourunderstanding our contribution in terms of ABM lies onthe fact that a new and simple model is proposed to dealwith the material and informational dimensions of socialorganization not just as an illustration but as a proof ofconcept space can have causal presence if agents generatepreferences as detailed in a domain that coevolves with them

Now does the simulation model corroborate the theoryintroduced in this paper Our proposition cannot be empir-ically validated at this stage given that it is very hard to

assess peoplersquos choices and the whole panorama of actionsin a city Entropy may well be beyond observation In thesecircumstances simulations of the behaviour of agents underdifferent spatial conditions become useful to assess entitiesevents and entropic forces at play One expected result fromthe model is that space matters for coordinating actions butsimulations surprisingly showed that space is a necessary butnot sufficient condition for the reduction of entropy Neithermaterial cognitive resources nor individual powers of agentsare enough Information in activity places and a bit of agentsrsquopersonal history also play roles in the way social systemsdeal with their own entropy That would mean a causal butnondeterministic presence of space

Substantively this approach explores a subtle but funda-mental presence of cities in social organizationWe argue thaturban space materializes gradients of difference in potentialinteractions from less to more recognizable costly or likelyPlacing simple criteria including distance as a factor in select-ing activities our ABM showed that space becomes a meansfor producing differences in the probabilities of interactionincreasing chances of certain selections and convergences incollective action

In reality people recognize the social information ofactivity places along with uses of busy streets or local central-ities We are able to retrieve these differences from the builtenvironment and relate to them One of the key points of ourargument is that these differences help us make selectionsEvery time we move through the city and select places toperform we participate in the large-scale coordination ofaction We enact space as a ldquoreferential systemrdquo [49 50]a set of semantic bits indexing social activities informingand guiding daily decisions Space becomes inherently partof the conversion of orientations into interactions that isconnections that crisscross sequences of actions recreatingthe interaction system By means of a structure that bothmaterializes and restricts the quasi-endless combinatorial pos-sibilities of interaction this system can acquire sufficientinternal guidance to make its own reproduction possibleBy distributing sufficiently recognizable differences in theprobability of interaction cities express and release localforces of reproduction of social systems The interfaces ofaction cognitive and spatial systems transform entropiccomplexity into structured complexity constantly reorderingactions in time

14 Complexity

This materialistic viewpoint can be integrated with otherviews of social organization Further work may include therelational roles of cultures language social norms and othersteering media along with explorations of possibly similareffects of communication technologies over entropy At thisstage we attempted to integrate a social dimension (agentsengaged in coordination) a spatial dimension (an extensiveenvironment) and an informational dimension (as differ-ences in action and in the environment itself) a demandinginterdisciplinary effort The approach identifies a central rolefor coevolving agents actualizing aims as they coordinate (cf[68]) and a central role for the environment exterior butresponsive to agents leading to cumulative changes alongtheir historyThemodel also identifies roles for cognition andmemory in the ways agents deal with their own orientationsand choose their actions and activity places changing theirspatial environment with consequences in the overall levelsof entropy All this suggests that the behaviour of the modelcannot be reduced to or fully predicted from the behaviour ofindividual variables which is a key aspect of complexity

So one of the main aims of this paper was to assesswhether space as information environment had any effecton the coordination and entropy of interaction However ifphysical distance and social information in activity placesmatter what about city size and the internal spatial struc-tures of cities What about rural areas or rarefied suburbsSince our model explores an abstract circular city it onlybegins to answer such questions Our findings indicate thatshorter distances between activity places tend to reducesocial entropy Density seems to matter denser spatialities(as opposed to rarefied ones) have a role to play A pathfor further development here is Jacobsrsquos [69] idea that thediversity of activities generates positive externalities Anddiversity as spatial economics shows us has to do withthe size of cities and density of population (eg [70 71])Entropy may also be influenced by diversity Batty et al [66]have shown that information increases as cities get biggerThe remaining question is whether larger denser and moreinternally structured cities could create process and reduceentropy with more intensity than smaller cities transformingbig pools of activities into differences in the probability ofinteraction Although we proposed a unidimensional modelour agentsrsquo behaviour can be explored in more realisticrepresentations of cities testing the roles of density topologyand diversity while going beyond perfectly informed agentsBy proposing the ring model we intended to simplify acity to a minimum system where the spatial dimensionof social organization becomes intelligible without losingfundamental properties

Shouldwe conclude fromall this that entropy is a problemfor social systems Our approach suggests that it is notEntropy is a necessary force It means that novel orientationsare entering the interaction system and must be dealt withEntropy only becomes a problem if not converted into orga-nization only if numerous potential actions are not convertedinto smaller sets of actual interactions The effects of newactions over entropy would be positive once actualizedhaving to dowith the diversity of agencyConservative systemsare likely to face less entropy but also likely to produce

less novelty having homogeneity as a problematic horizonFluctuations of entropy seem vital to complex societies

Finally our intention was to explore entropy as a meansto think about the conditions of social organization ldquofromanother anglerdquo so to speak from the viewpoint of challengesinvolved in social reproduction Hence the approach focusedon the idea of uncertainty surrounding agents dealing withaims and decisions and the different probabilities of inter-action that follow an attempt to bring social cognitiveinformational and spatial theories under a single roof Ourapproach sees cities as connective systems produced to createthe delicate fabric of interaction that keeps large numbersof agents coherently living together It also suggests that thevarying states of entropy reveal deep connections betweensocial informational and spatial systems

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

Theauthors would like to thank the Ecole Polytechnique Fed-erale de Lausanne (EPFL) CAPES FAPEMIG FAPESP andCNPq for financial support Maıra Pinheiro Henrique Loreaand colleagues at ABMUS2017 for earlier discussions andMike Batty for exchanges and inspiration Finally the authorsthank Romulo Krafta maverick of the complexity studiesof cities in Brazil for his input and support throughout theyears This work is dedicated to him

References

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[2] M Castells ldquoThe information age economy society and cul-turerdquo inThe Rise of the Network Blackwell London 1996

[3] C Hidalgo Why Information Grows The Evolution of Orderfrom Atoms to Economies Basic Books New York NY USA2015

[4] D Helbing The Automation of Society Is Next ETH ZurichSwitzerland 2016

[5] N Luhmann Social Systems University Press Stanford 1995[6] I Prigogine and I Stengers Order out of Chaos Manrsquos New

Dialogue with Nature Bantam Books 1984[7] C E Shannon ldquoAmathematical theory of communicationrdquo Bell

System Technical Journal vol 27 no 4 pp 623ndash656 1948[8] W Buckley Sociology and Modern Systems Theory Prentice-

Hall Englewood Cliffs NJ USA 1967[9] D D McFarland ldquoMeasuring the Permeability of Occupational

Structures An Information-Theoretic Approachrdquo AmericanJournal of Sociology vol 75 no 1 pp 41ndash61 1969

[10] F Charvat ldquoOn philosophical aspects of the system conceptionin contemporary sociological knowledgerdquo Quality and Quan-tity vol 6 no 1 pp 3ndash16 1972

[11] F Charvat J Kucera and M Soukup ldquoToward the systemtheory of dependence further general theoretical remarksrdquoQuality and Quantity vol 7 no 1 pp 69ndash90 1973

Complexity 15

[12] P M Horan ldquoInformation-theoretic measures and the analysisof social structuresrdquo Sociological Methods and Research vol 3no 3 pp 321ndash340 1975

[13] O E Klapp ldquoOpening and closing in open systemsrdquo BehavioralScience vol 20 no 4 pp 251ndash257 1975

[14] J Galtung ldquoEntropy and the general theory of peacerdquo in Essaysin Peace Research J Galtung Ed vol L Humanities PressAtlantic Highlands NJ USA 1975

[15] T Parsons The System of Modern Societies Englewood CliffsNew Jersey NJ USA 1972

[16] K D Bailey ldquoSociological entropy theory Toward a statisticaland verbal congruencerdquo Quality amp Quantity vol 18 no 1 pp113ndash133 1983

[17] K D Bailey Social EntropyTheory State University of NewYorkPress Albany NY USA 1990

[18] J G Miller Living Systems McGraw-Hill New York NY USA1978

[19] K D Bailey Sociology and the New Systems Theory Toward aTheoretical Synthesis State University of New York PressAlbany NY USA 1994

[20] K D Bailey ldquoSociocybernetics and social entropy theoryrdquoKybernetes vol 35 no 3-4 pp 375ndash384 2006

[21] F Geyer ldquoThe challenge of sociocyberneticsrdquo Kybernetes vol24 no 4 pp 6ndash32 1995

[22] F Geyer ldquoThe march of self-referencerdquo Kybernetes vol 31 no7-8 pp 1021ndash1042 2002

[23] J HabermasTheTheory of Communicative Action Polity PressCambridge UK 1987

[24] H Haken and J Portugali Information Adaptation The Inter-play between Shannon Information and Semantic Information inCognition Springer New York NY USA 2015

[25] M Weber Economy and Society vol 1 Berkeley University ofCalifornia Press 1978

[26] L Wittgenstein Philosophical Investigations Blackwell Publish-ers Oxford London 3rd edition 2001

[27] L Vygotsky Mind in Society The Development of HigherPsychological Processes Harvard University Press CambridgeMA 1978

[28] M Wilson ldquoSix Views of Embodied Cognitionrdquo PsychonomicBulletin amp Review vol 9 no 4 pp 625ndash636 2002

[29] A Clark and D Chalmers ldquoThe extended mindrdquo Analysis vol58 no 1 pp 7ndash19 1998

[30] R PassiniWayfinding in Architecture Van Nostrand ReinholdNew York NY USA 1992

[31] E Rosch ldquoPrinciples of categorizationrdquo in Cognition and Cate-gorization E Rosch and B Lloyd Eds Hillsdale LawrenceErlbaum Ass 1978

[32] W KintschMemory and Cognition John Wiley and Sons NewYork NY USA 1970

[33] U Neisser ldquoMultiple Systems a New Approach to CognitiveTheoryrdquo European Journal of Cognitive Psychology vol 6 no 3pp 225ndash241 1994

[34] A Faria and R Krafta ldquoRepresenting urban cognitive structurethrough spatial differentiationrdquo in Proceedings of 4th Space Syn-tax International Symposium pp 531-518 UCL Press London2003

[35] J Portugali Complexity Cognition and the City Springer NewYork NY USA 2011

[36] D Kirsh and P Maglio ldquoOn distinguishing epistemic frompragmatic actionrdquo Cognitive Science vol 18 no 4 pp 513ndash5491994

[37] S M Kosslyn Image and Brain The Resolution of the ImageryDebate MIT Press Cambridge MA

[38] R Brooks Cambrian Intelligence The Early History of the NewAI MIT Press Cambridge MA 1999

[39] A Clark BeinGThere Putting Brain Body AndWorld TogetherAgain MIT Press Cambridge MA 1997

[40] R Pfeifer and C ScheierUnderstanding Intelligence MIT PressCambridge MA 1999

[41] A M Glenberg and D A Robertson ldquoIndexical understandingof instructionsrdquoDiscourse Processes vol 28 no 1 pp 1ndash26 1999

[42] JM Iverson and SGoldin-Meadow ldquoWhypeople gesturewhenthey speakrdquo Nature vol 396 no 6708 p 228 1998

[43] R M Krauss ldquoWhy do we gesture when we speakrdquo CurrentDirections in Psychological Science vol 7 no 2 pp 54ndash60 1998

[44] S Franklin Artificial Minds MIT Press Cambridge MA 1995[45] J Gibson The Ecological Approach to Visual Perception

Houghton-Mifflin Boston 1979[46] G Lakoff andM JohnsonPhilosophy in the FleshTheEmbodied

Mind and Its Challenge to Western Thought Basic Books NewYork NY USA 1999

[47] E HutchinsCognition in theWild MIT Press CambridgeMA1995

[48] E Hutchins ldquoThe cultural ecosystem of human cognitionrdquoPhilosophical Psychology vol 27 no 1 pp 34ndash49 2014

[49] V M Netto ldquoPractice space and the duality of meaningrdquoEnvironment and Planning D Society and Space vol 26 no 2pp 359ndash379 2008

[50] V M Netto The Social Fabric of Cities Routledge New YorkNY USA 2017

[51] T Parsons The Structure of Social Action The Free Press NewYork NY USA 1968

[52] T Allen Managing the Flow of Technology MIT Press Cam-bridge MA 1977

[53] F L Ribeiro J Meirelles F F Ferreira and R C Neto ldquoAmodel of urban scaling laws based on distance-dependent inter-actionsrdquo Royal Society Open Science vol 4 Article ID 1609262017

[54] W Alonso Location and Land Use Toward a General Theory ofLand Rent Harvard University Press Cambridge MA 1964

[55] WGHansen ldquoHow accessibility shapes land userdquo Journal of theAmerican Institute of Planners vol 25 no 2 pp 73ndash76 1959

[56] B Hillier Space Is the Machine Cambridge University PressCambridge UK 1996

[57] S Farber M OrsquoKelly H J Miller and T Neutens ldquoMeasuringsegregation using patterns of daily travel behavior a socialinteraction based model of exposurerdquo Journal of TransportGeography vol 49 pp 26ndash38 2015

[58] T C Schelling ldquoDynamic models of segregationrdquo Journal ofMathematical Sociology vol 1 no 2 pp 143ndash186 1971

[59] R Krafta V M Netto and L Lima ldquoUrban built form growscriticalrdquo CyberGeo vol 2011 article 565 2011

[60] N Lanchier and S Scarlatos ldquoFixation in the one-dimensionalAxelrod modelrdquo The Annals of Applied Probability vol 23 no6 pp 2538ndash2559 2013

[61] F L Ribeiro and K N Ribeiro ldquoA one dimensional modelof population growthrdquo Physica A Statistical Mechanics and itsApplications vol 434 pp 201ndash210 2015

[62] F L Ribeiro ldquoA non-phenomenological model of competitionand cooperation to explain population growth behaviorsrdquo Bul-letin of Mathematical Biology vol 77 no 3 pp 409ndash433 2015

16 Complexity

[63] R V dos Santos F L Ribeiro and A S Martinez ldquoModels forAllee effect based on physical principlesrdquo Journal of TheoreticalBiology vol 385 pp 143ndash152 2015

[64] M BattyThe New Science of Cities The MIT Press CambridgeMA 2013

[65] A G Wilson Entropy in Urban and Regional Modelling PionLondon 1970

[66] M Batty R Morphet P Masucci and K Stanilov ldquoEntropycomplexity and spatial informationrdquo Journal of GeographicalSystems vol 16 no 4 pp 363ndash385 2014

[67] R Axelrod ldquoThe dissemination of culture a model with localconvergence and global polarizationrdquo Journal of Conflict Reso-lution vol 41 no 2 pp 203ndash226 1997

[68] B H Hodges and RM Baron ldquoValues as constraints on afford-ances perceiving and acting properlyrdquo Journal for the Theory ofSocial Behaviour vol 22 no 3 pp 263ndash294 1992

[69] J JacobsThe Economy of Cities Random house New York NYUSA 1969

[70] E Glaeser H Kallal J Scheinkman and A Shleifer ldquoGrowthin Citiesrdquo Journal of Political Economy vol 100 no 6 pp 1126ndash1152 1992

[71] S Rosenthal and W Strange ldquoEvidence on the nature andsources of agglomeration economiesrdquo in Handbook of Urbanand Regional Economics J V Henderson and J-F Thisse Edsvol 4 pp 2119-2171 NorthHollandNewYork NYUSA 2004

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Stochastic AnalysisInternational Journal of

Complexity 3

actions between agents pursuing changing goals mediated byan informational space

Other approaches take societies as collections of agentsinteracting in space-time within geographic boundaries [18]Bailey [19 20] also related entropy to measures of city sizepopulation size and territory Features from social cybernet-ics were also introduced such as context-dependency andthe self-reflexivity of agents and applied to issues like thedivision of labour problematic communication adaptationin societies of growing complexity the role of informationin decision making and the reduction of entropy [21 22]However such conceptualizations have a thin spatiality Inthem space is mostly a background not a part of the problemof entropy or its resolution in time Cities are virtually absentas part of the environment of social systems and as complexsystems in their own right In turn we will not find muchsupport in spatial disciplines either they havemostly ignoredthe conditions of production of actions and thereforethe place of cities in it Beyond other means to stimulatesocial organization usually seen in social theory (see [23])our aim is to deal with space as an active environmentalcondition and a steering medium in the coordination ofaction In order to include cities systemically we suggestgetting back to our initial question how can seemingly unpre-dictable individual acts amount to interconnected systems ofaction

We find a way to answer this question in Luhmann [5]a social system faces a number of possibilities of actionlarger than it can convert into actual actions which inturn imposes the need of selection Luhmann understandssocieties as a network of interconnected subsystems whereevents are communicatively formed The structural elementsthat enable these subsystems are fragile the fleeting butsuccessive moments of selection and connection betweenactionsThe reproduction of a social system requires the abil-ity to produce these connections In societies with growingamounts of information and agency the number of possibleinteractions grows exponentially Accordingly we have agrowing difficulty of knowing these possibilities and choosingamong them We have an increase in a type of complexitythat Luhmann calls ldquounstructuredrdquo an entropic informationalcomplexity that may at any moment lead into semantic andorganizational loss Rather than describing the actual courseof events this is an approach designed to throw light oncounterfactual possibilities and risks that social systems faceall the time Luhmann uncovers the effort we do every daychoosing agents and activities to interact with in order toreduce risks of not performing at all [5 page 287]

In order to fully appreciate this scenario of social sys-tems prone to entropy let us get back to the pioneeringdefinition of Shannon [7] Shannon defined entropy in thecontext of communication The fundamental problem ofcommunication is that of reproducing at one point a messageselected at another point either exactly or approximatelyMessages have meanings correlated with certain physicalor conceptual entities according to some system If thenumber of messages in a set is finite entropy is a measureof the distribution of probabilities that certain messages willoccur

An example will help clarify what this means as shown inthe following

Reduction of Entropy in the Construction of Language [7]

(1) Zero-order approximation (symbols independentand equiprobable)

XFOML RXKHRJFFJUJ ZLPWCFWKCYJ FFJ-EYVKCQSGHYD QPAAMKBZAACIBZL-HJ-QD

(2) First-order approximation (symbols independent butwith frequencies of English text)

OCRO HLI RGWR NMIELWIS EU LL NBNE-SEBYA TH EEI ALHENHTTPA OOBTTVANAH BRL

(3) Second-order approximation (diagram structure as inEnglish)

ON IE ANTSOUTINYS ARE T INCTORE STBE S DEAMY ACHIN D ILONASIVE TUCO-OWE AT TEASONARE FUSO TIZIN ANDYTOBE SEACE CTISBE

(4) Third-order approximation (trigram structure as inEnglish)

IN NO IST LAT WHEY CRATICT FROUREBIRS GROCID PONDENOME OF DEMONS-TURES OF THE REPTAGIN IS REGOAC-TIONA OF CRE

(5) First-order word approximation Rather than con-tinue with tetragram 119899-gram structure it is easierand better to jump at this point to word unitsHere words are chosen independently but with theirappropriate frequencies

REPRESENTING AND SPEEDILY IS ANGOOD APT OR COME CAN DIFFERENTNAT-URAL HERE HE THE A IN CAMETHE TO OF TO EXPERT GRAY COME TOFURNISHES THE LINE MESSAGE HAD BETHESE

(6) Second-order word approximation The word transi-tion probabilities are correct but no further structureis included

THE HEADAND IN FRONTAL ATTACKONAN ENGLISH WRITER THAT THE CHAR-ACTER OF THIS POINT IS THEREFOREANOTHER METHOD FOR THE LETTERSTHATTHETIMEOFWHOEVERTOLDTHEPROBLEM FOR AN UNEXPECTED

In Shannonrsquos diagram line (1) shows independent letterswith the same probability of occurrence On a ldquofirst-orderrdquoapproximation (line (2)) letters are independent of eachother and have the same probability of appearing in sentences

4 Complexity

that they have in English In line (3) we have the additionof the frequency with which letters follow other letters inpairs in English On the third-order approximation (line (4))letters are randomly ordered according to the likelihood ofassociation with others in trios Line (5) shows words chosenindependent to each other but with the same probability ofappearing in sentences that they have in English Finallyline (6) comprises the probability that a word is followed byanother specific word Even without decoding the meaningsof words a nearly intelligible message is emerging Differ-ences in frequency and sequence in letters andwords producedifferences in the probability of occurrence and an increasedprobability of certain combinations In practice this meansthat the message components become more predictable andintelligible The reduction of entropy allows communication

One could argue that Shannon entropy is anything butsemantic This is a crucial point Shannon indeed definesinformation irrespective of meaning However Shannonrsquoscoauthor in their 1949 book The Mathematical Theory ofCommunication Warren Weaver had already attempted toincorporate semantic information into Shannonrsquos theoryTheargument recently expanded by Haken and Portugali [24]shows that Shannon information participates in semanticinformation and vice versa as themind relates to the environ-ment deflatinginflating information variations in Shannoninformation entail differentmeanings Furthermore differentmeanings affect the quantity of information as they areproduced and retrieved through differences in events orentities however subject to ambiguities Aswe shall see belowfor the sake of assessing entropy pragmatically differencesin semantic information can be sufficiently captured throughShannon information This allows one to estimate semanticentropy through Shannon entropy

But what does this have to do with social entropy Soci-eties are interaction systems Peoplersquos actions are mediated byinformation they have meanings from personal orientation[25] to informational [5] and practical contents [26] Mean-ings produce informational difference We recognize actionsthrough such informational differences say in the contentsof utterances or in the tasks we perform with someone Inprinciple the greater the diversity of orientations and actionsthe harder it is to predict what someonersquos next action willbe But there is more to this temporal condition A state ofhigh entropy surrounds actions in a potential state before theycome into being when a number of possibilities of action lieahead Bringing Shannonrsquos and Luhmannrsquos insights togetherthe informational problem that societies face on a daily basislies in dealing with the selection of actions

We experience entropy whenever we deal with uncer-tainty options decisions unpredictable situations or toomuch information but we rarely realize we do so Entropy is aphenomenon ldquobeyond observationrdquo we cannot see or touchentropy even though we experience it on a daily basis Weare not used to think about the challenges we face in orderto make choices or put our actions together in any workablesense The key question here is how such a heightened fieldof choices and the taken-for-granted process of selectioninvolve our environment We need to understand how citiesare part of the semantic exchanges that constitute collective

action And the first step here is to understand city space asinformation in its own right

3 The City as Information Environment

Luhmann shows that social systems are immersed in seman-tic production meaning becomes their environment Wewish to further develop this idea exploring urban space aspart of such an environment In fact the idea of space as anenvironment able to contain social information has gainedgreat support recently FromVygotsky [27] toWilson [28] andHaken and Portugali [24] a number of cognitive properties ofspace and spatial properties of cognition have been identified

(i) Cognition Is Situated and Extended Our cognitive activityoccurs in the context of a real environment and inherentlyinvolves perception Internal cognitive processes are shapedby their coordination with external resources [27] Whilecognitive processes occur perceptual information continuesto be captured so as to affect our actions The ways in whichwe dive in cognitive activity are linked to our continuousinteraction with the environment [28] Theories of extendedmind assert a causal flow as the mind uses resources inthe environment and vice versa a two-way interaction in acoupled cognitive system [29] Order and systematicity inhuman cognition and action derive in part from the stabilityof our environment (Michaelian and Sutton 2013)

(ii) We Load the Built Environment with Information Cer-tain cognitive processes trigger associations with elementsof the environment through the incorporation of sociallyacquired information [30] In turn information is classifiedinto potentially shared categories [31] Clustering processesconnect pieces of information through similarities of physicalaspects of the environment or through relations between theirmeanings [32] Environmental elements (say buildings orplaces) are interpreted through such socially sharedmeanings[33] Visible elements of a city convey different amounts ofinformation [24] Small details (like a building entrance)can be related to higher orders of information and refer tolarger spatial formations (such as a main street or a citycentre) evolving to rich multilevel hierarchical structuresin which the interactions between information units in thelower levels generate patterns of environmental elements inhigher levels and vice versa The probability of a building orplace evoking a mental representation shared by people isenhanced by its physical appearance and visual identity alongwith its visibility and location in the environment and thesocial information associated with activities performed there([34] cf [35])

(iii) Spatial Information Relieves Our Cognitive Work Wedeal with the environment also through memory Howeverour short-term memory is constrained in its capability toprocess information [24] Luckily our knowledge of spatialproperties and patterns can integrate semantic visual andconfigurational aspects projected into an urban environmentas symbolic off-loading The spatial environment becomesan external memory carrying information about activities

Complexity 5

and agencies found in it for future retrieval [28] Insteadof trying to keep all the relevant details about activities inour short-term memory we retrieve these details from theenvironment itself as an extension memory and informationsource [36] Environmental information is a means to reducethe effort of memorization Other mnemonic features such askinesthetic images have been shown to functionally preservespatial and semantic properties of the external world [37]This external semantic resource relieves the cognitive loadand what Wilson [28] calls ldquorepresentational bottleneckrdquo thelimits faced by our internal memory which lead us to extendour mnemonics to the environment in the first place Theroles of long-term and short-term memories for agents ableto retrieve social information from space will be of particularinterest to our approach

(iv) Cognition Is Pressed for Time Daily life requires responsecapacity More cognitive capacity can be built up fromsuccessive layers of real-time interaction with our spatialcontext We create mental models of the environment fromwhich we create action plans Sophisticated forms of situatedcognition happen in any activity that involves the continuousupdating of plans in response to changing conditions ofaction Time pressure demands spatial decisions [38ndash40]

(v) Spatial Information Serves Action Agents build instruc-tions about events in the environment in an indexical form[41] The practical aims of agents can thus be directlyconnected to their situation [42 43] We manipulate theenvironment as a way of dealing with our practical problemsEmbodied cognition includes mechanisms serving adaptiveactivities [44] Our cognitive memory evolved in the serviceof perception and action in a three-dimensional environ-ment [28] Cognition serves action through a flexible andsophisticated strategy where information is stored for futureuse without firm commitments on what the future use maybe Spatial information can be absorbed through a varietyof uses for which it was not originally encoded This meansthat new uses may be derived from a stored representationof space They need not be triggered by direct observationof the environment and its affordances [45] Environmentalrepresentations either schematic or detailed appear to belargely free of a particular purpose or at least contain infor-mation beyond what is necessary for a specific action This iscertainly an adaptive cognitive strategyThe fact that humansencode the physical world using spatially and semanticallystructured mental models offer a huge advantage in solvingproblems

This nondiscursive knowledge includes spatial propertiesand heterogeneities and enables us to build inferences forexample when we try to imagine a likely street where tofind certain activity In turn Lakoff and Johnson [46] arguethat mental models are based on a modelling of the physicalworld and bolster in analogies between abstract and concretedomains Other works tie semantics with imagetic schemesable to incorporate knowledge of the physical world as away to encode relationships between events In perceivingsomething we perceive not only its observed form butalso the potential information enfolded in it [24] This very

property is crucial to the presence of urban space in theselection of actions to be performed

So our actions encode space with social informationand such informationally differentiated space plays a partin easing the representational bottleneck in our memoriesWe are able to retrieve information from this semanticarrangement that takes the form of the city before and duringour actions and experiences Such semantic layer is storedin activity places and in the spatial relations between themandmay be related to physical and functional heterogeneitiessuch as accessibility and centrality patterns [34] Such layersof spatial information can be evoked in our memories andused when we need to make inferences such as where wecould find a particular social situation or place Howevermost theories of situated cognition still do not recognize howthis informational space is part of the way we create actionsystems Let us see how this could be the case

4 The City as a Frame of Reference

Our aim has strong parallels with Hutchinsrsquos [47 48] viewon cultural ecosystems operating cognitively at larger spatialand temporal scales than an individual person According toHutchins all instances of cognition can be seen as emergingfrom distributed processes that is from interactions betweenelements in a system Cognitive properties are not predictablefrom individual cognitive capacities Humans create theircognitive powers by creating the environment in which theyexercise those powers This socially distributed cognition is acollective operation produced by interactions among agentsin active relation to their environment in contexts of ongoingactivity Possibilities for individual learning depend on thestructure of the environment to the point that the distributionof cognitive skills is determined by the distribution of prac-tices engaged in by people As Hutchins we are concernedwith the tasks people confront in their everyday world takingpart in processes of coordination of action as they movethrough physical spaces Ultimately our aim relates well tohis proposition that ldquocultural practices decrease entropy andincrease the predictability of experiencerdquo [48 page 46] Ourwork follows parallel lines through a systems approach ableto encompass large-scale systems of action and spaces

We propose to explore the informational role of city spacein how agents build connections between actions We allknow that urban space is produced to express human activityBut theories of cognition suggest much more space wouldbe semanticized by our actions That means that activityplaces are informationally differentiated and acquire a levelof semantic definition similar to types of action [49 50]This differentiation is enacted in and associated with the verystructure of urban space Space is differentiated in at leastthree levels (i) its physical properties of extensity and config-uration such as accessibility and centrality patterns in a city(ii) recognizable shapes in its visual and tridimensional form[24] and (iii) the semantic contents of activities performed inbuildings and places (cf [34 35]) (Figure 1)

How can this informational space become a part ofactionThere is a remarkable lack of theoretical and empiricalwork on this problem First we seem to retrieve meanings

6 Complexity

(a) (b) (c)

Figure 1 Differential space like a code of instructions for collective action urban space finds differentiation both physically as hierarchiesof accessibility and distribution of built form (a) and semantically through actions performed in buildings and places ((b) (c))

associated with our actions in architectural and urban spacesWe arguably grasp useful information about the activityperformed in a place not only by visual features but alsoinferring what people do there Space not only representsactivity it is actuated and thus loaded with meanings relatedto performance This is a semantic dimension of space spaceldquomeansrdquo as much as our actions precisely because it issemanticized by our actionsThis enacted information is veryimportant Action is temporally related to spatial locations

Second these locations allow us to access potential coun-terparts in interaction As we shall see below knowing acity means that we can find places where certain activitiesare performed Recognizing spatial differences and patternsallows us to infer such locations Apart from other sources ofinformation such as linguistic exchanges we do so becausewecan retrieve environmental information about activities andlocation patterns

Third places are means of connection we can actualizeconnections with certain people or activities during a periodof time These spatial links allow us to create complexes ofinteraction In practice places ldquodrawrdquo courses of action withdifferent orientations and contents and tie them togethermomentarily Flows of convergent and divergent actionshappen all the time when people leave their homes to workto seek services or go about to socialize and do so oftenwithout prior knowledge of activities counterparts or placesthey wish to visit (Figure 2) Urban space becomes whatParsons [51] would call an ldquoaction frame of referencerdquo aconnective fabric which both enables and constrains choicesmaterially and socially

5 Entropy as a Way into Social Organization

In order to be a part of selection the urban environmentneeds to be informational enough and not just to suggest

t3

t2

t1

L1 L2 L3 L4

Figure 2 As a reference system urban space converges courses ofaction in time (119905

1 1199052 and 119905

3) and space (119871

1 1198712 1198713 and 119871

4)

particular options but allow unforeseen changes in the courseof our actions Luhmann suggests that a desired social activitymust be within reach ldquoBecause complete interdependence isunattainable however interdependencies only come aboutby selection [ ] Successfully established interdependenciesthen serve as perspectives for and constraints on the struc-tural selections that connect onto themrdquo [5 page 284] As asocial event this connection is produced through communi-cation But before communication comes into being certainconditions have to be there spatially there Places must bechosen as conjunctive pieces that materialize connections

Complexity 7

Could cities stimulate connections If they could andgiven the fact that they are formed by space and thatspace is not fully malleable is it reasonable to supposethat certain spatial formations could ease such momentaryconnections We know that proximity increases the intensityof interaction [52 53] But what form should urbanized spacetake in order to meet such systemic expectations We mayunderstand such spatial conditions through a counterfactualscenario Imagine a fully chaotic urban formation with norecognizable concentrations or paths Visual elements couldnot convey information as spatial differences would notdevelop into structures emerging out of absolute heterogene-ity References to activities of interest would have to relyonly on our memory Selection of actions would becomedependent on very detailed mental representations of such achaotic formation Inferences about more likely places to findparticular activities would be very hard tomakeWewould beimmersed in a spatial environment without patterns to guideour choices reduce our efforts and amplify our interactionsIn a world of unstructured spaces actions would face lackof information noise and relentless entropy The possibilitythat space may have an informational presence in interactionexpresses the possibility that the physical environment mayhave informational properties and that the world is rich ininformation

Now not only our daily experience denies that cities aresuch completely chaotic structureless places but also spatialeconomics and urban studies offer plenty of theories andevidences that as complex as they may be cities are (at leastpartially) structured places From location patterns describedsince Alonso [54] and accessibility patterns described fromHansen [55] to Hillier [56] we know that urban patternsmatter they imply that activities can be arranged in ways thatare recognizable and accessible And from the point of view ofselection the realization that activities are somehow arrangedalong a spatial structure is a form of preselection Activitiestend to be distributed according to levels of accessibilityland values and practical dependencies facilitating comple-mentary actions Intentionally or not and however subject tocontingencies urban space is also structured (physically andinformationally) which allows possibilities of action to bemore easily known selected and viable In short at first theurban structure affords a range of possibilities for selectionIn a second moment the structure itself suggests gradients ofattractive possibilities

In a view of social systems constituted by connectionsof actions this means that space can be part of the constanttransition from individual acts to complexes of actions (indi-vidual act rarr places as connections rarr action systems) Ourhypothesis here to be tested in computational experimentsin the following section is that cities become a crucial part ofthis connectivity They would reduce risks such as the lack ofinformation or courses of action too costly to emerge

What does this say about the effect of cities on entropyBeginning with Luhmann interactions have structural valuebecause they represent selections from combinatorial possi-bilities If we relate this to Shannonrsquos entropy the very emer-gence of urban structures eliminates spatial scenarios whereevery possible connection would have the same probability

Another counterfactual scenario might help here Entirelyhomogeneous cities (or entirely heterogeneous for thatmatter) free from structure would present even probabilitiesof choice between actions Of course this would be a problemfor social systems connections potentially interesting tomostpeople would be as difficult to find as any If urban spacewas purely homogeneous or purely random the probabilityof finding an activity would be distributed homogeneouslyin space That is a scenario of maximum entropy The spatialenvironment would be useless in stimulating certain sought-after connections which could otherwise be there say whenwe concentrate activities like final supply or input-outputexchanges in production Effort time spent and economiccosts involved in finding activities of interest would rise

On the other hand when urban space finds differentiatedcontents and structures say location patterns along acces-sible paths it naturally creates differences in the probabilityof interaction and decreases in entropy Location patterns arein fact material and informational expressions of interactionsystems The differences engendered by such patterns gener-ate differences in the chances of finding an agent a service orcommerce If a city did not have recognizable internal forma-tions such as a centralities or high streets agents would havemore difficulties in knowingwhere they aremost likely to findwhat they are after A partially structured environment allowsus to build inferences about actions and agencies available in asocial systemWhen these agencies are also related temporallythrough proximity and functional complementarity there isan even greater potential for differences in probability toemerge suggesting new connections and sequences Whatall this means is that cities internally differentiated yet farfrom absolute heterogeneity have the effect of reducing socialentropy

Before experimenting with these relationships in silicolet us attempt an introductory description of what thesedifferences in probability entail in theway social systems han-dle entropy (Figure 3) Linearizing for the sake of simplicityprocesses that in fact occur simultaneously the reductionof social entropy would take the following form (a) In aninitial state free of space think of the actions as lines movingin time Agents can do anything and we cannot foreseewhat they will do a potential state of high entropy Colorsof the lines in Figure 4 represent different informationalcontents in personal orientations (b) Then agents startselecting actions and converging into different positions in aspatial system arranged as places or buildings (representedby colors in the vertical strip) in order to connect withothers Action lines might have subtle differences in relationto the places they approach so they converge into placesby informational proximity Connections can only happenif all other possibilities of connection are excluded (c)As these convergences happen the initially chaotic mazebecomes a momentarily coordinated system where agentscooperate The overall entropy of actions is reduced (d)After each spatially held event actions may change againaccording to changing orientations Accordingly the colorsof lines change Entropy increases as actions go into anothermomentary state of unpredictability as new possibilities arepresented to agents and somemust be selected if actions are to

8 Complexity

Momentarilycoherent systems

State of unpredictabledecisions

Action systems

Reduction of entropy

Coordination of actions

Selectionof actions

High entropy state

Information growth(Hidalgo 2015)

Possibilities of action(Luhmann 1995)

Differences in theprobability ofinteractions

(Shannon 1948)

Agents interact Orientations align inactivity places

Actions mediated byinformation

(Luhmann 1995)

City as informationenvironment(Haken and Portugali 2015)

Figure 3 Recursive process that conducts interaction systems from chaotic states (high entropy) into coherence (low entropy)

(a) (b) (c)

(e) (d)

Figure 4 The cycle of social entropy and space clockwise from the high unpredictability of potential actions to a momentarily orderedsystem as actions converge into places only to dive in entropy again

Complexity 9

Actionagent x

Actionagent y

Activity place

Cooperation

Change insubjectiveorientation

Figure 5 Aligning different actions in the cooperation process

be actualized (e) And then actions ldquoreferrdquo again to meaningsand positions in space starting a new cycle of entropy(Figure 4)

These distinct moments are theoretical of course Inreality reminding Prigogine and Stengers [6 page 124] thisldquomultitude of eventsrdquo takes place simultaneously and mayrender any clearly emergent cycle invisible But that neithermeans that individual cycles do not happen or aggregate inbunches nor they compensate one another like peaks andtroughs in colliding sound waves In fact routinization indaily life leads to rhythms and cycles unfolding from orfolding over the seemingly chaotic maze of actions But evenif actions were far from synchronic the fabric of interactionstill faces the challenge of entropy that is the challenge ofproducing differences in the probability of connections as a wayto ease organization whenever agents feel compelled to createnew interactions and this process is remarkably recursiveWe converge to places to act with others and do so on adaily basis Once engaged in an activity place interactionrequires levels of cooperationWemay bring our own distincttendencies or latent orientations in the form of long-termmemories of how to act along with short-term memoriesof a previous action both shaping our next actions As weact socially we bring our orientations into contact in away to open them in the communication process regulatingcollective action A result of this coordination is to reduceinitial differences Our actions ldquoalignrdquowithin an activity place(Figure 5)

All this suggests that social life involves a continuousalthough unconscious handling of entropy a vital oscillationbetween unpredictability and predictability As this processengages an enormous number of agents and events andtranscends places and contexts and therefore cannot beobserved as a whole entropy can only be fully grasped intheoretical representations A useful way to examine space asan information environment for action is through a compu-tational model a means to examine a counterfactual worldwhere other steering media such as language or social ruleswere ldquoswitched offrdquo keeping only properties of space activealong with agentsrsquo cognitive abilities to align orientationswith activities Once we abstract from other media wemay assess whether space could have any presence in theself-organization of action Let us examine this possibilitythrough some experiments

LBuildingactivity type

Figure 6The theoretical one-dimensional citywith a set of differentactivity places Here there are 119871 buildings distributed in a ringconstituting the city space Colors represent activity types

6 Digital ExperimentsA Unidimensional Model

61 Design Concepts An agent based model (ABM) is pro-posed Agents are defined to perform actions at each time-step and places by corresponding activity types The decisionon which action to perform next may be influenced bythree different conditions namely (i) latent orientation120590in

119894(119905)

representing the tendency of a single agent to act arounda particular type of action initially randomly distributedthis condition remains over time (ii) current action 120590

119894(119905)

performed by an agent as heshe selects an activity placein order to perform a new action this means an influenceof the current action while allowing gradual changes oforientation in time influenced by other agents and activitiesand (iii) types of activity places 120590

1199091015840(119905) where agents perform

supporting specific types of action Agents follow simplerules selecting their activity places closer to their latentorientations and current actions while being influencedby those activities Activity places are also influenced byvisiting agents but they change at a slower rate In shortagents coevolve with their spatial and social environmentThe model simulates different situations under this basicstructure Differences in the weight of these factors overthe next action may lead to quite different levels of socialentropy suggesting new possibilities for understanding thephenomenonAlthough there is a tradition inmodelling dailyactivities including a growing literature using digital loca-tional data (eg [57]) our ABM is not designed to representempirical sequences of actions or actual routines Instead itfocuses on trends that might emerge from the interfaces ofsimplified systems of action information and space

62 Variables and Scale Consider a unidimensional cityformed by agents and places Places form a ring with length(or perimeter) 119871 as shown in Figure 6 Our choice forrepresenting the city by a ring is justified by the minimalsufficient representation of spatial distance as a factor whose

10 Complexity

role in social organization is to be assessed This is allowedby a unidimensional model (eg [58ndash60]) The ring formallows continuous movement across a linear sequence oflocations eliminating centrality factors while taking intoaccount periodic boundary conditions in order to reduceborder effects eliminating the role of topologywhile isolatingthe problem of distance (see [61ndash63]) This stylized cityhouses 119873 agents which in each time-step 119905 select and visita specific place located at the position 119909 within the cityConsider that the position of the 119894th agent at the time 119905 isrepresented by 119909

119894(119905) The position of an agent can assume the

integer values 1 2 119871 according to the place that shehechooses to visit This means that at every time-step eachagent will choose one activity place to perform her nextaction

We considered 120590119894(119905) as the current action of the 119894-th agent

and 120590119909(119905) the activity performed in a place located at the

position 119909 = (1 2 119871) of the city both in time 119905 Wequantify action orientations weighting the three variables(latent orientation current action and activity place) Firstvariables are translated into different numeric values Let usassume that an orientation is an integer number between 0and 1000 each number represents a particular type of actionand numbers in their vicinity mean similar actions Theresulting orientation in time 119905 for each agent is the weightedaverage of those three parametersThedifference |120590

119894(119905)minus120590119909(119905)|

gives us the difference in orientation between an agent andthe activity place At every time-step each agent assesses thedifference between her orientation and every activity placein the city An agent chooses a specific place based on herorientation affinity with this place minimum difference

As we hypothesized above the urban structure has a rolein this selection In this experiment we reduced ldquourban struc-turerdquo to distance In order to investigate how distance may ornot interfere in this selection we propose two scenarios Inthe first one the agent selects her next activity based only onthe similarity in orientations between herself and the activityplace Agents can move across a space free of friction Spaceis not a constraintThis scenario considers only differences inorientation The parameter 119864

119894(119909 119905) estimates the interaction

between agent 119894 and the place located at 119909 at the time 119905 Theagent selects activity locations that minimize this quantityWe propose

119864119894 (119909 119905) =

1003816100381610038161003816120590119894 (119905) minus 1205901199091003816100381610038161003816 (1)

In the second more realistic scenario space imposesfriction to movement The agent takes into account thephysical distance between her current location in the city andthe location of her next activity and opts to minimize thisdistance along with the difference between her orientationand that of the activity place We propose

119864119894 (119909 119905) =

1003816100381610038161003816(119909119894 (119905) minus 119909) (120590119894 (119905) minus 120590119909)1003816100381610038161003816 (2)

Summing up an agent 119894 selects at time 119905 a particularactivity place located at119909 thatminimizes the function119864

119894(119909 119905)

In the first scenario she will choose activity places closerto her orientation In the second she will choose activityplaces closer in both orientation and physical distance This

brings a form of ldquoenergy costrdquo into the factors consideredSpace takes on a function role because of the energy used inldquovisitingrdquo Agents move in this linear city willing to minimizethis energy while searching for social information latentin places close to their changing orientations Changes inentropy levels are derived from this energy function In itsspatial version the model assesses effects of the effort toreduce energy in action minimizing informational differ-ences (between orientations and activity places) and spatialdistances (between agents and activity places)

63TheEvolution of ActionOrientations Orientation evolvesin time according to the following rule When an agentchooses and joins a particular place both agent and placebecome a little closer in terms of orientation This meansthat activity places are not immune to what agents performin them and that the urban activity system also changes intime To illustrate how orientations are constantly updatedconsider that agent 119894 selects an activity place located at 1199091015840Orientations will be updated according the rule

120590119894 (119905 + 1) =

1120572 + 120573 + 120574

(120572120590in119894(119905) + 120573120590119894 (119905) + 1205741205901199091015840 (119905)) (3)

Selection also depends on three additional parameterslatent orientation 120590in

119894(119905) weighted by parameter 120572 current

orientation120590119894(119905)weighted by parameter120573 and type of activity

place 1205901199091015840(119905) weighted by parameter 120574 This means that agents

will consider the three factors previously described (120590in119894(119905)

120590119894(119905) 1205901199091015840(119905)) with different weights (120572 120573 120574) to estimate which

activity she wants to perform in the next time-stepThe latent orientation evolves from an initial randomly

distributed behaviour 1205900119894 We consider that 120590in

119894(119905) is normally

distributed around 1205900119894 meaning that every agent has an

ldquoaveragerdquo latent orientation whose value will be derived ateach time-step from a normal distribution peaking at theagentrsquos average orientation If 120572 is sufficiently small in relationto 120573 and 120574 then the latent orientation does not play a rolein the agentrsquos action However if 120572 is sufficiently large (oncein relation to 120573 and 120574) then the action becomes stronglydependent on the latent orientation This would lead to apretty conservative city where inhabitants are not open tochanges in behaviour

The current action of an agent may influence a newaction with an intensity that depends on parameter 120573 If thisparameter is small in comparison to 120572 and 120574 then the agenthas no memory of an ongoing orientation However if 120573 issufficiently large then the new action is strongly dependenton a previous one (a Markovian behaviour)

The intensity of the activity placersquos influence on the agentrsquosaction depends on the value of 120574 If this parameter is smallin comparison to 120572 and 120573 then the activity place does notaffect her next actionmeaning that the place does not changebehaviours However if this parameter is sufficiently largethen the place plays a strong influence on future actions Theactivity place will be updated according to the rule

1205901199091015840 (119905 + 1) = 1205901199091015840 (119905) + 120579sum

119894isin1199091015840

120590119894 (119905) (4)

Complexity 11

where 120579 is a parameter sufficiently small That is placesare less influenced by agents than the other way roundThis means that at every time-step a place would have itstype of activity closer to the average of orientations of itsvisitors Therefore the main activity of the place will changeslightly according to visiting agents while changing theirorientations

64 Frequency of Orientations and Entropy Levels We cal-culate entropy assessing the distribution of resulting orien-tations of all agents in the social system at each time-stepusing Shannon entropy Our quantitative interpretation ofinformation removes ambiguities from the relation betweenShannon information and semantic information as differentorientations actions and types of activity places are repre-sented by numerical values This allows us to estimate socialentropy through Shannon entropy The entropy of actions iscalculated as follows consider119873(120590 119905) as the number of agentswith orientation 120590 at the time 119905 (note that the total populationis 119873 = sum

120590119873(120590 119905)) We compute the frequency (or density)

120588(120590 119905) of this orientation within a population with

120588 (120590 119905) = 1119873119873 (120590 119905) (5)

Equation (5) calculates the probability of observing anorientation 120590 at time 119905 We compute the entropy level for anydistribution of orientations with

119878 (119905) = minussum120590

120588 (120590 119905) ln (120588 (120590 119905)) (6)

Equation (6) describes how uneven is the probability of find-ing different orientations Higher values mean that differentorientations have almost the same probability to happenwhile lower values indicate a system with clear orientationtrends The reduction of entropy implies that the probabilityof certain actions increases that is actions grow in similarityIn the limit as entropy falls to zero all agents in the systemwould reach the same orientation

65Model Procedure Themodel performs the following pro-cedures

(1) The city is created by generating the following

(a) The type of activity in places that is (120590119909=1(119905 =

0) 120590119909=2(119905 = 0) 120590

119909=119871(119905 = 0)) also generated

from a random distribution This step createsactivity places in the ring-city and assign themrandomly distributed activity values 120590

1199091015840(119905)

(b) The initial position of 119873 agents that is (1199091

1199092 119909

119873) and their respective orientations

(1205901(119905 = 0) 120590

2(119905 = 0) 120590

119873(119905 = 0)) generated

from a random distribution This step createsmoving agents in random locations and assignthem latent orientations based on the initialrandomly distributed orientation 120590in

119894(119905) = 120590

119894(0)

(2) At each time-step 119905 every agent chooses a place toact This place should minimize the function 119864

119894(119909 119905)

Suppose that the chosen place by 119894 is defined as 1199091015840Then agent and place update their orientations 120590

119894and

1205901199091015840 as described above

(3) Update the distribution of orientations 120588(120590 119905)(4) Compute Shannon entropy 119878(119905)

66 Results We have developed a number of observationson the behaviour of social systems working under differentparameters

(i) Entropy reduction is only found in scenarios wherespace imposes friction to movement that is wherethe spatial distance between agents and activity placesis considered as an active factor in selection Entropyrequires an extensive space in order to be reduced Butspace cannot do it alone Surprisingly space is a nec-essary but not sufficient condition Figure 7 comparestwo scenarios In the first one distance is an issue inthe selection of activities (red line)Thedistribution oforientations changes from a homogeneous one at thestart of simulation (a) to a nearly normal distributionat the end of simulation (b) where certain kindsof action are more likely to happen Informationalcontents in extensive space help aligning contents inactions In the second scenario agents move free ofspatial friction (blue line) Orientations are initiallyrandomly distributed (a) and continue to be so at theend of simulation (b) as a similar number of agentsare distributed along different orientationsThis resultsuggests that a materially active space becomes ameans for increasing the probability of certain inter-actions easing the collective coordination of action

Nowassessing the relative influence of social and personalfactors on the selection of a next action namely the socialinformation in activity places latent orientation and the cur-rent action under the influence of spatial friction (Figure 8)we can say the following

(i) There is structure in the relation between selectionfactors and the reduction of entropy Entropy beginsto fall and fluctuate around specific values accordingto different combinations of factors For some com-binations there is a great reduction of entropy Forothers the reduction is minimal almost negligibleFluctuations derive from the random component inthe decisionmaking of the agent (under the influenceof latent orientation)

(ii) A strong latent orientation (120572 sufficiently large) leadsto increasing entropy (blue lines in Figures 8 and9) since agents cannot align their actions with otheragents and activity places Like a long-term memorysystems whose actions are dictated by latent orienta-tions are likely to preserve initial orientations whichwere randomly distributed and therefore homoge-neous This condition is responsible for limiting thereduction of entropy

(iii) Current action alone does not shape new actions inany specific way (green lines in Figures 8 and 9)

12 Complexity

Start of simulation

00000

00005

00010

00015

00020

o

f tot

al ac

tions

250 500 750 10000Type of action

Materially inactive spaceMaterially active space

t = 0

(a)

End of simulation

00000

00005

00010

00015

00020

o

f tot

al ac

tions

250 500 750 10000Type of action

Materially inactive spaceMaterially active space

t = 50

(b)

Figure 7 Histogram of probability distribution of actors performing different types of action at the start (a) and at the end (b) of simulationsin scenarios where distance is not considered (blue) and distance is considered aligning orientations in action coordination (red) Resultsare averaged for 30 runs for 125 parameter combinations

activityplace

currentaction

latentorientation

225

250

275

300

325

Entr

opy

10 20 30 40 500Time

Figure 8 Evolution of entropy colors in a RGB scale refer toparameter weights in decisions on actions in a next time-step andtheir impact on entropy levels in different simulations (under thefriction of distance) In the palette triangle each color representsa specific combination of the three parameters In each vertex aparameter reaches maximum value while the others have minimumvalue Results are averaged for 30 different runs for 125 parametercombinations

It just keeps what is already happening workingas ldquoreinforcement feedbackrdquo for either direction theagent (and the system) is going Current actions aremeans to conserve tendencies in the system

(iv) Activity places play a key role in the reduction ofentropy (red lines in Figures 8 and 9)The social infor-mation in space ldquocontaminatesrdquo agents they aligntheir actions through the social contents of placesWe have seen that space matters as extension Nowwe see that space also has a very active informationalpresence

(v) Different combinations of factors have differenteffects over entropy Figure 8 shows the transitions

between the three parameters and how nuancedweights resulting from blending factors matter inthe reduction of entropy For instance blending theweight of activity places and latent orientation on thenext action displays a strong reduction of entropy(pink lines in Figure 8) That means that when agentsdo not keep information from previous selections(ie 120573 asymp 0) the action system reduces more entropythan in other parameter combinations A strongshort-term memory leads the system into conservingitself and into a poorer capability to coordinateactions Also when current actions 120573 and activityplaces 120574 share a similar weight over the next action(orange lines on top) the social system does notexperience a great reduction of entropy as agents tendto reproduce their actions

(vi) Finally the reduction of entropy implies that theprobability of certain actions and interactionsincreases In practical terms this means morealignments between agents and more connectionsbetween actions (interactions) However if all agentsin the system reached the same orientation andentropy dropped to zero the system would lose inter-nal differentiation Agents would behave in the sameway say in a world with no personal differentiationspecialization or division of labour New orientations(therefore entropy) are necessary if the social systemis to keep differentiated agencies On the otherhand a system with maximum entropy would createdifference in actions to a point where no action coor-dination or interaction would be possible Clearly thissituation cannot be the case A social system requiresbalances neither full entropy nor total predictability

These results also suggest that we are dealing with distinctldquosocial memoriesrdquo a long-term memory is active in what we

Complexity 13

20

25

30En

trop

y

04 0602-latent orientation

04 0602-current action

20

25

30

Entr

opy

20

25

30

Entr

opy

04 0602-activity place

Figure 9 Parameter exploration isolated importance of each parameter in entropy (under the friction of distance) Each vertical set of dotsrepresents one fixed value for the highlighted parameter with the two other parameters assuming every possible combination within thetested range Each dot corresponds to the entropy value when a sufficiently large time-frame is used

call latent orientations Short-term memory is active in theinfluence of a current orientation over a new action In turnthe social information latent in activity places is a result ofsocial arrangements (say as firms a local economy and soon) which find certain stability changing much slower thenactions It is an extension of the social system projected ontourban space stabilizing the system to some extent

7 Cities and Social Interaction Conclusions

What does this approach bring to the state of the art on therelations of social interaction information and space and onsocial entropy and ABM in particular We have seen thatprevious approaches overlooked the coordination of actionas a key empirical and analytical problem Furthermorethey tend to have the thin spatiality of a passive territorialbackground (eg [16 20]) In turn although the conceptof ldquocollective actionrdquo is finally getting attention in urbanstudies [64] the problem of social organization is still largelyunderestimated The problem of entropy has been dealt withsince Wilsonrsquos [65] work on spatial interaction but has notreached the mainstream of the discipline and deals mostlywith urban form (eg [24]) and spatial distributions [66]not with social information and action Regarding agentbased models we are not aware of an approach that dealswith the problem of how cities and space are part of socialorganization In a sense our model is closer to Axelrodrsquos[67] ABM model of dissemination of culture His agentsexchange ldquoculturerdquo through direct contact with neighboursin a cellular automata model whereas our agents are mobileand actively deal with distance and social information inspace recognizing types of activities in places The intensityof exchanges between agents in Axelrodrsquos model is a functionof similarities between them In our model exchanges aremediated by the selection and interaction with space In ourunderstanding our contribution in terms of ABM lies onthe fact that a new and simple model is proposed to dealwith the material and informational dimensions of socialorganization not just as an illustration but as a proof ofconcept space can have causal presence if agents generatepreferences as detailed in a domain that coevolves with them

Now does the simulation model corroborate the theoryintroduced in this paper Our proposition cannot be empir-ically validated at this stage given that it is very hard to

assess peoplersquos choices and the whole panorama of actionsin a city Entropy may well be beyond observation In thesecircumstances simulations of the behaviour of agents underdifferent spatial conditions become useful to assess entitiesevents and entropic forces at play One expected result fromthe model is that space matters for coordinating actions butsimulations surprisingly showed that space is a necessary butnot sufficient condition for the reduction of entropy Neithermaterial cognitive resources nor individual powers of agentsare enough Information in activity places and a bit of agentsrsquopersonal history also play roles in the way social systemsdeal with their own entropy That would mean a causal butnondeterministic presence of space

Substantively this approach explores a subtle but funda-mental presence of cities in social organizationWe argue thaturban space materializes gradients of difference in potentialinteractions from less to more recognizable costly or likelyPlacing simple criteria including distance as a factor in select-ing activities our ABM showed that space becomes a meansfor producing differences in the probabilities of interactionincreasing chances of certain selections and convergences incollective action

In reality people recognize the social information ofactivity places along with uses of busy streets or local central-ities We are able to retrieve these differences from the builtenvironment and relate to them One of the key points of ourargument is that these differences help us make selectionsEvery time we move through the city and select places toperform we participate in the large-scale coordination ofaction We enact space as a ldquoreferential systemrdquo [49 50]a set of semantic bits indexing social activities informingand guiding daily decisions Space becomes inherently partof the conversion of orientations into interactions that isconnections that crisscross sequences of actions recreatingthe interaction system By means of a structure that bothmaterializes and restricts the quasi-endless combinatorial pos-sibilities of interaction this system can acquire sufficientinternal guidance to make its own reproduction possibleBy distributing sufficiently recognizable differences in theprobability of interaction cities express and release localforces of reproduction of social systems The interfaces ofaction cognitive and spatial systems transform entropiccomplexity into structured complexity constantly reorderingactions in time

14 Complexity

This materialistic viewpoint can be integrated with otherviews of social organization Further work may include therelational roles of cultures language social norms and othersteering media along with explorations of possibly similareffects of communication technologies over entropy At thisstage we attempted to integrate a social dimension (agentsengaged in coordination) a spatial dimension (an extensiveenvironment) and an informational dimension (as differ-ences in action and in the environment itself) a demandinginterdisciplinary effort The approach identifies a central rolefor coevolving agents actualizing aims as they coordinate (cf[68]) and a central role for the environment exterior butresponsive to agents leading to cumulative changes alongtheir historyThemodel also identifies roles for cognition andmemory in the ways agents deal with their own orientationsand choose their actions and activity places changing theirspatial environment with consequences in the overall levelsof entropy All this suggests that the behaviour of the modelcannot be reduced to or fully predicted from the behaviour ofindividual variables which is a key aspect of complexity

So one of the main aims of this paper was to assesswhether space as information environment had any effecton the coordination and entropy of interaction However ifphysical distance and social information in activity placesmatter what about city size and the internal spatial struc-tures of cities What about rural areas or rarefied suburbsSince our model explores an abstract circular city it onlybegins to answer such questions Our findings indicate thatshorter distances between activity places tend to reducesocial entropy Density seems to matter denser spatialities(as opposed to rarefied ones) have a role to play A pathfor further development here is Jacobsrsquos [69] idea that thediversity of activities generates positive externalities Anddiversity as spatial economics shows us has to do withthe size of cities and density of population (eg [70 71])Entropy may also be influenced by diversity Batty et al [66]have shown that information increases as cities get biggerThe remaining question is whether larger denser and moreinternally structured cities could create process and reduceentropy with more intensity than smaller cities transformingbig pools of activities into differences in the probability ofinteraction Although we proposed a unidimensional modelour agentsrsquo behaviour can be explored in more realisticrepresentations of cities testing the roles of density topologyand diversity while going beyond perfectly informed agentsBy proposing the ring model we intended to simplify acity to a minimum system where the spatial dimensionof social organization becomes intelligible without losingfundamental properties

Shouldwe conclude fromall this that entropy is a problemfor social systems Our approach suggests that it is notEntropy is a necessary force It means that novel orientationsare entering the interaction system and must be dealt withEntropy only becomes a problem if not converted into orga-nization only if numerous potential actions are not convertedinto smaller sets of actual interactions The effects of newactions over entropy would be positive once actualizedhaving to dowith the diversity of agencyConservative systemsare likely to face less entropy but also likely to produce

less novelty having homogeneity as a problematic horizonFluctuations of entropy seem vital to complex societies

Finally our intention was to explore entropy as a meansto think about the conditions of social organization ldquofromanother anglerdquo so to speak from the viewpoint of challengesinvolved in social reproduction Hence the approach focusedon the idea of uncertainty surrounding agents dealing withaims and decisions and the different probabilities of inter-action that follow an attempt to bring social cognitiveinformational and spatial theories under a single roof Ourapproach sees cities as connective systems produced to createthe delicate fabric of interaction that keeps large numbersof agents coherently living together It also suggests that thevarying states of entropy reveal deep connections betweensocial informational and spatial systems

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

Theauthors would like to thank the Ecole Polytechnique Fed-erale de Lausanne (EPFL) CAPES FAPEMIG FAPESP andCNPq for financial support Maıra Pinheiro Henrique Loreaand colleagues at ABMUS2017 for earlier discussions andMike Batty for exchanges and inspiration Finally the authorsthank Romulo Krafta maverick of the complexity studiesof cities in Brazil for his input and support throughout theyears This work is dedicated to him

References

[1] C W Mills The Sociological Imagination Oxford UniversityPress New York NY USA 1959

[2] M Castells ldquoThe information age economy society and cul-turerdquo inThe Rise of the Network Blackwell London 1996

[3] C Hidalgo Why Information Grows The Evolution of Orderfrom Atoms to Economies Basic Books New York NY USA2015

[4] D Helbing The Automation of Society Is Next ETH ZurichSwitzerland 2016

[5] N Luhmann Social Systems University Press Stanford 1995[6] I Prigogine and I Stengers Order out of Chaos Manrsquos New

Dialogue with Nature Bantam Books 1984[7] C E Shannon ldquoAmathematical theory of communicationrdquo Bell

System Technical Journal vol 27 no 4 pp 623ndash656 1948[8] W Buckley Sociology and Modern Systems Theory Prentice-

Hall Englewood Cliffs NJ USA 1967[9] D D McFarland ldquoMeasuring the Permeability of Occupational

Structures An Information-Theoretic Approachrdquo AmericanJournal of Sociology vol 75 no 1 pp 41ndash61 1969

[10] F Charvat ldquoOn philosophical aspects of the system conceptionin contemporary sociological knowledgerdquo Quality and Quan-tity vol 6 no 1 pp 3ndash16 1972

[11] F Charvat J Kucera and M Soukup ldquoToward the systemtheory of dependence further general theoretical remarksrdquoQuality and Quantity vol 7 no 1 pp 69ndash90 1973

Complexity 15

[12] P M Horan ldquoInformation-theoretic measures and the analysisof social structuresrdquo Sociological Methods and Research vol 3no 3 pp 321ndash340 1975

[13] O E Klapp ldquoOpening and closing in open systemsrdquo BehavioralScience vol 20 no 4 pp 251ndash257 1975

[14] J Galtung ldquoEntropy and the general theory of peacerdquo in Essaysin Peace Research J Galtung Ed vol L Humanities PressAtlantic Highlands NJ USA 1975

[15] T Parsons The System of Modern Societies Englewood CliffsNew Jersey NJ USA 1972

[16] K D Bailey ldquoSociological entropy theory Toward a statisticaland verbal congruencerdquo Quality amp Quantity vol 18 no 1 pp113ndash133 1983

[17] K D Bailey Social EntropyTheory State University of NewYorkPress Albany NY USA 1990

[18] J G Miller Living Systems McGraw-Hill New York NY USA1978

[19] K D Bailey Sociology and the New Systems Theory Toward aTheoretical Synthesis State University of New York PressAlbany NY USA 1994

[20] K D Bailey ldquoSociocybernetics and social entropy theoryrdquoKybernetes vol 35 no 3-4 pp 375ndash384 2006

[21] F Geyer ldquoThe challenge of sociocyberneticsrdquo Kybernetes vol24 no 4 pp 6ndash32 1995

[22] F Geyer ldquoThe march of self-referencerdquo Kybernetes vol 31 no7-8 pp 1021ndash1042 2002

[23] J HabermasTheTheory of Communicative Action Polity PressCambridge UK 1987

[24] H Haken and J Portugali Information Adaptation The Inter-play between Shannon Information and Semantic Information inCognition Springer New York NY USA 2015

[25] M Weber Economy and Society vol 1 Berkeley University ofCalifornia Press 1978

[26] L Wittgenstein Philosophical Investigations Blackwell Publish-ers Oxford London 3rd edition 2001

[27] L Vygotsky Mind in Society The Development of HigherPsychological Processes Harvard University Press CambridgeMA 1978

[28] M Wilson ldquoSix Views of Embodied Cognitionrdquo PsychonomicBulletin amp Review vol 9 no 4 pp 625ndash636 2002

[29] A Clark and D Chalmers ldquoThe extended mindrdquo Analysis vol58 no 1 pp 7ndash19 1998

[30] R PassiniWayfinding in Architecture Van Nostrand ReinholdNew York NY USA 1992

[31] E Rosch ldquoPrinciples of categorizationrdquo in Cognition and Cate-gorization E Rosch and B Lloyd Eds Hillsdale LawrenceErlbaum Ass 1978

[32] W KintschMemory and Cognition John Wiley and Sons NewYork NY USA 1970

[33] U Neisser ldquoMultiple Systems a New Approach to CognitiveTheoryrdquo European Journal of Cognitive Psychology vol 6 no 3pp 225ndash241 1994

[34] A Faria and R Krafta ldquoRepresenting urban cognitive structurethrough spatial differentiationrdquo in Proceedings of 4th Space Syn-tax International Symposium pp 531-518 UCL Press London2003

[35] J Portugali Complexity Cognition and the City Springer NewYork NY USA 2011

[36] D Kirsh and P Maglio ldquoOn distinguishing epistemic frompragmatic actionrdquo Cognitive Science vol 18 no 4 pp 513ndash5491994

[37] S M Kosslyn Image and Brain The Resolution of the ImageryDebate MIT Press Cambridge MA

[38] R Brooks Cambrian Intelligence The Early History of the NewAI MIT Press Cambridge MA 1999

[39] A Clark BeinGThere Putting Brain Body AndWorld TogetherAgain MIT Press Cambridge MA 1997

[40] R Pfeifer and C ScheierUnderstanding Intelligence MIT PressCambridge MA 1999

[41] A M Glenberg and D A Robertson ldquoIndexical understandingof instructionsrdquoDiscourse Processes vol 28 no 1 pp 1ndash26 1999

[42] JM Iverson and SGoldin-Meadow ldquoWhypeople gesturewhenthey speakrdquo Nature vol 396 no 6708 p 228 1998

[43] R M Krauss ldquoWhy do we gesture when we speakrdquo CurrentDirections in Psychological Science vol 7 no 2 pp 54ndash60 1998

[44] S Franklin Artificial Minds MIT Press Cambridge MA 1995[45] J Gibson The Ecological Approach to Visual Perception

Houghton-Mifflin Boston 1979[46] G Lakoff andM JohnsonPhilosophy in the FleshTheEmbodied

Mind and Its Challenge to Western Thought Basic Books NewYork NY USA 1999

[47] E HutchinsCognition in theWild MIT Press CambridgeMA1995

[48] E Hutchins ldquoThe cultural ecosystem of human cognitionrdquoPhilosophical Psychology vol 27 no 1 pp 34ndash49 2014

[49] V M Netto ldquoPractice space and the duality of meaningrdquoEnvironment and Planning D Society and Space vol 26 no 2pp 359ndash379 2008

[50] V M Netto The Social Fabric of Cities Routledge New YorkNY USA 2017

[51] T Parsons The Structure of Social Action The Free Press NewYork NY USA 1968

[52] T Allen Managing the Flow of Technology MIT Press Cam-bridge MA 1977

[53] F L Ribeiro J Meirelles F F Ferreira and R C Neto ldquoAmodel of urban scaling laws based on distance-dependent inter-actionsrdquo Royal Society Open Science vol 4 Article ID 1609262017

[54] W Alonso Location and Land Use Toward a General Theory ofLand Rent Harvard University Press Cambridge MA 1964

[55] WGHansen ldquoHow accessibility shapes land userdquo Journal of theAmerican Institute of Planners vol 25 no 2 pp 73ndash76 1959

[56] B Hillier Space Is the Machine Cambridge University PressCambridge UK 1996

[57] S Farber M OrsquoKelly H J Miller and T Neutens ldquoMeasuringsegregation using patterns of daily travel behavior a socialinteraction based model of exposurerdquo Journal of TransportGeography vol 49 pp 26ndash38 2015

[58] T C Schelling ldquoDynamic models of segregationrdquo Journal ofMathematical Sociology vol 1 no 2 pp 143ndash186 1971

[59] R Krafta V M Netto and L Lima ldquoUrban built form growscriticalrdquo CyberGeo vol 2011 article 565 2011

[60] N Lanchier and S Scarlatos ldquoFixation in the one-dimensionalAxelrod modelrdquo The Annals of Applied Probability vol 23 no6 pp 2538ndash2559 2013

[61] F L Ribeiro and K N Ribeiro ldquoA one dimensional modelof population growthrdquo Physica A Statistical Mechanics and itsApplications vol 434 pp 201ndash210 2015

[62] F L Ribeiro ldquoA non-phenomenological model of competitionand cooperation to explain population growth behaviorsrdquo Bul-letin of Mathematical Biology vol 77 no 3 pp 409ndash433 2015

16 Complexity

[63] R V dos Santos F L Ribeiro and A S Martinez ldquoModels forAllee effect based on physical principlesrdquo Journal of TheoreticalBiology vol 385 pp 143ndash152 2015

[64] M BattyThe New Science of Cities The MIT Press CambridgeMA 2013

[65] A G Wilson Entropy in Urban and Regional Modelling PionLondon 1970

[66] M Batty R Morphet P Masucci and K Stanilov ldquoEntropycomplexity and spatial informationrdquo Journal of GeographicalSystems vol 16 no 4 pp 363ndash385 2014

[67] R Axelrod ldquoThe dissemination of culture a model with localconvergence and global polarizationrdquo Journal of Conflict Reso-lution vol 41 no 2 pp 203ndash226 1997

[68] B H Hodges and RM Baron ldquoValues as constraints on afford-ances perceiving and acting properlyrdquo Journal for the Theory ofSocial Behaviour vol 22 no 3 pp 263ndash294 1992

[69] J JacobsThe Economy of Cities Random house New York NYUSA 1969

[70] E Glaeser H Kallal J Scheinkman and A Shleifer ldquoGrowthin Citiesrdquo Journal of Political Economy vol 100 no 6 pp 1126ndash1152 1992

[71] S Rosenthal and W Strange ldquoEvidence on the nature andsources of agglomeration economiesrdquo in Handbook of Urbanand Regional Economics J V Henderson and J-F Thisse Edsvol 4 pp 2119-2171 NorthHollandNewYork NYUSA 2004

Submit your manuscripts athttpswwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Complexity

that they have in English In line (3) we have the additionof the frequency with which letters follow other letters inpairs in English On the third-order approximation (line (4))letters are randomly ordered according to the likelihood ofassociation with others in trios Line (5) shows words chosenindependent to each other but with the same probability ofappearing in sentences that they have in English Finallyline (6) comprises the probability that a word is followed byanother specific word Even without decoding the meaningsof words a nearly intelligible message is emerging Differ-ences in frequency and sequence in letters andwords producedifferences in the probability of occurrence and an increasedprobability of certain combinations In practice this meansthat the message components become more predictable andintelligible The reduction of entropy allows communication

One could argue that Shannon entropy is anything butsemantic This is a crucial point Shannon indeed definesinformation irrespective of meaning However Shannonrsquoscoauthor in their 1949 book The Mathematical Theory ofCommunication Warren Weaver had already attempted toincorporate semantic information into Shannonrsquos theoryTheargument recently expanded by Haken and Portugali [24]shows that Shannon information participates in semanticinformation and vice versa as themind relates to the environ-ment deflatinginflating information variations in Shannoninformation entail differentmeanings Furthermore differentmeanings affect the quantity of information as they areproduced and retrieved through differences in events orentities however subject to ambiguities Aswe shall see belowfor the sake of assessing entropy pragmatically differencesin semantic information can be sufficiently captured throughShannon information This allows one to estimate semanticentropy through Shannon entropy

But what does this have to do with social entropy Soci-eties are interaction systems Peoplersquos actions are mediated byinformation they have meanings from personal orientation[25] to informational [5] and practical contents [26] Mean-ings produce informational difference We recognize actionsthrough such informational differences say in the contentsof utterances or in the tasks we perform with someone Inprinciple the greater the diversity of orientations and actionsthe harder it is to predict what someonersquos next action willbe But there is more to this temporal condition A state ofhigh entropy surrounds actions in a potential state before theycome into being when a number of possibilities of action lieahead Bringing Shannonrsquos and Luhmannrsquos insights togetherthe informational problem that societies face on a daily basislies in dealing with the selection of actions

We experience entropy whenever we deal with uncer-tainty options decisions unpredictable situations or toomuch information but we rarely realize we do so Entropy is aphenomenon ldquobeyond observationrdquo we cannot see or touchentropy even though we experience it on a daily basis Weare not used to think about the challenges we face in orderto make choices or put our actions together in any workablesense The key question here is how such a heightened fieldof choices and the taken-for-granted process of selectioninvolve our environment We need to understand how citiesare part of the semantic exchanges that constitute collective

action And the first step here is to understand city space asinformation in its own right

3 The City as Information Environment

Luhmann shows that social systems are immersed in seman-tic production meaning becomes their environment Wewish to further develop this idea exploring urban space aspart of such an environment In fact the idea of space as anenvironment able to contain social information has gainedgreat support recently FromVygotsky [27] toWilson [28] andHaken and Portugali [24] a number of cognitive properties ofspace and spatial properties of cognition have been identified

(i) Cognition Is Situated and Extended Our cognitive activityoccurs in the context of a real environment and inherentlyinvolves perception Internal cognitive processes are shapedby their coordination with external resources [27] Whilecognitive processes occur perceptual information continuesto be captured so as to affect our actions The ways in whichwe dive in cognitive activity are linked to our continuousinteraction with the environment [28] Theories of extendedmind assert a causal flow as the mind uses resources inthe environment and vice versa a two-way interaction in acoupled cognitive system [29] Order and systematicity inhuman cognition and action derive in part from the stabilityof our environment (Michaelian and Sutton 2013)

(ii) We Load the Built Environment with Information Cer-tain cognitive processes trigger associations with elementsof the environment through the incorporation of sociallyacquired information [30] In turn information is classifiedinto potentially shared categories [31] Clustering processesconnect pieces of information through similarities of physicalaspects of the environment or through relations between theirmeanings [32] Environmental elements (say buildings orplaces) are interpreted through such socially sharedmeanings[33] Visible elements of a city convey different amounts ofinformation [24] Small details (like a building entrance)can be related to higher orders of information and refer tolarger spatial formations (such as a main street or a citycentre) evolving to rich multilevel hierarchical structuresin which the interactions between information units in thelower levels generate patterns of environmental elements inhigher levels and vice versa The probability of a building orplace evoking a mental representation shared by people isenhanced by its physical appearance and visual identity alongwith its visibility and location in the environment and thesocial information associated with activities performed there([34] cf [35])

(iii) Spatial Information Relieves Our Cognitive Work Wedeal with the environment also through memory Howeverour short-term memory is constrained in its capability toprocess information [24] Luckily our knowledge of spatialproperties and patterns can integrate semantic visual andconfigurational aspects projected into an urban environmentas symbolic off-loading The spatial environment becomesan external memory carrying information about activities

Complexity 5

and agencies found in it for future retrieval [28] Insteadof trying to keep all the relevant details about activities inour short-term memory we retrieve these details from theenvironment itself as an extension memory and informationsource [36] Environmental information is a means to reducethe effort of memorization Other mnemonic features such askinesthetic images have been shown to functionally preservespatial and semantic properties of the external world [37]This external semantic resource relieves the cognitive loadand what Wilson [28] calls ldquorepresentational bottleneckrdquo thelimits faced by our internal memory which lead us to extendour mnemonics to the environment in the first place Theroles of long-term and short-term memories for agents ableto retrieve social information from space will be of particularinterest to our approach

(iv) Cognition Is Pressed for Time Daily life requires responsecapacity More cognitive capacity can be built up fromsuccessive layers of real-time interaction with our spatialcontext We create mental models of the environment fromwhich we create action plans Sophisticated forms of situatedcognition happen in any activity that involves the continuousupdating of plans in response to changing conditions ofaction Time pressure demands spatial decisions [38ndash40]

(v) Spatial Information Serves Action Agents build instruc-tions about events in the environment in an indexical form[41] The practical aims of agents can thus be directlyconnected to their situation [42 43] We manipulate theenvironment as a way of dealing with our practical problemsEmbodied cognition includes mechanisms serving adaptiveactivities [44] Our cognitive memory evolved in the serviceof perception and action in a three-dimensional environ-ment [28] Cognition serves action through a flexible andsophisticated strategy where information is stored for futureuse without firm commitments on what the future use maybe Spatial information can be absorbed through a varietyof uses for which it was not originally encoded This meansthat new uses may be derived from a stored representationof space They need not be triggered by direct observationof the environment and its affordances [45] Environmentalrepresentations either schematic or detailed appear to belargely free of a particular purpose or at least contain infor-mation beyond what is necessary for a specific action This iscertainly an adaptive cognitive strategyThe fact that humansencode the physical world using spatially and semanticallystructured mental models offer a huge advantage in solvingproblems

This nondiscursive knowledge includes spatial propertiesand heterogeneities and enables us to build inferences forexample when we try to imagine a likely street where tofind certain activity In turn Lakoff and Johnson [46] arguethat mental models are based on a modelling of the physicalworld and bolster in analogies between abstract and concretedomains Other works tie semantics with imagetic schemesable to incorporate knowledge of the physical world as away to encode relationships between events In perceivingsomething we perceive not only its observed form butalso the potential information enfolded in it [24] This very

property is crucial to the presence of urban space in theselection of actions to be performed

So our actions encode space with social informationand such informationally differentiated space plays a partin easing the representational bottleneck in our memoriesWe are able to retrieve information from this semanticarrangement that takes the form of the city before and duringour actions and experiences Such semantic layer is storedin activity places and in the spatial relations between themandmay be related to physical and functional heterogeneitiessuch as accessibility and centrality patterns [34] Such layersof spatial information can be evoked in our memories andused when we need to make inferences such as where wecould find a particular social situation or place Howevermost theories of situated cognition still do not recognize howthis informational space is part of the way we create actionsystems Let us see how this could be the case

4 The City as a Frame of Reference

Our aim has strong parallels with Hutchinsrsquos [47 48] viewon cultural ecosystems operating cognitively at larger spatialand temporal scales than an individual person According toHutchins all instances of cognition can be seen as emergingfrom distributed processes that is from interactions betweenelements in a system Cognitive properties are not predictablefrom individual cognitive capacities Humans create theircognitive powers by creating the environment in which theyexercise those powers This socially distributed cognition is acollective operation produced by interactions among agentsin active relation to their environment in contexts of ongoingactivity Possibilities for individual learning depend on thestructure of the environment to the point that the distributionof cognitive skills is determined by the distribution of prac-tices engaged in by people As Hutchins we are concernedwith the tasks people confront in their everyday world takingpart in processes of coordination of action as they movethrough physical spaces Ultimately our aim relates well tohis proposition that ldquocultural practices decrease entropy andincrease the predictability of experiencerdquo [48 page 46] Ourwork follows parallel lines through a systems approach ableto encompass large-scale systems of action and spaces

We propose to explore the informational role of city spacein how agents build connections between actions We allknow that urban space is produced to express human activityBut theories of cognition suggest much more space wouldbe semanticized by our actions That means that activityplaces are informationally differentiated and acquire a levelof semantic definition similar to types of action [49 50]This differentiation is enacted in and associated with the verystructure of urban space Space is differentiated in at leastthree levels (i) its physical properties of extensity and config-uration such as accessibility and centrality patterns in a city(ii) recognizable shapes in its visual and tridimensional form[24] and (iii) the semantic contents of activities performed inbuildings and places (cf [34 35]) (Figure 1)

How can this informational space become a part ofactionThere is a remarkable lack of theoretical and empiricalwork on this problem First we seem to retrieve meanings

6 Complexity

(a) (b) (c)

Figure 1 Differential space like a code of instructions for collective action urban space finds differentiation both physically as hierarchiesof accessibility and distribution of built form (a) and semantically through actions performed in buildings and places ((b) (c))

associated with our actions in architectural and urban spacesWe arguably grasp useful information about the activityperformed in a place not only by visual features but alsoinferring what people do there Space not only representsactivity it is actuated and thus loaded with meanings relatedto performance This is a semantic dimension of space spaceldquomeansrdquo as much as our actions precisely because it issemanticized by our actionsThis enacted information is veryimportant Action is temporally related to spatial locations

Second these locations allow us to access potential coun-terparts in interaction As we shall see below knowing acity means that we can find places where certain activitiesare performed Recognizing spatial differences and patternsallows us to infer such locations Apart from other sources ofinformation such as linguistic exchanges we do so becausewecan retrieve environmental information about activities andlocation patterns

Third places are means of connection we can actualizeconnections with certain people or activities during a periodof time These spatial links allow us to create complexes ofinteraction In practice places ldquodrawrdquo courses of action withdifferent orientations and contents and tie them togethermomentarily Flows of convergent and divergent actionshappen all the time when people leave their homes to workto seek services or go about to socialize and do so oftenwithout prior knowledge of activities counterparts or placesthey wish to visit (Figure 2) Urban space becomes whatParsons [51] would call an ldquoaction frame of referencerdquo aconnective fabric which both enables and constrains choicesmaterially and socially

5 Entropy as a Way into Social Organization

In order to be a part of selection the urban environmentneeds to be informational enough and not just to suggest

t3

t2

t1

L1 L2 L3 L4

Figure 2 As a reference system urban space converges courses ofaction in time (119905

1 1199052 and 119905

3) and space (119871

1 1198712 1198713 and 119871

4)

particular options but allow unforeseen changes in the courseof our actions Luhmann suggests that a desired social activitymust be within reach ldquoBecause complete interdependence isunattainable however interdependencies only come aboutby selection [ ] Successfully established interdependenciesthen serve as perspectives for and constraints on the struc-tural selections that connect onto themrdquo [5 page 284] As asocial event this connection is produced through communi-cation But before communication comes into being certainconditions have to be there spatially there Places must bechosen as conjunctive pieces that materialize connections

Complexity 7

Could cities stimulate connections If they could andgiven the fact that they are formed by space and thatspace is not fully malleable is it reasonable to supposethat certain spatial formations could ease such momentaryconnections We know that proximity increases the intensityof interaction [52 53] But what form should urbanized spacetake in order to meet such systemic expectations We mayunderstand such spatial conditions through a counterfactualscenario Imagine a fully chaotic urban formation with norecognizable concentrations or paths Visual elements couldnot convey information as spatial differences would notdevelop into structures emerging out of absolute heterogene-ity References to activities of interest would have to relyonly on our memory Selection of actions would becomedependent on very detailed mental representations of such achaotic formation Inferences about more likely places to findparticular activities would be very hard tomakeWewould beimmersed in a spatial environment without patterns to guideour choices reduce our efforts and amplify our interactionsIn a world of unstructured spaces actions would face lackof information noise and relentless entropy The possibilitythat space may have an informational presence in interactionexpresses the possibility that the physical environment mayhave informational properties and that the world is rich ininformation

Now not only our daily experience denies that cities aresuch completely chaotic structureless places but also spatialeconomics and urban studies offer plenty of theories andevidences that as complex as they may be cities are (at leastpartially) structured places From location patterns describedsince Alonso [54] and accessibility patterns described fromHansen [55] to Hillier [56] we know that urban patternsmatter they imply that activities can be arranged in ways thatare recognizable and accessible And from the point of view ofselection the realization that activities are somehow arrangedalong a spatial structure is a form of preselection Activitiestend to be distributed according to levels of accessibilityland values and practical dependencies facilitating comple-mentary actions Intentionally or not and however subject tocontingencies urban space is also structured (physically andinformationally) which allows possibilities of action to bemore easily known selected and viable In short at first theurban structure affords a range of possibilities for selectionIn a second moment the structure itself suggests gradients ofattractive possibilities

In a view of social systems constituted by connectionsof actions this means that space can be part of the constanttransition from individual acts to complexes of actions (indi-vidual act rarr places as connections rarr action systems) Ourhypothesis here to be tested in computational experimentsin the following section is that cities become a crucial part ofthis connectivity They would reduce risks such as the lack ofinformation or courses of action too costly to emerge

What does this say about the effect of cities on entropyBeginning with Luhmann interactions have structural valuebecause they represent selections from combinatorial possi-bilities If we relate this to Shannonrsquos entropy the very emer-gence of urban structures eliminates spatial scenarios whereevery possible connection would have the same probability

Another counterfactual scenario might help here Entirelyhomogeneous cities (or entirely heterogeneous for thatmatter) free from structure would present even probabilitiesof choice between actions Of course this would be a problemfor social systems connections potentially interesting tomostpeople would be as difficult to find as any If urban spacewas purely homogeneous or purely random the probabilityof finding an activity would be distributed homogeneouslyin space That is a scenario of maximum entropy The spatialenvironment would be useless in stimulating certain sought-after connections which could otherwise be there say whenwe concentrate activities like final supply or input-outputexchanges in production Effort time spent and economiccosts involved in finding activities of interest would rise

On the other hand when urban space finds differentiatedcontents and structures say location patterns along acces-sible paths it naturally creates differences in the probabilityof interaction and decreases in entropy Location patterns arein fact material and informational expressions of interactionsystems The differences engendered by such patterns gener-ate differences in the chances of finding an agent a service orcommerce If a city did not have recognizable internal forma-tions such as a centralities or high streets agents would havemore difficulties in knowingwhere they aremost likely to findwhat they are after A partially structured environment allowsus to build inferences about actions and agencies available in asocial systemWhen these agencies are also related temporallythrough proximity and functional complementarity there isan even greater potential for differences in probability toemerge suggesting new connections and sequences Whatall this means is that cities internally differentiated yet farfrom absolute heterogeneity have the effect of reducing socialentropy

Before experimenting with these relationships in silicolet us attempt an introductory description of what thesedifferences in probability entail in theway social systems han-dle entropy (Figure 3) Linearizing for the sake of simplicityprocesses that in fact occur simultaneously the reductionof social entropy would take the following form (a) In aninitial state free of space think of the actions as lines movingin time Agents can do anything and we cannot foreseewhat they will do a potential state of high entropy Colorsof the lines in Figure 4 represent different informationalcontents in personal orientations (b) Then agents startselecting actions and converging into different positions in aspatial system arranged as places or buildings (representedby colors in the vertical strip) in order to connect withothers Action lines might have subtle differences in relationto the places they approach so they converge into placesby informational proximity Connections can only happenif all other possibilities of connection are excluded (c)As these convergences happen the initially chaotic mazebecomes a momentarily coordinated system where agentscooperate The overall entropy of actions is reduced (d)After each spatially held event actions may change againaccording to changing orientations Accordingly the colorsof lines change Entropy increases as actions go into anothermomentary state of unpredictability as new possibilities arepresented to agents and somemust be selected if actions are to

8 Complexity

Momentarilycoherent systems

State of unpredictabledecisions

Action systems

Reduction of entropy

Coordination of actions

Selectionof actions

High entropy state

Information growth(Hidalgo 2015)

Possibilities of action(Luhmann 1995)

Differences in theprobability ofinteractions

(Shannon 1948)

Agents interact Orientations align inactivity places

Actions mediated byinformation

(Luhmann 1995)

City as informationenvironment(Haken and Portugali 2015)

Figure 3 Recursive process that conducts interaction systems from chaotic states (high entropy) into coherence (low entropy)

(a) (b) (c)

(e) (d)

Figure 4 The cycle of social entropy and space clockwise from the high unpredictability of potential actions to a momentarily orderedsystem as actions converge into places only to dive in entropy again

Complexity 9

Actionagent x

Actionagent y

Activity place

Cooperation

Change insubjectiveorientation

Figure 5 Aligning different actions in the cooperation process

be actualized (e) And then actions ldquoreferrdquo again to meaningsand positions in space starting a new cycle of entropy(Figure 4)

These distinct moments are theoretical of course Inreality reminding Prigogine and Stengers [6 page 124] thisldquomultitude of eventsrdquo takes place simultaneously and mayrender any clearly emergent cycle invisible But that neithermeans that individual cycles do not happen or aggregate inbunches nor they compensate one another like peaks andtroughs in colliding sound waves In fact routinization indaily life leads to rhythms and cycles unfolding from orfolding over the seemingly chaotic maze of actions But evenif actions were far from synchronic the fabric of interactionstill faces the challenge of entropy that is the challenge ofproducing differences in the probability of connections as a wayto ease organization whenever agents feel compelled to createnew interactions and this process is remarkably recursiveWe converge to places to act with others and do so on adaily basis Once engaged in an activity place interactionrequires levels of cooperationWemay bring our own distincttendencies or latent orientations in the form of long-termmemories of how to act along with short-term memoriesof a previous action both shaping our next actions As weact socially we bring our orientations into contact in away to open them in the communication process regulatingcollective action A result of this coordination is to reduceinitial differences Our actions ldquoalignrdquowithin an activity place(Figure 5)

All this suggests that social life involves a continuousalthough unconscious handling of entropy a vital oscillationbetween unpredictability and predictability As this processengages an enormous number of agents and events andtranscends places and contexts and therefore cannot beobserved as a whole entropy can only be fully grasped intheoretical representations A useful way to examine space asan information environment for action is through a compu-tational model a means to examine a counterfactual worldwhere other steering media such as language or social ruleswere ldquoswitched offrdquo keeping only properties of space activealong with agentsrsquo cognitive abilities to align orientationswith activities Once we abstract from other media wemay assess whether space could have any presence in theself-organization of action Let us examine this possibilitythrough some experiments

LBuildingactivity type

Figure 6The theoretical one-dimensional citywith a set of differentactivity places Here there are 119871 buildings distributed in a ringconstituting the city space Colors represent activity types

6 Digital ExperimentsA Unidimensional Model

61 Design Concepts An agent based model (ABM) is pro-posed Agents are defined to perform actions at each time-step and places by corresponding activity types The decisionon which action to perform next may be influenced bythree different conditions namely (i) latent orientation120590in

119894(119905)

representing the tendency of a single agent to act arounda particular type of action initially randomly distributedthis condition remains over time (ii) current action 120590

119894(119905)

performed by an agent as heshe selects an activity placein order to perform a new action this means an influenceof the current action while allowing gradual changes oforientation in time influenced by other agents and activitiesand (iii) types of activity places 120590

1199091015840(119905) where agents perform

supporting specific types of action Agents follow simplerules selecting their activity places closer to their latentorientations and current actions while being influencedby those activities Activity places are also influenced byvisiting agents but they change at a slower rate In shortagents coevolve with their spatial and social environmentThe model simulates different situations under this basicstructure Differences in the weight of these factors overthe next action may lead to quite different levels of socialentropy suggesting new possibilities for understanding thephenomenonAlthough there is a tradition inmodelling dailyactivities including a growing literature using digital loca-tional data (eg [57]) our ABM is not designed to representempirical sequences of actions or actual routines Instead itfocuses on trends that might emerge from the interfaces ofsimplified systems of action information and space

62 Variables and Scale Consider a unidimensional cityformed by agents and places Places form a ring with length(or perimeter) 119871 as shown in Figure 6 Our choice forrepresenting the city by a ring is justified by the minimalsufficient representation of spatial distance as a factor whose

10 Complexity

role in social organization is to be assessed This is allowedby a unidimensional model (eg [58ndash60]) The ring formallows continuous movement across a linear sequence oflocations eliminating centrality factors while taking intoaccount periodic boundary conditions in order to reduceborder effects eliminating the role of topologywhile isolatingthe problem of distance (see [61ndash63]) This stylized cityhouses 119873 agents which in each time-step 119905 select and visita specific place located at the position 119909 within the cityConsider that the position of the 119894th agent at the time 119905 isrepresented by 119909

119894(119905) The position of an agent can assume the

integer values 1 2 119871 according to the place that shehechooses to visit This means that at every time-step eachagent will choose one activity place to perform her nextaction

We considered 120590119894(119905) as the current action of the 119894-th agent

and 120590119909(119905) the activity performed in a place located at the

position 119909 = (1 2 119871) of the city both in time 119905 Wequantify action orientations weighting the three variables(latent orientation current action and activity place) Firstvariables are translated into different numeric values Let usassume that an orientation is an integer number between 0and 1000 each number represents a particular type of actionand numbers in their vicinity mean similar actions Theresulting orientation in time 119905 for each agent is the weightedaverage of those three parametersThedifference |120590

119894(119905)minus120590119909(119905)|

gives us the difference in orientation between an agent andthe activity place At every time-step each agent assesses thedifference between her orientation and every activity placein the city An agent chooses a specific place based on herorientation affinity with this place minimum difference

As we hypothesized above the urban structure has a rolein this selection In this experiment we reduced ldquourban struc-turerdquo to distance In order to investigate how distance may ornot interfere in this selection we propose two scenarios Inthe first one the agent selects her next activity based only onthe similarity in orientations between herself and the activityplace Agents can move across a space free of friction Spaceis not a constraintThis scenario considers only differences inorientation The parameter 119864

119894(119909 119905) estimates the interaction

between agent 119894 and the place located at 119909 at the time 119905 Theagent selects activity locations that minimize this quantityWe propose

119864119894 (119909 119905) =

1003816100381610038161003816120590119894 (119905) minus 1205901199091003816100381610038161003816 (1)

In the second more realistic scenario space imposesfriction to movement The agent takes into account thephysical distance between her current location in the city andthe location of her next activity and opts to minimize thisdistance along with the difference between her orientationand that of the activity place We propose

119864119894 (119909 119905) =

1003816100381610038161003816(119909119894 (119905) minus 119909) (120590119894 (119905) minus 120590119909)1003816100381610038161003816 (2)

Summing up an agent 119894 selects at time 119905 a particularactivity place located at119909 thatminimizes the function119864

119894(119909 119905)

In the first scenario she will choose activity places closerto her orientation In the second she will choose activityplaces closer in both orientation and physical distance This

brings a form of ldquoenergy costrdquo into the factors consideredSpace takes on a function role because of the energy used inldquovisitingrdquo Agents move in this linear city willing to minimizethis energy while searching for social information latentin places close to their changing orientations Changes inentropy levels are derived from this energy function In itsspatial version the model assesses effects of the effort toreduce energy in action minimizing informational differ-ences (between orientations and activity places) and spatialdistances (between agents and activity places)

63TheEvolution of ActionOrientations Orientation evolvesin time according to the following rule When an agentchooses and joins a particular place both agent and placebecome a little closer in terms of orientation This meansthat activity places are not immune to what agents performin them and that the urban activity system also changes intime To illustrate how orientations are constantly updatedconsider that agent 119894 selects an activity place located at 1199091015840Orientations will be updated according the rule

120590119894 (119905 + 1) =

1120572 + 120573 + 120574

(120572120590in119894(119905) + 120573120590119894 (119905) + 1205741205901199091015840 (119905)) (3)

Selection also depends on three additional parameterslatent orientation 120590in

119894(119905) weighted by parameter 120572 current

orientation120590119894(119905)weighted by parameter120573 and type of activity

place 1205901199091015840(119905) weighted by parameter 120574 This means that agents

will consider the three factors previously described (120590in119894(119905)

120590119894(119905) 1205901199091015840(119905)) with different weights (120572 120573 120574) to estimate which

activity she wants to perform in the next time-stepThe latent orientation evolves from an initial randomly

distributed behaviour 1205900119894 We consider that 120590in

119894(119905) is normally

distributed around 1205900119894 meaning that every agent has an

ldquoaveragerdquo latent orientation whose value will be derived ateach time-step from a normal distribution peaking at theagentrsquos average orientation If 120572 is sufficiently small in relationto 120573 and 120574 then the latent orientation does not play a rolein the agentrsquos action However if 120572 is sufficiently large (oncein relation to 120573 and 120574) then the action becomes stronglydependent on the latent orientation This would lead to apretty conservative city where inhabitants are not open tochanges in behaviour

The current action of an agent may influence a newaction with an intensity that depends on parameter 120573 If thisparameter is small in comparison to 120572 and 120574 then the agenthas no memory of an ongoing orientation However if 120573 issufficiently large then the new action is strongly dependenton a previous one (a Markovian behaviour)

The intensity of the activity placersquos influence on the agentrsquosaction depends on the value of 120574 If this parameter is smallin comparison to 120572 and 120573 then the activity place does notaffect her next actionmeaning that the place does not changebehaviours However if this parameter is sufficiently largethen the place plays a strong influence on future actions Theactivity place will be updated according to the rule

1205901199091015840 (119905 + 1) = 1205901199091015840 (119905) + 120579sum

119894isin1199091015840

120590119894 (119905) (4)

Complexity 11

where 120579 is a parameter sufficiently small That is placesare less influenced by agents than the other way roundThis means that at every time-step a place would have itstype of activity closer to the average of orientations of itsvisitors Therefore the main activity of the place will changeslightly according to visiting agents while changing theirorientations

64 Frequency of Orientations and Entropy Levels We cal-culate entropy assessing the distribution of resulting orien-tations of all agents in the social system at each time-stepusing Shannon entropy Our quantitative interpretation ofinformation removes ambiguities from the relation betweenShannon information and semantic information as differentorientations actions and types of activity places are repre-sented by numerical values This allows us to estimate socialentropy through Shannon entropy The entropy of actions iscalculated as follows consider119873(120590 119905) as the number of agentswith orientation 120590 at the time 119905 (note that the total populationis 119873 = sum

120590119873(120590 119905)) We compute the frequency (or density)

120588(120590 119905) of this orientation within a population with

120588 (120590 119905) = 1119873119873 (120590 119905) (5)

Equation (5) calculates the probability of observing anorientation 120590 at time 119905 We compute the entropy level for anydistribution of orientations with

119878 (119905) = minussum120590

120588 (120590 119905) ln (120588 (120590 119905)) (6)

Equation (6) describes how uneven is the probability of find-ing different orientations Higher values mean that differentorientations have almost the same probability to happenwhile lower values indicate a system with clear orientationtrends The reduction of entropy implies that the probabilityof certain actions increases that is actions grow in similarityIn the limit as entropy falls to zero all agents in the systemwould reach the same orientation

65Model Procedure Themodel performs the following pro-cedures

(1) The city is created by generating the following

(a) The type of activity in places that is (120590119909=1(119905 =

0) 120590119909=2(119905 = 0) 120590

119909=119871(119905 = 0)) also generated

from a random distribution This step createsactivity places in the ring-city and assign themrandomly distributed activity values 120590

1199091015840(119905)

(b) The initial position of 119873 agents that is (1199091

1199092 119909

119873) and their respective orientations

(1205901(119905 = 0) 120590

2(119905 = 0) 120590

119873(119905 = 0)) generated

from a random distribution This step createsmoving agents in random locations and assignthem latent orientations based on the initialrandomly distributed orientation 120590in

119894(119905) = 120590

119894(0)

(2) At each time-step 119905 every agent chooses a place toact This place should minimize the function 119864

119894(119909 119905)

Suppose that the chosen place by 119894 is defined as 1199091015840Then agent and place update their orientations 120590

119894and

1205901199091015840 as described above

(3) Update the distribution of orientations 120588(120590 119905)(4) Compute Shannon entropy 119878(119905)

66 Results We have developed a number of observationson the behaviour of social systems working under differentparameters

(i) Entropy reduction is only found in scenarios wherespace imposes friction to movement that is wherethe spatial distance between agents and activity placesis considered as an active factor in selection Entropyrequires an extensive space in order to be reduced Butspace cannot do it alone Surprisingly space is a nec-essary but not sufficient condition Figure 7 comparestwo scenarios In the first one distance is an issue inthe selection of activities (red line)Thedistribution oforientations changes from a homogeneous one at thestart of simulation (a) to a nearly normal distributionat the end of simulation (b) where certain kindsof action are more likely to happen Informationalcontents in extensive space help aligning contents inactions In the second scenario agents move free ofspatial friction (blue line) Orientations are initiallyrandomly distributed (a) and continue to be so at theend of simulation (b) as a similar number of agentsare distributed along different orientationsThis resultsuggests that a materially active space becomes ameans for increasing the probability of certain inter-actions easing the collective coordination of action

Nowassessing the relative influence of social and personalfactors on the selection of a next action namely the socialinformation in activity places latent orientation and the cur-rent action under the influence of spatial friction (Figure 8)we can say the following

(i) There is structure in the relation between selectionfactors and the reduction of entropy Entropy beginsto fall and fluctuate around specific values accordingto different combinations of factors For some com-binations there is a great reduction of entropy Forothers the reduction is minimal almost negligibleFluctuations derive from the random component inthe decisionmaking of the agent (under the influenceof latent orientation)

(ii) A strong latent orientation (120572 sufficiently large) leadsto increasing entropy (blue lines in Figures 8 and9) since agents cannot align their actions with otheragents and activity places Like a long-term memorysystems whose actions are dictated by latent orienta-tions are likely to preserve initial orientations whichwere randomly distributed and therefore homoge-neous This condition is responsible for limiting thereduction of entropy

(iii) Current action alone does not shape new actions inany specific way (green lines in Figures 8 and 9)

12 Complexity

Start of simulation

00000

00005

00010

00015

00020

o

f tot

al ac

tions

250 500 750 10000Type of action

Materially inactive spaceMaterially active space

t = 0

(a)

End of simulation

00000

00005

00010

00015

00020

o

f tot

al ac

tions

250 500 750 10000Type of action

Materially inactive spaceMaterially active space

t = 50

(b)

Figure 7 Histogram of probability distribution of actors performing different types of action at the start (a) and at the end (b) of simulationsin scenarios where distance is not considered (blue) and distance is considered aligning orientations in action coordination (red) Resultsare averaged for 30 runs for 125 parameter combinations

activityplace

currentaction

latentorientation

225

250

275

300

325

Entr

opy

10 20 30 40 500Time

Figure 8 Evolution of entropy colors in a RGB scale refer toparameter weights in decisions on actions in a next time-step andtheir impact on entropy levels in different simulations (under thefriction of distance) In the palette triangle each color representsa specific combination of the three parameters In each vertex aparameter reaches maximum value while the others have minimumvalue Results are averaged for 30 different runs for 125 parametercombinations

It just keeps what is already happening workingas ldquoreinforcement feedbackrdquo for either direction theagent (and the system) is going Current actions aremeans to conserve tendencies in the system

(iv) Activity places play a key role in the reduction ofentropy (red lines in Figures 8 and 9)The social infor-mation in space ldquocontaminatesrdquo agents they aligntheir actions through the social contents of placesWe have seen that space matters as extension Nowwe see that space also has a very active informationalpresence

(v) Different combinations of factors have differenteffects over entropy Figure 8 shows the transitions

between the three parameters and how nuancedweights resulting from blending factors matter inthe reduction of entropy For instance blending theweight of activity places and latent orientation on thenext action displays a strong reduction of entropy(pink lines in Figure 8) That means that when agentsdo not keep information from previous selections(ie 120573 asymp 0) the action system reduces more entropythan in other parameter combinations A strongshort-term memory leads the system into conservingitself and into a poorer capability to coordinateactions Also when current actions 120573 and activityplaces 120574 share a similar weight over the next action(orange lines on top) the social system does notexperience a great reduction of entropy as agents tendto reproduce their actions

(vi) Finally the reduction of entropy implies that theprobability of certain actions and interactionsincreases In practical terms this means morealignments between agents and more connectionsbetween actions (interactions) However if all agentsin the system reached the same orientation andentropy dropped to zero the system would lose inter-nal differentiation Agents would behave in the sameway say in a world with no personal differentiationspecialization or division of labour New orientations(therefore entropy) are necessary if the social systemis to keep differentiated agencies On the otherhand a system with maximum entropy would createdifference in actions to a point where no action coor-dination or interaction would be possible Clearly thissituation cannot be the case A social system requiresbalances neither full entropy nor total predictability

These results also suggest that we are dealing with distinctldquosocial memoriesrdquo a long-term memory is active in what we

Complexity 13

20

25

30En

trop

y

04 0602-latent orientation

04 0602-current action

20

25

30

Entr

opy

20

25

30

Entr

opy

04 0602-activity place

Figure 9 Parameter exploration isolated importance of each parameter in entropy (under the friction of distance) Each vertical set of dotsrepresents one fixed value for the highlighted parameter with the two other parameters assuming every possible combination within thetested range Each dot corresponds to the entropy value when a sufficiently large time-frame is used

call latent orientations Short-term memory is active in theinfluence of a current orientation over a new action In turnthe social information latent in activity places is a result ofsocial arrangements (say as firms a local economy and soon) which find certain stability changing much slower thenactions It is an extension of the social system projected ontourban space stabilizing the system to some extent

7 Cities and Social Interaction Conclusions

What does this approach bring to the state of the art on therelations of social interaction information and space and onsocial entropy and ABM in particular We have seen thatprevious approaches overlooked the coordination of actionas a key empirical and analytical problem Furthermorethey tend to have the thin spatiality of a passive territorialbackground (eg [16 20]) In turn although the conceptof ldquocollective actionrdquo is finally getting attention in urbanstudies [64] the problem of social organization is still largelyunderestimated The problem of entropy has been dealt withsince Wilsonrsquos [65] work on spatial interaction but has notreached the mainstream of the discipline and deals mostlywith urban form (eg [24]) and spatial distributions [66]not with social information and action Regarding agentbased models we are not aware of an approach that dealswith the problem of how cities and space are part of socialorganization In a sense our model is closer to Axelrodrsquos[67] ABM model of dissemination of culture His agentsexchange ldquoculturerdquo through direct contact with neighboursin a cellular automata model whereas our agents are mobileand actively deal with distance and social information inspace recognizing types of activities in places The intensityof exchanges between agents in Axelrodrsquos model is a functionof similarities between them In our model exchanges aremediated by the selection and interaction with space In ourunderstanding our contribution in terms of ABM lies onthe fact that a new and simple model is proposed to dealwith the material and informational dimensions of socialorganization not just as an illustration but as a proof ofconcept space can have causal presence if agents generatepreferences as detailed in a domain that coevolves with them

Now does the simulation model corroborate the theoryintroduced in this paper Our proposition cannot be empir-ically validated at this stage given that it is very hard to

assess peoplersquos choices and the whole panorama of actionsin a city Entropy may well be beyond observation In thesecircumstances simulations of the behaviour of agents underdifferent spatial conditions become useful to assess entitiesevents and entropic forces at play One expected result fromthe model is that space matters for coordinating actions butsimulations surprisingly showed that space is a necessary butnot sufficient condition for the reduction of entropy Neithermaterial cognitive resources nor individual powers of agentsare enough Information in activity places and a bit of agentsrsquopersonal history also play roles in the way social systemsdeal with their own entropy That would mean a causal butnondeterministic presence of space

Substantively this approach explores a subtle but funda-mental presence of cities in social organizationWe argue thaturban space materializes gradients of difference in potentialinteractions from less to more recognizable costly or likelyPlacing simple criteria including distance as a factor in select-ing activities our ABM showed that space becomes a meansfor producing differences in the probabilities of interactionincreasing chances of certain selections and convergences incollective action

In reality people recognize the social information ofactivity places along with uses of busy streets or local central-ities We are able to retrieve these differences from the builtenvironment and relate to them One of the key points of ourargument is that these differences help us make selectionsEvery time we move through the city and select places toperform we participate in the large-scale coordination ofaction We enact space as a ldquoreferential systemrdquo [49 50]a set of semantic bits indexing social activities informingand guiding daily decisions Space becomes inherently partof the conversion of orientations into interactions that isconnections that crisscross sequences of actions recreatingthe interaction system By means of a structure that bothmaterializes and restricts the quasi-endless combinatorial pos-sibilities of interaction this system can acquire sufficientinternal guidance to make its own reproduction possibleBy distributing sufficiently recognizable differences in theprobability of interaction cities express and release localforces of reproduction of social systems The interfaces ofaction cognitive and spatial systems transform entropiccomplexity into structured complexity constantly reorderingactions in time

14 Complexity

This materialistic viewpoint can be integrated with otherviews of social organization Further work may include therelational roles of cultures language social norms and othersteering media along with explorations of possibly similareffects of communication technologies over entropy At thisstage we attempted to integrate a social dimension (agentsengaged in coordination) a spatial dimension (an extensiveenvironment) and an informational dimension (as differ-ences in action and in the environment itself) a demandinginterdisciplinary effort The approach identifies a central rolefor coevolving agents actualizing aims as they coordinate (cf[68]) and a central role for the environment exterior butresponsive to agents leading to cumulative changes alongtheir historyThemodel also identifies roles for cognition andmemory in the ways agents deal with their own orientationsand choose their actions and activity places changing theirspatial environment with consequences in the overall levelsof entropy All this suggests that the behaviour of the modelcannot be reduced to or fully predicted from the behaviour ofindividual variables which is a key aspect of complexity

So one of the main aims of this paper was to assesswhether space as information environment had any effecton the coordination and entropy of interaction However ifphysical distance and social information in activity placesmatter what about city size and the internal spatial struc-tures of cities What about rural areas or rarefied suburbsSince our model explores an abstract circular city it onlybegins to answer such questions Our findings indicate thatshorter distances between activity places tend to reducesocial entropy Density seems to matter denser spatialities(as opposed to rarefied ones) have a role to play A pathfor further development here is Jacobsrsquos [69] idea that thediversity of activities generates positive externalities Anddiversity as spatial economics shows us has to do withthe size of cities and density of population (eg [70 71])Entropy may also be influenced by diversity Batty et al [66]have shown that information increases as cities get biggerThe remaining question is whether larger denser and moreinternally structured cities could create process and reduceentropy with more intensity than smaller cities transformingbig pools of activities into differences in the probability ofinteraction Although we proposed a unidimensional modelour agentsrsquo behaviour can be explored in more realisticrepresentations of cities testing the roles of density topologyand diversity while going beyond perfectly informed agentsBy proposing the ring model we intended to simplify acity to a minimum system where the spatial dimensionof social organization becomes intelligible without losingfundamental properties

Shouldwe conclude fromall this that entropy is a problemfor social systems Our approach suggests that it is notEntropy is a necessary force It means that novel orientationsare entering the interaction system and must be dealt withEntropy only becomes a problem if not converted into orga-nization only if numerous potential actions are not convertedinto smaller sets of actual interactions The effects of newactions over entropy would be positive once actualizedhaving to dowith the diversity of agencyConservative systemsare likely to face less entropy but also likely to produce

less novelty having homogeneity as a problematic horizonFluctuations of entropy seem vital to complex societies

Finally our intention was to explore entropy as a meansto think about the conditions of social organization ldquofromanother anglerdquo so to speak from the viewpoint of challengesinvolved in social reproduction Hence the approach focusedon the idea of uncertainty surrounding agents dealing withaims and decisions and the different probabilities of inter-action that follow an attempt to bring social cognitiveinformational and spatial theories under a single roof Ourapproach sees cities as connective systems produced to createthe delicate fabric of interaction that keeps large numbersof agents coherently living together It also suggests that thevarying states of entropy reveal deep connections betweensocial informational and spatial systems

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

Theauthors would like to thank the Ecole Polytechnique Fed-erale de Lausanne (EPFL) CAPES FAPEMIG FAPESP andCNPq for financial support Maıra Pinheiro Henrique Loreaand colleagues at ABMUS2017 for earlier discussions andMike Batty for exchanges and inspiration Finally the authorsthank Romulo Krafta maverick of the complexity studiesof cities in Brazil for his input and support throughout theyears This work is dedicated to him

References

[1] C W Mills The Sociological Imagination Oxford UniversityPress New York NY USA 1959

[2] M Castells ldquoThe information age economy society and cul-turerdquo inThe Rise of the Network Blackwell London 1996

[3] C Hidalgo Why Information Grows The Evolution of Orderfrom Atoms to Economies Basic Books New York NY USA2015

[4] D Helbing The Automation of Society Is Next ETH ZurichSwitzerland 2016

[5] N Luhmann Social Systems University Press Stanford 1995[6] I Prigogine and I Stengers Order out of Chaos Manrsquos New

Dialogue with Nature Bantam Books 1984[7] C E Shannon ldquoAmathematical theory of communicationrdquo Bell

System Technical Journal vol 27 no 4 pp 623ndash656 1948[8] W Buckley Sociology and Modern Systems Theory Prentice-

Hall Englewood Cliffs NJ USA 1967[9] D D McFarland ldquoMeasuring the Permeability of Occupational

Structures An Information-Theoretic Approachrdquo AmericanJournal of Sociology vol 75 no 1 pp 41ndash61 1969

[10] F Charvat ldquoOn philosophical aspects of the system conceptionin contemporary sociological knowledgerdquo Quality and Quan-tity vol 6 no 1 pp 3ndash16 1972

[11] F Charvat J Kucera and M Soukup ldquoToward the systemtheory of dependence further general theoretical remarksrdquoQuality and Quantity vol 7 no 1 pp 69ndash90 1973

Complexity 15

[12] P M Horan ldquoInformation-theoretic measures and the analysisof social structuresrdquo Sociological Methods and Research vol 3no 3 pp 321ndash340 1975

[13] O E Klapp ldquoOpening and closing in open systemsrdquo BehavioralScience vol 20 no 4 pp 251ndash257 1975

[14] J Galtung ldquoEntropy and the general theory of peacerdquo in Essaysin Peace Research J Galtung Ed vol L Humanities PressAtlantic Highlands NJ USA 1975

[15] T Parsons The System of Modern Societies Englewood CliffsNew Jersey NJ USA 1972

[16] K D Bailey ldquoSociological entropy theory Toward a statisticaland verbal congruencerdquo Quality amp Quantity vol 18 no 1 pp113ndash133 1983

[17] K D Bailey Social EntropyTheory State University of NewYorkPress Albany NY USA 1990

[18] J G Miller Living Systems McGraw-Hill New York NY USA1978

[19] K D Bailey Sociology and the New Systems Theory Toward aTheoretical Synthesis State University of New York PressAlbany NY USA 1994

[20] K D Bailey ldquoSociocybernetics and social entropy theoryrdquoKybernetes vol 35 no 3-4 pp 375ndash384 2006

[21] F Geyer ldquoThe challenge of sociocyberneticsrdquo Kybernetes vol24 no 4 pp 6ndash32 1995

[22] F Geyer ldquoThe march of self-referencerdquo Kybernetes vol 31 no7-8 pp 1021ndash1042 2002

[23] J HabermasTheTheory of Communicative Action Polity PressCambridge UK 1987

[24] H Haken and J Portugali Information Adaptation The Inter-play between Shannon Information and Semantic Information inCognition Springer New York NY USA 2015

[25] M Weber Economy and Society vol 1 Berkeley University ofCalifornia Press 1978

[26] L Wittgenstein Philosophical Investigations Blackwell Publish-ers Oxford London 3rd edition 2001

[27] L Vygotsky Mind in Society The Development of HigherPsychological Processes Harvard University Press CambridgeMA 1978

[28] M Wilson ldquoSix Views of Embodied Cognitionrdquo PsychonomicBulletin amp Review vol 9 no 4 pp 625ndash636 2002

[29] A Clark and D Chalmers ldquoThe extended mindrdquo Analysis vol58 no 1 pp 7ndash19 1998

[30] R PassiniWayfinding in Architecture Van Nostrand ReinholdNew York NY USA 1992

[31] E Rosch ldquoPrinciples of categorizationrdquo in Cognition and Cate-gorization E Rosch and B Lloyd Eds Hillsdale LawrenceErlbaum Ass 1978

[32] W KintschMemory and Cognition John Wiley and Sons NewYork NY USA 1970

[33] U Neisser ldquoMultiple Systems a New Approach to CognitiveTheoryrdquo European Journal of Cognitive Psychology vol 6 no 3pp 225ndash241 1994

[34] A Faria and R Krafta ldquoRepresenting urban cognitive structurethrough spatial differentiationrdquo in Proceedings of 4th Space Syn-tax International Symposium pp 531-518 UCL Press London2003

[35] J Portugali Complexity Cognition and the City Springer NewYork NY USA 2011

[36] D Kirsh and P Maglio ldquoOn distinguishing epistemic frompragmatic actionrdquo Cognitive Science vol 18 no 4 pp 513ndash5491994

[37] S M Kosslyn Image and Brain The Resolution of the ImageryDebate MIT Press Cambridge MA

[38] R Brooks Cambrian Intelligence The Early History of the NewAI MIT Press Cambridge MA 1999

[39] A Clark BeinGThere Putting Brain Body AndWorld TogetherAgain MIT Press Cambridge MA 1997

[40] R Pfeifer and C ScheierUnderstanding Intelligence MIT PressCambridge MA 1999

[41] A M Glenberg and D A Robertson ldquoIndexical understandingof instructionsrdquoDiscourse Processes vol 28 no 1 pp 1ndash26 1999

[42] JM Iverson and SGoldin-Meadow ldquoWhypeople gesturewhenthey speakrdquo Nature vol 396 no 6708 p 228 1998

[43] R M Krauss ldquoWhy do we gesture when we speakrdquo CurrentDirections in Psychological Science vol 7 no 2 pp 54ndash60 1998

[44] S Franklin Artificial Minds MIT Press Cambridge MA 1995[45] J Gibson The Ecological Approach to Visual Perception

Houghton-Mifflin Boston 1979[46] G Lakoff andM JohnsonPhilosophy in the FleshTheEmbodied

Mind and Its Challenge to Western Thought Basic Books NewYork NY USA 1999

[47] E HutchinsCognition in theWild MIT Press CambridgeMA1995

[48] E Hutchins ldquoThe cultural ecosystem of human cognitionrdquoPhilosophical Psychology vol 27 no 1 pp 34ndash49 2014

[49] V M Netto ldquoPractice space and the duality of meaningrdquoEnvironment and Planning D Society and Space vol 26 no 2pp 359ndash379 2008

[50] V M Netto The Social Fabric of Cities Routledge New YorkNY USA 2017

[51] T Parsons The Structure of Social Action The Free Press NewYork NY USA 1968

[52] T Allen Managing the Flow of Technology MIT Press Cam-bridge MA 1977

[53] F L Ribeiro J Meirelles F F Ferreira and R C Neto ldquoAmodel of urban scaling laws based on distance-dependent inter-actionsrdquo Royal Society Open Science vol 4 Article ID 1609262017

[54] W Alonso Location and Land Use Toward a General Theory ofLand Rent Harvard University Press Cambridge MA 1964

[55] WGHansen ldquoHow accessibility shapes land userdquo Journal of theAmerican Institute of Planners vol 25 no 2 pp 73ndash76 1959

[56] B Hillier Space Is the Machine Cambridge University PressCambridge UK 1996

[57] S Farber M OrsquoKelly H J Miller and T Neutens ldquoMeasuringsegregation using patterns of daily travel behavior a socialinteraction based model of exposurerdquo Journal of TransportGeography vol 49 pp 26ndash38 2015

[58] T C Schelling ldquoDynamic models of segregationrdquo Journal ofMathematical Sociology vol 1 no 2 pp 143ndash186 1971

[59] R Krafta V M Netto and L Lima ldquoUrban built form growscriticalrdquo CyberGeo vol 2011 article 565 2011

[60] N Lanchier and S Scarlatos ldquoFixation in the one-dimensionalAxelrod modelrdquo The Annals of Applied Probability vol 23 no6 pp 2538ndash2559 2013

[61] F L Ribeiro and K N Ribeiro ldquoA one dimensional modelof population growthrdquo Physica A Statistical Mechanics and itsApplications vol 434 pp 201ndash210 2015

[62] F L Ribeiro ldquoA non-phenomenological model of competitionand cooperation to explain population growth behaviorsrdquo Bul-letin of Mathematical Biology vol 77 no 3 pp 409ndash433 2015

16 Complexity

[63] R V dos Santos F L Ribeiro and A S Martinez ldquoModels forAllee effect based on physical principlesrdquo Journal of TheoreticalBiology vol 385 pp 143ndash152 2015

[64] M BattyThe New Science of Cities The MIT Press CambridgeMA 2013

[65] A G Wilson Entropy in Urban and Regional Modelling PionLondon 1970

[66] M Batty R Morphet P Masucci and K Stanilov ldquoEntropycomplexity and spatial informationrdquo Journal of GeographicalSystems vol 16 no 4 pp 363ndash385 2014

[67] R Axelrod ldquoThe dissemination of culture a model with localconvergence and global polarizationrdquo Journal of Conflict Reso-lution vol 41 no 2 pp 203ndash226 1997

[68] B H Hodges and RM Baron ldquoValues as constraints on afford-ances perceiving and acting properlyrdquo Journal for the Theory ofSocial Behaviour vol 22 no 3 pp 263ndash294 1992

[69] J JacobsThe Economy of Cities Random house New York NYUSA 1969

[70] E Glaeser H Kallal J Scheinkman and A Shleifer ldquoGrowthin Citiesrdquo Journal of Political Economy vol 100 no 6 pp 1126ndash1152 1992

[71] S Rosenthal and W Strange ldquoEvidence on the nature andsources of agglomeration economiesrdquo in Handbook of Urbanand Regional Economics J V Henderson and J-F Thisse Edsvol 4 pp 2119-2171 NorthHollandNewYork NYUSA 2004

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Complexity 5

and agencies found in it for future retrieval [28] Insteadof trying to keep all the relevant details about activities inour short-term memory we retrieve these details from theenvironment itself as an extension memory and informationsource [36] Environmental information is a means to reducethe effort of memorization Other mnemonic features such askinesthetic images have been shown to functionally preservespatial and semantic properties of the external world [37]This external semantic resource relieves the cognitive loadand what Wilson [28] calls ldquorepresentational bottleneckrdquo thelimits faced by our internal memory which lead us to extendour mnemonics to the environment in the first place Theroles of long-term and short-term memories for agents ableto retrieve social information from space will be of particularinterest to our approach

(iv) Cognition Is Pressed for Time Daily life requires responsecapacity More cognitive capacity can be built up fromsuccessive layers of real-time interaction with our spatialcontext We create mental models of the environment fromwhich we create action plans Sophisticated forms of situatedcognition happen in any activity that involves the continuousupdating of plans in response to changing conditions ofaction Time pressure demands spatial decisions [38ndash40]

(v) Spatial Information Serves Action Agents build instruc-tions about events in the environment in an indexical form[41] The practical aims of agents can thus be directlyconnected to their situation [42 43] We manipulate theenvironment as a way of dealing with our practical problemsEmbodied cognition includes mechanisms serving adaptiveactivities [44] Our cognitive memory evolved in the serviceof perception and action in a three-dimensional environ-ment [28] Cognition serves action through a flexible andsophisticated strategy where information is stored for futureuse without firm commitments on what the future use maybe Spatial information can be absorbed through a varietyof uses for which it was not originally encoded This meansthat new uses may be derived from a stored representationof space They need not be triggered by direct observationof the environment and its affordances [45] Environmentalrepresentations either schematic or detailed appear to belargely free of a particular purpose or at least contain infor-mation beyond what is necessary for a specific action This iscertainly an adaptive cognitive strategyThe fact that humansencode the physical world using spatially and semanticallystructured mental models offer a huge advantage in solvingproblems

This nondiscursive knowledge includes spatial propertiesand heterogeneities and enables us to build inferences forexample when we try to imagine a likely street where tofind certain activity In turn Lakoff and Johnson [46] arguethat mental models are based on a modelling of the physicalworld and bolster in analogies between abstract and concretedomains Other works tie semantics with imagetic schemesable to incorporate knowledge of the physical world as away to encode relationships between events In perceivingsomething we perceive not only its observed form butalso the potential information enfolded in it [24] This very

property is crucial to the presence of urban space in theselection of actions to be performed

So our actions encode space with social informationand such informationally differentiated space plays a partin easing the representational bottleneck in our memoriesWe are able to retrieve information from this semanticarrangement that takes the form of the city before and duringour actions and experiences Such semantic layer is storedin activity places and in the spatial relations between themandmay be related to physical and functional heterogeneitiessuch as accessibility and centrality patterns [34] Such layersof spatial information can be evoked in our memories andused when we need to make inferences such as where wecould find a particular social situation or place Howevermost theories of situated cognition still do not recognize howthis informational space is part of the way we create actionsystems Let us see how this could be the case

4 The City as a Frame of Reference

Our aim has strong parallels with Hutchinsrsquos [47 48] viewon cultural ecosystems operating cognitively at larger spatialand temporal scales than an individual person According toHutchins all instances of cognition can be seen as emergingfrom distributed processes that is from interactions betweenelements in a system Cognitive properties are not predictablefrom individual cognitive capacities Humans create theircognitive powers by creating the environment in which theyexercise those powers This socially distributed cognition is acollective operation produced by interactions among agentsin active relation to their environment in contexts of ongoingactivity Possibilities for individual learning depend on thestructure of the environment to the point that the distributionof cognitive skills is determined by the distribution of prac-tices engaged in by people As Hutchins we are concernedwith the tasks people confront in their everyday world takingpart in processes of coordination of action as they movethrough physical spaces Ultimately our aim relates well tohis proposition that ldquocultural practices decrease entropy andincrease the predictability of experiencerdquo [48 page 46] Ourwork follows parallel lines through a systems approach ableto encompass large-scale systems of action and spaces

We propose to explore the informational role of city spacein how agents build connections between actions We allknow that urban space is produced to express human activityBut theories of cognition suggest much more space wouldbe semanticized by our actions That means that activityplaces are informationally differentiated and acquire a levelof semantic definition similar to types of action [49 50]This differentiation is enacted in and associated with the verystructure of urban space Space is differentiated in at leastthree levels (i) its physical properties of extensity and config-uration such as accessibility and centrality patterns in a city(ii) recognizable shapes in its visual and tridimensional form[24] and (iii) the semantic contents of activities performed inbuildings and places (cf [34 35]) (Figure 1)

How can this informational space become a part ofactionThere is a remarkable lack of theoretical and empiricalwork on this problem First we seem to retrieve meanings

6 Complexity

(a) (b) (c)

Figure 1 Differential space like a code of instructions for collective action urban space finds differentiation both physically as hierarchiesof accessibility and distribution of built form (a) and semantically through actions performed in buildings and places ((b) (c))

associated with our actions in architectural and urban spacesWe arguably grasp useful information about the activityperformed in a place not only by visual features but alsoinferring what people do there Space not only representsactivity it is actuated and thus loaded with meanings relatedto performance This is a semantic dimension of space spaceldquomeansrdquo as much as our actions precisely because it issemanticized by our actionsThis enacted information is veryimportant Action is temporally related to spatial locations

Second these locations allow us to access potential coun-terparts in interaction As we shall see below knowing acity means that we can find places where certain activitiesare performed Recognizing spatial differences and patternsallows us to infer such locations Apart from other sources ofinformation such as linguistic exchanges we do so becausewecan retrieve environmental information about activities andlocation patterns

Third places are means of connection we can actualizeconnections with certain people or activities during a periodof time These spatial links allow us to create complexes ofinteraction In practice places ldquodrawrdquo courses of action withdifferent orientations and contents and tie them togethermomentarily Flows of convergent and divergent actionshappen all the time when people leave their homes to workto seek services or go about to socialize and do so oftenwithout prior knowledge of activities counterparts or placesthey wish to visit (Figure 2) Urban space becomes whatParsons [51] would call an ldquoaction frame of referencerdquo aconnective fabric which both enables and constrains choicesmaterially and socially

5 Entropy as a Way into Social Organization

In order to be a part of selection the urban environmentneeds to be informational enough and not just to suggest

t3

t2

t1

L1 L2 L3 L4

Figure 2 As a reference system urban space converges courses ofaction in time (119905

1 1199052 and 119905

3) and space (119871

1 1198712 1198713 and 119871

4)

particular options but allow unforeseen changes in the courseof our actions Luhmann suggests that a desired social activitymust be within reach ldquoBecause complete interdependence isunattainable however interdependencies only come aboutby selection [ ] Successfully established interdependenciesthen serve as perspectives for and constraints on the struc-tural selections that connect onto themrdquo [5 page 284] As asocial event this connection is produced through communi-cation But before communication comes into being certainconditions have to be there spatially there Places must bechosen as conjunctive pieces that materialize connections

Complexity 7

Could cities stimulate connections If they could andgiven the fact that they are formed by space and thatspace is not fully malleable is it reasonable to supposethat certain spatial formations could ease such momentaryconnections We know that proximity increases the intensityof interaction [52 53] But what form should urbanized spacetake in order to meet such systemic expectations We mayunderstand such spatial conditions through a counterfactualscenario Imagine a fully chaotic urban formation with norecognizable concentrations or paths Visual elements couldnot convey information as spatial differences would notdevelop into structures emerging out of absolute heterogene-ity References to activities of interest would have to relyonly on our memory Selection of actions would becomedependent on very detailed mental representations of such achaotic formation Inferences about more likely places to findparticular activities would be very hard tomakeWewould beimmersed in a spatial environment without patterns to guideour choices reduce our efforts and amplify our interactionsIn a world of unstructured spaces actions would face lackof information noise and relentless entropy The possibilitythat space may have an informational presence in interactionexpresses the possibility that the physical environment mayhave informational properties and that the world is rich ininformation

Now not only our daily experience denies that cities aresuch completely chaotic structureless places but also spatialeconomics and urban studies offer plenty of theories andevidences that as complex as they may be cities are (at leastpartially) structured places From location patterns describedsince Alonso [54] and accessibility patterns described fromHansen [55] to Hillier [56] we know that urban patternsmatter they imply that activities can be arranged in ways thatare recognizable and accessible And from the point of view ofselection the realization that activities are somehow arrangedalong a spatial structure is a form of preselection Activitiestend to be distributed according to levels of accessibilityland values and practical dependencies facilitating comple-mentary actions Intentionally or not and however subject tocontingencies urban space is also structured (physically andinformationally) which allows possibilities of action to bemore easily known selected and viable In short at first theurban structure affords a range of possibilities for selectionIn a second moment the structure itself suggests gradients ofattractive possibilities

In a view of social systems constituted by connectionsof actions this means that space can be part of the constanttransition from individual acts to complexes of actions (indi-vidual act rarr places as connections rarr action systems) Ourhypothesis here to be tested in computational experimentsin the following section is that cities become a crucial part ofthis connectivity They would reduce risks such as the lack ofinformation or courses of action too costly to emerge

What does this say about the effect of cities on entropyBeginning with Luhmann interactions have structural valuebecause they represent selections from combinatorial possi-bilities If we relate this to Shannonrsquos entropy the very emer-gence of urban structures eliminates spatial scenarios whereevery possible connection would have the same probability

Another counterfactual scenario might help here Entirelyhomogeneous cities (or entirely heterogeneous for thatmatter) free from structure would present even probabilitiesof choice between actions Of course this would be a problemfor social systems connections potentially interesting tomostpeople would be as difficult to find as any If urban spacewas purely homogeneous or purely random the probabilityof finding an activity would be distributed homogeneouslyin space That is a scenario of maximum entropy The spatialenvironment would be useless in stimulating certain sought-after connections which could otherwise be there say whenwe concentrate activities like final supply or input-outputexchanges in production Effort time spent and economiccosts involved in finding activities of interest would rise

On the other hand when urban space finds differentiatedcontents and structures say location patterns along acces-sible paths it naturally creates differences in the probabilityof interaction and decreases in entropy Location patterns arein fact material and informational expressions of interactionsystems The differences engendered by such patterns gener-ate differences in the chances of finding an agent a service orcommerce If a city did not have recognizable internal forma-tions such as a centralities or high streets agents would havemore difficulties in knowingwhere they aremost likely to findwhat they are after A partially structured environment allowsus to build inferences about actions and agencies available in asocial systemWhen these agencies are also related temporallythrough proximity and functional complementarity there isan even greater potential for differences in probability toemerge suggesting new connections and sequences Whatall this means is that cities internally differentiated yet farfrom absolute heterogeneity have the effect of reducing socialentropy

Before experimenting with these relationships in silicolet us attempt an introductory description of what thesedifferences in probability entail in theway social systems han-dle entropy (Figure 3) Linearizing for the sake of simplicityprocesses that in fact occur simultaneously the reductionof social entropy would take the following form (a) In aninitial state free of space think of the actions as lines movingin time Agents can do anything and we cannot foreseewhat they will do a potential state of high entropy Colorsof the lines in Figure 4 represent different informationalcontents in personal orientations (b) Then agents startselecting actions and converging into different positions in aspatial system arranged as places or buildings (representedby colors in the vertical strip) in order to connect withothers Action lines might have subtle differences in relationto the places they approach so they converge into placesby informational proximity Connections can only happenif all other possibilities of connection are excluded (c)As these convergences happen the initially chaotic mazebecomes a momentarily coordinated system where agentscooperate The overall entropy of actions is reduced (d)After each spatially held event actions may change againaccording to changing orientations Accordingly the colorsof lines change Entropy increases as actions go into anothermomentary state of unpredictability as new possibilities arepresented to agents and somemust be selected if actions are to

8 Complexity

Momentarilycoherent systems

State of unpredictabledecisions

Action systems

Reduction of entropy

Coordination of actions

Selectionof actions

High entropy state

Information growth(Hidalgo 2015)

Possibilities of action(Luhmann 1995)

Differences in theprobability ofinteractions

(Shannon 1948)

Agents interact Orientations align inactivity places

Actions mediated byinformation

(Luhmann 1995)

City as informationenvironment(Haken and Portugali 2015)

Figure 3 Recursive process that conducts interaction systems from chaotic states (high entropy) into coherence (low entropy)

(a) (b) (c)

(e) (d)

Figure 4 The cycle of social entropy and space clockwise from the high unpredictability of potential actions to a momentarily orderedsystem as actions converge into places only to dive in entropy again

Complexity 9

Actionagent x

Actionagent y

Activity place

Cooperation

Change insubjectiveorientation

Figure 5 Aligning different actions in the cooperation process

be actualized (e) And then actions ldquoreferrdquo again to meaningsand positions in space starting a new cycle of entropy(Figure 4)

These distinct moments are theoretical of course Inreality reminding Prigogine and Stengers [6 page 124] thisldquomultitude of eventsrdquo takes place simultaneously and mayrender any clearly emergent cycle invisible But that neithermeans that individual cycles do not happen or aggregate inbunches nor they compensate one another like peaks andtroughs in colliding sound waves In fact routinization indaily life leads to rhythms and cycles unfolding from orfolding over the seemingly chaotic maze of actions But evenif actions were far from synchronic the fabric of interactionstill faces the challenge of entropy that is the challenge ofproducing differences in the probability of connections as a wayto ease organization whenever agents feel compelled to createnew interactions and this process is remarkably recursiveWe converge to places to act with others and do so on adaily basis Once engaged in an activity place interactionrequires levels of cooperationWemay bring our own distincttendencies or latent orientations in the form of long-termmemories of how to act along with short-term memoriesof a previous action both shaping our next actions As weact socially we bring our orientations into contact in away to open them in the communication process regulatingcollective action A result of this coordination is to reduceinitial differences Our actions ldquoalignrdquowithin an activity place(Figure 5)

All this suggests that social life involves a continuousalthough unconscious handling of entropy a vital oscillationbetween unpredictability and predictability As this processengages an enormous number of agents and events andtranscends places and contexts and therefore cannot beobserved as a whole entropy can only be fully grasped intheoretical representations A useful way to examine space asan information environment for action is through a compu-tational model a means to examine a counterfactual worldwhere other steering media such as language or social ruleswere ldquoswitched offrdquo keeping only properties of space activealong with agentsrsquo cognitive abilities to align orientationswith activities Once we abstract from other media wemay assess whether space could have any presence in theself-organization of action Let us examine this possibilitythrough some experiments

LBuildingactivity type

Figure 6The theoretical one-dimensional citywith a set of differentactivity places Here there are 119871 buildings distributed in a ringconstituting the city space Colors represent activity types

6 Digital ExperimentsA Unidimensional Model

61 Design Concepts An agent based model (ABM) is pro-posed Agents are defined to perform actions at each time-step and places by corresponding activity types The decisionon which action to perform next may be influenced bythree different conditions namely (i) latent orientation120590in

119894(119905)

representing the tendency of a single agent to act arounda particular type of action initially randomly distributedthis condition remains over time (ii) current action 120590

119894(119905)

performed by an agent as heshe selects an activity placein order to perform a new action this means an influenceof the current action while allowing gradual changes oforientation in time influenced by other agents and activitiesand (iii) types of activity places 120590

1199091015840(119905) where agents perform

supporting specific types of action Agents follow simplerules selecting their activity places closer to their latentorientations and current actions while being influencedby those activities Activity places are also influenced byvisiting agents but they change at a slower rate In shortagents coevolve with their spatial and social environmentThe model simulates different situations under this basicstructure Differences in the weight of these factors overthe next action may lead to quite different levels of socialentropy suggesting new possibilities for understanding thephenomenonAlthough there is a tradition inmodelling dailyactivities including a growing literature using digital loca-tional data (eg [57]) our ABM is not designed to representempirical sequences of actions or actual routines Instead itfocuses on trends that might emerge from the interfaces ofsimplified systems of action information and space

62 Variables and Scale Consider a unidimensional cityformed by agents and places Places form a ring with length(or perimeter) 119871 as shown in Figure 6 Our choice forrepresenting the city by a ring is justified by the minimalsufficient representation of spatial distance as a factor whose

10 Complexity

role in social organization is to be assessed This is allowedby a unidimensional model (eg [58ndash60]) The ring formallows continuous movement across a linear sequence oflocations eliminating centrality factors while taking intoaccount periodic boundary conditions in order to reduceborder effects eliminating the role of topologywhile isolatingthe problem of distance (see [61ndash63]) This stylized cityhouses 119873 agents which in each time-step 119905 select and visita specific place located at the position 119909 within the cityConsider that the position of the 119894th agent at the time 119905 isrepresented by 119909

119894(119905) The position of an agent can assume the

integer values 1 2 119871 according to the place that shehechooses to visit This means that at every time-step eachagent will choose one activity place to perform her nextaction

We considered 120590119894(119905) as the current action of the 119894-th agent

and 120590119909(119905) the activity performed in a place located at the

position 119909 = (1 2 119871) of the city both in time 119905 Wequantify action orientations weighting the three variables(latent orientation current action and activity place) Firstvariables are translated into different numeric values Let usassume that an orientation is an integer number between 0and 1000 each number represents a particular type of actionand numbers in their vicinity mean similar actions Theresulting orientation in time 119905 for each agent is the weightedaverage of those three parametersThedifference |120590

119894(119905)minus120590119909(119905)|

gives us the difference in orientation between an agent andthe activity place At every time-step each agent assesses thedifference between her orientation and every activity placein the city An agent chooses a specific place based on herorientation affinity with this place minimum difference

As we hypothesized above the urban structure has a rolein this selection In this experiment we reduced ldquourban struc-turerdquo to distance In order to investigate how distance may ornot interfere in this selection we propose two scenarios Inthe first one the agent selects her next activity based only onthe similarity in orientations between herself and the activityplace Agents can move across a space free of friction Spaceis not a constraintThis scenario considers only differences inorientation The parameter 119864

119894(119909 119905) estimates the interaction

between agent 119894 and the place located at 119909 at the time 119905 Theagent selects activity locations that minimize this quantityWe propose

119864119894 (119909 119905) =

1003816100381610038161003816120590119894 (119905) minus 1205901199091003816100381610038161003816 (1)

In the second more realistic scenario space imposesfriction to movement The agent takes into account thephysical distance between her current location in the city andthe location of her next activity and opts to minimize thisdistance along with the difference between her orientationand that of the activity place We propose

119864119894 (119909 119905) =

1003816100381610038161003816(119909119894 (119905) minus 119909) (120590119894 (119905) minus 120590119909)1003816100381610038161003816 (2)

Summing up an agent 119894 selects at time 119905 a particularactivity place located at119909 thatminimizes the function119864

119894(119909 119905)

In the first scenario she will choose activity places closerto her orientation In the second she will choose activityplaces closer in both orientation and physical distance This

brings a form of ldquoenergy costrdquo into the factors consideredSpace takes on a function role because of the energy used inldquovisitingrdquo Agents move in this linear city willing to minimizethis energy while searching for social information latentin places close to their changing orientations Changes inentropy levels are derived from this energy function In itsspatial version the model assesses effects of the effort toreduce energy in action minimizing informational differ-ences (between orientations and activity places) and spatialdistances (between agents and activity places)

63TheEvolution of ActionOrientations Orientation evolvesin time according to the following rule When an agentchooses and joins a particular place both agent and placebecome a little closer in terms of orientation This meansthat activity places are not immune to what agents performin them and that the urban activity system also changes intime To illustrate how orientations are constantly updatedconsider that agent 119894 selects an activity place located at 1199091015840Orientations will be updated according the rule

120590119894 (119905 + 1) =

1120572 + 120573 + 120574

(120572120590in119894(119905) + 120573120590119894 (119905) + 1205741205901199091015840 (119905)) (3)

Selection also depends on three additional parameterslatent orientation 120590in

119894(119905) weighted by parameter 120572 current

orientation120590119894(119905)weighted by parameter120573 and type of activity

place 1205901199091015840(119905) weighted by parameter 120574 This means that agents

will consider the three factors previously described (120590in119894(119905)

120590119894(119905) 1205901199091015840(119905)) with different weights (120572 120573 120574) to estimate which

activity she wants to perform in the next time-stepThe latent orientation evolves from an initial randomly

distributed behaviour 1205900119894 We consider that 120590in

119894(119905) is normally

distributed around 1205900119894 meaning that every agent has an

ldquoaveragerdquo latent orientation whose value will be derived ateach time-step from a normal distribution peaking at theagentrsquos average orientation If 120572 is sufficiently small in relationto 120573 and 120574 then the latent orientation does not play a rolein the agentrsquos action However if 120572 is sufficiently large (oncein relation to 120573 and 120574) then the action becomes stronglydependent on the latent orientation This would lead to apretty conservative city where inhabitants are not open tochanges in behaviour

The current action of an agent may influence a newaction with an intensity that depends on parameter 120573 If thisparameter is small in comparison to 120572 and 120574 then the agenthas no memory of an ongoing orientation However if 120573 issufficiently large then the new action is strongly dependenton a previous one (a Markovian behaviour)

The intensity of the activity placersquos influence on the agentrsquosaction depends on the value of 120574 If this parameter is smallin comparison to 120572 and 120573 then the activity place does notaffect her next actionmeaning that the place does not changebehaviours However if this parameter is sufficiently largethen the place plays a strong influence on future actions Theactivity place will be updated according to the rule

1205901199091015840 (119905 + 1) = 1205901199091015840 (119905) + 120579sum

119894isin1199091015840

120590119894 (119905) (4)

Complexity 11

where 120579 is a parameter sufficiently small That is placesare less influenced by agents than the other way roundThis means that at every time-step a place would have itstype of activity closer to the average of orientations of itsvisitors Therefore the main activity of the place will changeslightly according to visiting agents while changing theirorientations

64 Frequency of Orientations and Entropy Levels We cal-culate entropy assessing the distribution of resulting orien-tations of all agents in the social system at each time-stepusing Shannon entropy Our quantitative interpretation ofinformation removes ambiguities from the relation betweenShannon information and semantic information as differentorientations actions and types of activity places are repre-sented by numerical values This allows us to estimate socialentropy through Shannon entropy The entropy of actions iscalculated as follows consider119873(120590 119905) as the number of agentswith orientation 120590 at the time 119905 (note that the total populationis 119873 = sum

120590119873(120590 119905)) We compute the frequency (or density)

120588(120590 119905) of this orientation within a population with

120588 (120590 119905) = 1119873119873 (120590 119905) (5)

Equation (5) calculates the probability of observing anorientation 120590 at time 119905 We compute the entropy level for anydistribution of orientations with

119878 (119905) = minussum120590

120588 (120590 119905) ln (120588 (120590 119905)) (6)

Equation (6) describes how uneven is the probability of find-ing different orientations Higher values mean that differentorientations have almost the same probability to happenwhile lower values indicate a system with clear orientationtrends The reduction of entropy implies that the probabilityof certain actions increases that is actions grow in similarityIn the limit as entropy falls to zero all agents in the systemwould reach the same orientation

65Model Procedure Themodel performs the following pro-cedures

(1) The city is created by generating the following

(a) The type of activity in places that is (120590119909=1(119905 =

0) 120590119909=2(119905 = 0) 120590

119909=119871(119905 = 0)) also generated

from a random distribution This step createsactivity places in the ring-city and assign themrandomly distributed activity values 120590

1199091015840(119905)

(b) The initial position of 119873 agents that is (1199091

1199092 119909

119873) and their respective orientations

(1205901(119905 = 0) 120590

2(119905 = 0) 120590

119873(119905 = 0)) generated

from a random distribution This step createsmoving agents in random locations and assignthem latent orientations based on the initialrandomly distributed orientation 120590in

119894(119905) = 120590

119894(0)

(2) At each time-step 119905 every agent chooses a place toact This place should minimize the function 119864

119894(119909 119905)

Suppose that the chosen place by 119894 is defined as 1199091015840Then agent and place update their orientations 120590

119894and

1205901199091015840 as described above

(3) Update the distribution of orientations 120588(120590 119905)(4) Compute Shannon entropy 119878(119905)

66 Results We have developed a number of observationson the behaviour of social systems working under differentparameters

(i) Entropy reduction is only found in scenarios wherespace imposes friction to movement that is wherethe spatial distance between agents and activity placesis considered as an active factor in selection Entropyrequires an extensive space in order to be reduced Butspace cannot do it alone Surprisingly space is a nec-essary but not sufficient condition Figure 7 comparestwo scenarios In the first one distance is an issue inthe selection of activities (red line)Thedistribution oforientations changes from a homogeneous one at thestart of simulation (a) to a nearly normal distributionat the end of simulation (b) where certain kindsof action are more likely to happen Informationalcontents in extensive space help aligning contents inactions In the second scenario agents move free ofspatial friction (blue line) Orientations are initiallyrandomly distributed (a) and continue to be so at theend of simulation (b) as a similar number of agentsare distributed along different orientationsThis resultsuggests that a materially active space becomes ameans for increasing the probability of certain inter-actions easing the collective coordination of action

Nowassessing the relative influence of social and personalfactors on the selection of a next action namely the socialinformation in activity places latent orientation and the cur-rent action under the influence of spatial friction (Figure 8)we can say the following

(i) There is structure in the relation between selectionfactors and the reduction of entropy Entropy beginsto fall and fluctuate around specific values accordingto different combinations of factors For some com-binations there is a great reduction of entropy Forothers the reduction is minimal almost negligibleFluctuations derive from the random component inthe decisionmaking of the agent (under the influenceof latent orientation)

(ii) A strong latent orientation (120572 sufficiently large) leadsto increasing entropy (blue lines in Figures 8 and9) since agents cannot align their actions with otheragents and activity places Like a long-term memorysystems whose actions are dictated by latent orienta-tions are likely to preserve initial orientations whichwere randomly distributed and therefore homoge-neous This condition is responsible for limiting thereduction of entropy

(iii) Current action alone does not shape new actions inany specific way (green lines in Figures 8 and 9)

12 Complexity

Start of simulation

00000

00005

00010

00015

00020

o

f tot

al ac

tions

250 500 750 10000Type of action

Materially inactive spaceMaterially active space

t = 0

(a)

End of simulation

00000

00005

00010

00015

00020

o

f tot

al ac

tions

250 500 750 10000Type of action

Materially inactive spaceMaterially active space

t = 50

(b)

Figure 7 Histogram of probability distribution of actors performing different types of action at the start (a) and at the end (b) of simulationsin scenarios where distance is not considered (blue) and distance is considered aligning orientations in action coordination (red) Resultsare averaged for 30 runs for 125 parameter combinations

activityplace

currentaction

latentorientation

225

250

275

300

325

Entr

opy

10 20 30 40 500Time

Figure 8 Evolution of entropy colors in a RGB scale refer toparameter weights in decisions on actions in a next time-step andtheir impact on entropy levels in different simulations (under thefriction of distance) In the palette triangle each color representsa specific combination of the three parameters In each vertex aparameter reaches maximum value while the others have minimumvalue Results are averaged for 30 different runs for 125 parametercombinations

It just keeps what is already happening workingas ldquoreinforcement feedbackrdquo for either direction theagent (and the system) is going Current actions aremeans to conserve tendencies in the system

(iv) Activity places play a key role in the reduction ofentropy (red lines in Figures 8 and 9)The social infor-mation in space ldquocontaminatesrdquo agents they aligntheir actions through the social contents of placesWe have seen that space matters as extension Nowwe see that space also has a very active informationalpresence

(v) Different combinations of factors have differenteffects over entropy Figure 8 shows the transitions

between the three parameters and how nuancedweights resulting from blending factors matter inthe reduction of entropy For instance blending theweight of activity places and latent orientation on thenext action displays a strong reduction of entropy(pink lines in Figure 8) That means that when agentsdo not keep information from previous selections(ie 120573 asymp 0) the action system reduces more entropythan in other parameter combinations A strongshort-term memory leads the system into conservingitself and into a poorer capability to coordinateactions Also when current actions 120573 and activityplaces 120574 share a similar weight over the next action(orange lines on top) the social system does notexperience a great reduction of entropy as agents tendto reproduce their actions

(vi) Finally the reduction of entropy implies that theprobability of certain actions and interactionsincreases In practical terms this means morealignments between agents and more connectionsbetween actions (interactions) However if all agentsin the system reached the same orientation andentropy dropped to zero the system would lose inter-nal differentiation Agents would behave in the sameway say in a world with no personal differentiationspecialization or division of labour New orientations(therefore entropy) are necessary if the social systemis to keep differentiated agencies On the otherhand a system with maximum entropy would createdifference in actions to a point where no action coor-dination or interaction would be possible Clearly thissituation cannot be the case A social system requiresbalances neither full entropy nor total predictability

These results also suggest that we are dealing with distinctldquosocial memoriesrdquo a long-term memory is active in what we

Complexity 13

20

25

30En

trop

y

04 0602-latent orientation

04 0602-current action

20

25

30

Entr

opy

20

25

30

Entr

opy

04 0602-activity place

Figure 9 Parameter exploration isolated importance of each parameter in entropy (under the friction of distance) Each vertical set of dotsrepresents one fixed value for the highlighted parameter with the two other parameters assuming every possible combination within thetested range Each dot corresponds to the entropy value when a sufficiently large time-frame is used

call latent orientations Short-term memory is active in theinfluence of a current orientation over a new action In turnthe social information latent in activity places is a result ofsocial arrangements (say as firms a local economy and soon) which find certain stability changing much slower thenactions It is an extension of the social system projected ontourban space stabilizing the system to some extent

7 Cities and Social Interaction Conclusions

What does this approach bring to the state of the art on therelations of social interaction information and space and onsocial entropy and ABM in particular We have seen thatprevious approaches overlooked the coordination of actionas a key empirical and analytical problem Furthermorethey tend to have the thin spatiality of a passive territorialbackground (eg [16 20]) In turn although the conceptof ldquocollective actionrdquo is finally getting attention in urbanstudies [64] the problem of social organization is still largelyunderestimated The problem of entropy has been dealt withsince Wilsonrsquos [65] work on spatial interaction but has notreached the mainstream of the discipline and deals mostlywith urban form (eg [24]) and spatial distributions [66]not with social information and action Regarding agentbased models we are not aware of an approach that dealswith the problem of how cities and space are part of socialorganization In a sense our model is closer to Axelrodrsquos[67] ABM model of dissemination of culture His agentsexchange ldquoculturerdquo through direct contact with neighboursin a cellular automata model whereas our agents are mobileand actively deal with distance and social information inspace recognizing types of activities in places The intensityof exchanges between agents in Axelrodrsquos model is a functionof similarities between them In our model exchanges aremediated by the selection and interaction with space In ourunderstanding our contribution in terms of ABM lies onthe fact that a new and simple model is proposed to dealwith the material and informational dimensions of socialorganization not just as an illustration but as a proof ofconcept space can have causal presence if agents generatepreferences as detailed in a domain that coevolves with them

Now does the simulation model corroborate the theoryintroduced in this paper Our proposition cannot be empir-ically validated at this stage given that it is very hard to

assess peoplersquos choices and the whole panorama of actionsin a city Entropy may well be beyond observation In thesecircumstances simulations of the behaviour of agents underdifferent spatial conditions become useful to assess entitiesevents and entropic forces at play One expected result fromthe model is that space matters for coordinating actions butsimulations surprisingly showed that space is a necessary butnot sufficient condition for the reduction of entropy Neithermaterial cognitive resources nor individual powers of agentsare enough Information in activity places and a bit of agentsrsquopersonal history also play roles in the way social systemsdeal with their own entropy That would mean a causal butnondeterministic presence of space

Substantively this approach explores a subtle but funda-mental presence of cities in social organizationWe argue thaturban space materializes gradients of difference in potentialinteractions from less to more recognizable costly or likelyPlacing simple criteria including distance as a factor in select-ing activities our ABM showed that space becomes a meansfor producing differences in the probabilities of interactionincreasing chances of certain selections and convergences incollective action

In reality people recognize the social information ofactivity places along with uses of busy streets or local central-ities We are able to retrieve these differences from the builtenvironment and relate to them One of the key points of ourargument is that these differences help us make selectionsEvery time we move through the city and select places toperform we participate in the large-scale coordination ofaction We enact space as a ldquoreferential systemrdquo [49 50]a set of semantic bits indexing social activities informingand guiding daily decisions Space becomes inherently partof the conversion of orientations into interactions that isconnections that crisscross sequences of actions recreatingthe interaction system By means of a structure that bothmaterializes and restricts the quasi-endless combinatorial pos-sibilities of interaction this system can acquire sufficientinternal guidance to make its own reproduction possibleBy distributing sufficiently recognizable differences in theprobability of interaction cities express and release localforces of reproduction of social systems The interfaces ofaction cognitive and spatial systems transform entropiccomplexity into structured complexity constantly reorderingactions in time

14 Complexity

This materialistic viewpoint can be integrated with otherviews of social organization Further work may include therelational roles of cultures language social norms and othersteering media along with explorations of possibly similareffects of communication technologies over entropy At thisstage we attempted to integrate a social dimension (agentsengaged in coordination) a spatial dimension (an extensiveenvironment) and an informational dimension (as differ-ences in action and in the environment itself) a demandinginterdisciplinary effort The approach identifies a central rolefor coevolving agents actualizing aims as they coordinate (cf[68]) and a central role for the environment exterior butresponsive to agents leading to cumulative changes alongtheir historyThemodel also identifies roles for cognition andmemory in the ways agents deal with their own orientationsand choose their actions and activity places changing theirspatial environment with consequences in the overall levelsof entropy All this suggests that the behaviour of the modelcannot be reduced to or fully predicted from the behaviour ofindividual variables which is a key aspect of complexity

So one of the main aims of this paper was to assesswhether space as information environment had any effecton the coordination and entropy of interaction However ifphysical distance and social information in activity placesmatter what about city size and the internal spatial struc-tures of cities What about rural areas or rarefied suburbsSince our model explores an abstract circular city it onlybegins to answer such questions Our findings indicate thatshorter distances between activity places tend to reducesocial entropy Density seems to matter denser spatialities(as opposed to rarefied ones) have a role to play A pathfor further development here is Jacobsrsquos [69] idea that thediversity of activities generates positive externalities Anddiversity as spatial economics shows us has to do withthe size of cities and density of population (eg [70 71])Entropy may also be influenced by diversity Batty et al [66]have shown that information increases as cities get biggerThe remaining question is whether larger denser and moreinternally structured cities could create process and reduceentropy with more intensity than smaller cities transformingbig pools of activities into differences in the probability ofinteraction Although we proposed a unidimensional modelour agentsrsquo behaviour can be explored in more realisticrepresentations of cities testing the roles of density topologyand diversity while going beyond perfectly informed agentsBy proposing the ring model we intended to simplify acity to a minimum system where the spatial dimensionof social organization becomes intelligible without losingfundamental properties

Shouldwe conclude fromall this that entropy is a problemfor social systems Our approach suggests that it is notEntropy is a necessary force It means that novel orientationsare entering the interaction system and must be dealt withEntropy only becomes a problem if not converted into orga-nization only if numerous potential actions are not convertedinto smaller sets of actual interactions The effects of newactions over entropy would be positive once actualizedhaving to dowith the diversity of agencyConservative systemsare likely to face less entropy but also likely to produce

less novelty having homogeneity as a problematic horizonFluctuations of entropy seem vital to complex societies

Finally our intention was to explore entropy as a meansto think about the conditions of social organization ldquofromanother anglerdquo so to speak from the viewpoint of challengesinvolved in social reproduction Hence the approach focusedon the idea of uncertainty surrounding agents dealing withaims and decisions and the different probabilities of inter-action that follow an attempt to bring social cognitiveinformational and spatial theories under a single roof Ourapproach sees cities as connective systems produced to createthe delicate fabric of interaction that keeps large numbersof agents coherently living together It also suggests that thevarying states of entropy reveal deep connections betweensocial informational and spatial systems

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

Theauthors would like to thank the Ecole Polytechnique Fed-erale de Lausanne (EPFL) CAPES FAPEMIG FAPESP andCNPq for financial support Maıra Pinheiro Henrique Loreaand colleagues at ABMUS2017 for earlier discussions andMike Batty for exchanges and inspiration Finally the authorsthank Romulo Krafta maverick of the complexity studiesof cities in Brazil for his input and support throughout theyears This work is dedicated to him

References

[1] C W Mills The Sociological Imagination Oxford UniversityPress New York NY USA 1959

[2] M Castells ldquoThe information age economy society and cul-turerdquo inThe Rise of the Network Blackwell London 1996

[3] C Hidalgo Why Information Grows The Evolution of Orderfrom Atoms to Economies Basic Books New York NY USA2015

[4] D Helbing The Automation of Society Is Next ETH ZurichSwitzerland 2016

[5] N Luhmann Social Systems University Press Stanford 1995[6] I Prigogine and I Stengers Order out of Chaos Manrsquos New

Dialogue with Nature Bantam Books 1984[7] C E Shannon ldquoAmathematical theory of communicationrdquo Bell

System Technical Journal vol 27 no 4 pp 623ndash656 1948[8] W Buckley Sociology and Modern Systems Theory Prentice-

Hall Englewood Cliffs NJ USA 1967[9] D D McFarland ldquoMeasuring the Permeability of Occupational

Structures An Information-Theoretic Approachrdquo AmericanJournal of Sociology vol 75 no 1 pp 41ndash61 1969

[10] F Charvat ldquoOn philosophical aspects of the system conceptionin contemporary sociological knowledgerdquo Quality and Quan-tity vol 6 no 1 pp 3ndash16 1972

[11] F Charvat J Kucera and M Soukup ldquoToward the systemtheory of dependence further general theoretical remarksrdquoQuality and Quantity vol 7 no 1 pp 69ndash90 1973

Complexity 15

[12] P M Horan ldquoInformation-theoretic measures and the analysisof social structuresrdquo Sociological Methods and Research vol 3no 3 pp 321ndash340 1975

[13] O E Klapp ldquoOpening and closing in open systemsrdquo BehavioralScience vol 20 no 4 pp 251ndash257 1975

[14] J Galtung ldquoEntropy and the general theory of peacerdquo in Essaysin Peace Research J Galtung Ed vol L Humanities PressAtlantic Highlands NJ USA 1975

[15] T Parsons The System of Modern Societies Englewood CliffsNew Jersey NJ USA 1972

[16] K D Bailey ldquoSociological entropy theory Toward a statisticaland verbal congruencerdquo Quality amp Quantity vol 18 no 1 pp113ndash133 1983

[17] K D Bailey Social EntropyTheory State University of NewYorkPress Albany NY USA 1990

[18] J G Miller Living Systems McGraw-Hill New York NY USA1978

[19] K D Bailey Sociology and the New Systems Theory Toward aTheoretical Synthesis State University of New York PressAlbany NY USA 1994

[20] K D Bailey ldquoSociocybernetics and social entropy theoryrdquoKybernetes vol 35 no 3-4 pp 375ndash384 2006

[21] F Geyer ldquoThe challenge of sociocyberneticsrdquo Kybernetes vol24 no 4 pp 6ndash32 1995

[22] F Geyer ldquoThe march of self-referencerdquo Kybernetes vol 31 no7-8 pp 1021ndash1042 2002

[23] J HabermasTheTheory of Communicative Action Polity PressCambridge UK 1987

[24] H Haken and J Portugali Information Adaptation The Inter-play between Shannon Information and Semantic Information inCognition Springer New York NY USA 2015

[25] M Weber Economy and Society vol 1 Berkeley University ofCalifornia Press 1978

[26] L Wittgenstein Philosophical Investigations Blackwell Publish-ers Oxford London 3rd edition 2001

[27] L Vygotsky Mind in Society The Development of HigherPsychological Processes Harvard University Press CambridgeMA 1978

[28] M Wilson ldquoSix Views of Embodied Cognitionrdquo PsychonomicBulletin amp Review vol 9 no 4 pp 625ndash636 2002

[29] A Clark and D Chalmers ldquoThe extended mindrdquo Analysis vol58 no 1 pp 7ndash19 1998

[30] R PassiniWayfinding in Architecture Van Nostrand ReinholdNew York NY USA 1992

[31] E Rosch ldquoPrinciples of categorizationrdquo in Cognition and Cate-gorization E Rosch and B Lloyd Eds Hillsdale LawrenceErlbaum Ass 1978

[32] W KintschMemory and Cognition John Wiley and Sons NewYork NY USA 1970

[33] U Neisser ldquoMultiple Systems a New Approach to CognitiveTheoryrdquo European Journal of Cognitive Psychology vol 6 no 3pp 225ndash241 1994

[34] A Faria and R Krafta ldquoRepresenting urban cognitive structurethrough spatial differentiationrdquo in Proceedings of 4th Space Syn-tax International Symposium pp 531-518 UCL Press London2003

[35] J Portugali Complexity Cognition and the City Springer NewYork NY USA 2011

[36] D Kirsh and P Maglio ldquoOn distinguishing epistemic frompragmatic actionrdquo Cognitive Science vol 18 no 4 pp 513ndash5491994

[37] S M Kosslyn Image and Brain The Resolution of the ImageryDebate MIT Press Cambridge MA

[38] R Brooks Cambrian Intelligence The Early History of the NewAI MIT Press Cambridge MA 1999

[39] A Clark BeinGThere Putting Brain Body AndWorld TogetherAgain MIT Press Cambridge MA 1997

[40] R Pfeifer and C ScheierUnderstanding Intelligence MIT PressCambridge MA 1999

[41] A M Glenberg and D A Robertson ldquoIndexical understandingof instructionsrdquoDiscourse Processes vol 28 no 1 pp 1ndash26 1999

[42] JM Iverson and SGoldin-Meadow ldquoWhypeople gesturewhenthey speakrdquo Nature vol 396 no 6708 p 228 1998

[43] R M Krauss ldquoWhy do we gesture when we speakrdquo CurrentDirections in Psychological Science vol 7 no 2 pp 54ndash60 1998

[44] S Franklin Artificial Minds MIT Press Cambridge MA 1995[45] J Gibson The Ecological Approach to Visual Perception

Houghton-Mifflin Boston 1979[46] G Lakoff andM JohnsonPhilosophy in the FleshTheEmbodied

Mind and Its Challenge to Western Thought Basic Books NewYork NY USA 1999

[47] E HutchinsCognition in theWild MIT Press CambridgeMA1995

[48] E Hutchins ldquoThe cultural ecosystem of human cognitionrdquoPhilosophical Psychology vol 27 no 1 pp 34ndash49 2014

[49] V M Netto ldquoPractice space and the duality of meaningrdquoEnvironment and Planning D Society and Space vol 26 no 2pp 359ndash379 2008

[50] V M Netto The Social Fabric of Cities Routledge New YorkNY USA 2017

[51] T Parsons The Structure of Social Action The Free Press NewYork NY USA 1968

[52] T Allen Managing the Flow of Technology MIT Press Cam-bridge MA 1977

[53] F L Ribeiro J Meirelles F F Ferreira and R C Neto ldquoAmodel of urban scaling laws based on distance-dependent inter-actionsrdquo Royal Society Open Science vol 4 Article ID 1609262017

[54] W Alonso Location and Land Use Toward a General Theory ofLand Rent Harvard University Press Cambridge MA 1964

[55] WGHansen ldquoHow accessibility shapes land userdquo Journal of theAmerican Institute of Planners vol 25 no 2 pp 73ndash76 1959

[56] B Hillier Space Is the Machine Cambridge University PressCambridge UK 1996

[57] S Farber M OrsquoKelly H J Miller and T Neutens ldquoMeasuringsegregation using patterns of daily travel behavior a socialinteraction based model of exposurerdquo Journal of TransportGeography vol 49 pp 26ndash38 2015

[58] T C Schelling ldquoDynamic models of segregationrdquo Journal ofMathematical Sociology vol 1 no 2 pp 143ndash186 1971

[59] R Krafta V M Netto and L Lima ldquoUrban built form growscriticalrdquo CyberGeo vol 2011 article 565 2011

[60] N Lanchier and S Scarlatos ldquoFixation in the one-dimensionalAxelrod modelrdquo The Annals of Applied Probability vol 23 no6 pp 2538ndash2559 2013

[61] F L Ribeiro and K N Ribeiro ldquoA one dimensional modelof population growthrdquo Physica A Statistical Mechanics and itsApplications vol 434 pp 201ndash210 2015

[62] F L Ribeiro ldquoA non-phenomenological model of competitionand cooperation to explain population growth behaviorsrdquo Bul-letin of Mathematical Biology vol 77 no 3 pp 409ndash433 2015

16 Complexity

[63] R V dos Santos F L Ribeiro and A S Martinez ldquoModels forAllee effect based on physical principlesrdquo Journal of TheoreticalBiology vol 385 pp 143ndash152 2015

[64] M BattyThe New Science of Cities The MIT Press CambridgeMA 2013

[65] A G Wilson Entropy in Urban and Regional Modelling PionLondon 1970

[66] M Batty R Morphet P Masucci and K Stanilov ldquoEntropycomplexity and spatial informationrdquo Journal of GeographicalSystems vol 16 no 4 pp 363ndash385 2014

[67] R Axelrod ldquoThe dissemination of culture a model with localconvergence and global polarizationrdquo Journal of Conflict Reso-lution vol 41 no 2 pp 203ndash226 1997

[68] B H Hodges and RM Baron ldquoValues as constraints on afford-ances perceiving and acting properlyrdquo Journal for the Theory ofSocial Behaviour vol 22 no 3 pp 263ndash294 1992

[69] J JacobsThe Economy of Cities Random house New York NYUSA 1969

[70] E Glaeser H Kallal J Scheinkman and A Shleifer ldquoGrowthin Citiesrdquo Journal of Political Economy vol 100 no 6 pp 1126ndash1152 1992

[71] S Rosenthal and W Strange ldquoEvidence on the nature andsources of agglomeration economiesrdquo in Handbook of Urbanand Regional Economics J V Henderson and J-F Thisse Edsvol 4 pp 2119-2171 NorthHollandNewYork NYUSA 2004

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Complexity

(a) (b) (c)

Figure 1 Differential space like a code of instructions for collective action urban space finds differentiation both physically as hierarchiesof accessibility and distribution of built form (a) and semantically through actions performed in buildings and places ((b) (c))

associated with our actions in architectural and urban spacesWe arguably grasp useful information about the activityperformed in a place not only by visual features but alsoinferring what people do there Space not only representsactivity it is actuated and thus loaded with meanings relatedto performance This is a semantic dimension of space spaceldquomeansrdquo as much as our actions precisely because it issemanticized by our actionsThis enacted information is veryimportant Action is temporally related to spatial locations

Second these locations allow us to access potential coun-terparts in interaction As we shall see below knowing acity means that we can find places where certain activitiesare performed Recognizing spatial differences and patternsallows us to infer such locations Apart from other sources ofinformation such as linguistic exchanges we do so becausewecan retrieve environmental information about activities andlocation patterns

Third places are means of connection we can actualizeconnections with certain people or activities during a periodof time These spatial links allow us to create complexes ofinteraction In practice places ldquodrawrdquo courses of action withdifferent orientations and contents and tie them togethermomentarily Flows of convergent and divergent actionshappen all the time when people leave their homes to workto seek services or go about to socialize and do so oftenwithout prior knowledge of activities counterparts or placesthey wish to visit (Figure 2) Urban space becomes whatParsons [51] would call an ldquoaction frame of referencerdquo aconnective fabric which both enables and constrains choicesmaterially and socially

5 Entropy as a Way into Social Organization

In order to be a part of selection the urban environmentneeds to be informational enough and not just to suggest

t3

t2

t1

L1 L2 L3 L4

Figure 2 As a reference system urban space converges courses ofaction in time (119905

1 1199052 and 119905

3) and space (119871

1 1198712 1198713 and 119871

4)

particular options but allow unforeseen changes in the courseof our actions Luhmann suggests that a desired social activitymust be within reach ldquoBecause complete interdependence isunattainable however interdependencies only come aboutby selection [ ] Successfully established interdependenciesthen serve as perspectives for and constraints on the struc-tural selections that connect onto themrdquo [5 page 284] As asocial event this connection is produced through communi-cation But before communication comes into being certainconditions have to be there spatially there Places must bechosen as conjunctive pieces that materialize connections

Complexity 7

Could cities stimulate connections If they could andgiven the fact that they are formed by space and thatspace is not fully malleable is it reasonable to supposethat certain spatial formations could ease such momentaryconnections We know that proximity increases the intensityof interaction [52 53] But what form should urbanized spacetake in order to meet such systemic expectations We mayunderstand such spatial conditions through a counterfactualscenario Imagine a fully chaotic urban formation with norecognizable concentrations or paths Visual elements couldnot convey information as spatial differences would notdevelop into structures emerging out of absolute heterogene-ity References to activities of interest would have to relyonly on our memory Selection of actions would becomedependent on very detailed mental representations of such achaotic formation Inferences about more likely places to findparticular activities would be very hard tomakeWewould beimmersed in a spatial environment without patterns to guideour choices reduce our efforts and amplify our interactionsIn a world of unstructured spaces actions would face lackof information noise and relentless entropy The possibilitythat space may have an informational presence in interactionexpresses the possibility that the physical environment mayhave informational properties and that the world is rich ininformation

Now not only our daily experience denies that cities aresuch completely chaotic structureless places but also spatialeconomics and urban studies offer plenty of theories andevidences that as complex as they may be cities are (at leastpartially) structured places From location patterns describedsince Alonso [54] and accessibility patterns described fromHansen [55] to Hillier [56] we know that urban patternsmatter they imply that activities can be arranged in ways thatare recognizable and accessible And from the point of view ofselection the realization that activities are somehow arrangedalong a spatial structure is a form of preselection Activitiestend to be distributed according to levels of accessibilityland values and practical dependencies facilitating comple-mentary actions Intentionally or not and however subject tocontingencies urban space is also structured (physically andinformationally) which allows possibilities of action to bemore easily known selected and viable In short at first theurban structure affords a range of possibilities for selectionIn a second moment the structure itself suggests gradients ofattractive possibilities

In a view of social systems constituted by connectionsof actions this means that space can be part of the constanttransition from individual acts to complexes of actions (indi-vidual act rarr places as connections rarr action systems) Ourhypothesis here to be tested in computational experimentsin the following section is that cities become a crucial part ofthis connectivity They would reduce risks such as the lack ofinformation or courses of action too costly to emerge

What does this say about the effect of cities on entropyBeginning with Luhmann interactions have structural valuebecause they represent selections from combinatorial possi-bilities If we relate this to Shannonrsquos entropy the very emer-gence of urban structures eliminates spatial scenarios whereevery possible connection would have the same probability

Another counterfactual scenario might help here Entirelyhomogeneous cities (or entirely heterogeneous for thatmatter) free from structure would present even probabilitiesof choice between actions Of course this would be a problemfor social systems connections potentially interesting tomostpeople would be as difficult to find as any If urban spacewas purely homogeneous or purely random the probabilityof finding an activity would be distributed homogeneouslyin space That is a scenario of maximum entropy The spatialenvironment would be useless in stimulating certain sought-after connections which could otherwise be there say whenwe concentrate activities like final supply or input-outputexchanges in production Effort time spent and economiccosts involved in finding activities of interest would rise

On the other hand when urban space finds differentiatedcontents and structures say location patterns along acces-sible paths it naturally creates differences in the probabilityof interaction and decreases in entropy Location patterns arein fact material and informational expressions of interactionsystems The differences engendered by such patterns gener-ate differences in the chances of finding an agent a service orcommerce If a city did not have recognizable internal forma-tions such as a centralities or high streets agents would havemore difficulties in knowingwhere they aremost likely to findwhat they are after A partially structured environment allowsus to build inferences about actions and agencies available in asocial systemWhen these agencies are also related temporallythrough proximity and functional complementarity there isan even greater potential for differences in probability toemerge suggesting new connections and sequences Whatall this means is that cities internally differentiated yet farfrom absolute heterogeneity have the effect of reducing socialentropy

Before experimenting with these relationships in silicolet us attempt an introductory description of what thesedifferences in probability entail in theway social systems han-dle entropy (Figure 3) Linearizing for the sake of simplicityprocesses that in fact occur simultaneously the reductionof social entropy would take the following form (a) In aninitial state free of space think of the actions as lines movingin time Agents can do anything and we cannot foreseewhat they will do a potential state of high entropy Colorsof the lines in Figure 4 represent different informationalcontents in personal orientations (b) Then agents startselecting actions and converging into different positions in aspatial system arranged as places or buildings (representedby colors in the vertical strip) in order to connect withothers Action lines might have subtle differences in relationto the places they approach so they converge into placesby informational proximity Connections can only happenif all other possibilities of connection are excluded (c)As these convergences happen the initially chaotic mazebecomes a momentarily coordinated system where agentscooperate The overall entropy of actions is reduced (d)After each spatially held event actions may change againaccording to changing orientations Accordingly the colorsof lines change Entropy increases as actions go into anothermomentary state of unpredictability as new possibilities arepresented to agents and somemust be selected if actions are to

8 Complexity

Momentarilycoherent systems

State of unpredictabledecisions

Action systems

Reduction of entropy

Coordination of actions

Selectionof actions

High entropy state

Information growth(Hidalgo 2015)

Possibilities of action(Luhmann 1995)

Differences in theprobability ofinteractions

(Shannon 1948)

Agents interact Orientations align inactivity places

Actions mediated byinformation

(Luhmann 1995)

City as informationenvironment(Haken and Portugali 2015)

Figure 3 Recursive process that conducts interaction systems from chaotic states (high entropy) into coherence (low entropy)

(a) (b) (c)

(e) (d)

Figure 4 The cycle of social entropy and space clockwise from the high unpredictability of potential actions to a momentarily orderedsystem as actions converge into places only to dive in entropy again

Complexity 9

Actionagent x

Actionagent y

Activity place

Cooperation

Change insubjectiveorientation

Figure 5 Aligning different actions in the cooperation process

be actualized (e) And then actions ldquoreferrdquo again to meaningsand positions in space starting a new cycle of entropy(Figure 4)

These distinct moments are theoretical of course Inreality reminding Prigogine and Stengers [6 page 124] thisldquomultitude of eventsrdquo takes place simultaneously and mayrender any clearly emergent cycle invisible But that neithermeans that individual cycles do not happen or aggregate inbunches nor they compensate one another like peaks andtroughs in colliding sound waves In fact routinization indaily life leads to rhythms and cycles unfolding from orfolding over the seemingly chaotic maze of actions But evenif actions were far from synchronic the fabric of interactionstill faces the challenge of entropy that is the challenge ofproducing differences in the probability of connections as a wayto ease organization whenever agents feel compelled to createnew interactions and this process is remarkably recursiveWe converge to places to act with others and do so on adaily basis Once engaged in an activity place interactionrequires levels of cooperationWemay bring our own distincttendencies or latent orientations in the form of long-termmemories of how to act along with short-term memoriesof a previous action both shaping our next actions As weact socially we bring our orientations into contact in away to open them in the communication process regulatingcollective action A result of this coordination is to reduceinitial differences Our actions ldquoalignrdquowithin an activity place(Figure 5)

All this suggests that social life involves a continuousalthough unconscious handling of entropy a vital oscillationbetween unpredictability and predictability As this processengages an enormous number of agents and events andtranscends places and contexts and therefore cannot beobserved as a whole entropy can only be fully grasped intheoretical representations A useful way to examine space asan information environment for action is through a compu-tational model a means to examine a counterfactual worldwhere other steering media such as language or social ruleswere ldquoswitched offrdquo keeping only properties of space activealong with agentsrsquo cognitive abilities to align orientationswith activities Once we abstract from other media wemay assess whether space could have any presence in theself-organization of action Let us examine this possibilitythrough some experiments

LBuildingactivity type

Figure 6The theoretical one-dimensional citywith a set of differentactivity places Here there are 119871 buildings distributed in a ringconstituting the city space Colors represent activity types

6 Digital ExperimentsA Unidimensional Model

61 Design Concepts An agent based model (ABM) is pro-posed Agents are defined to perform actions at each time-step and places by corresponding activity types The decisionon which action to perform next may be influenced bythree different conditions namely (i) latent orientation120590in

119894(119905)

representing the tendency of a single agent to act arounda particular type of action initially randomly distributedthis condition remains over time (ii) current action 120590

119894(119905)

performed by an agent as heshe selects an activity placein order to perform a new action this means an influenceof the current action while allowing gradual changes oforientation in time influenced by other agents and activitiesand (iii) types of activity places 120590

1199091015840(119905) where agents perform

supporting specific types of action Agents follow simplerules selecting their activity places closer to their latentorientations and current actions while being influencedby those activities Activity places are also influenced byvisiting agents but they change at a slower rate In shortagents coevolve with their spatial and social environmentThe model simulates different situations under this basicstructure Differences in the weight of these factors overthe next action may lead to quite different levels of socialentropy suggesting new possibilities for understanding thephenomenonAlthough there is a tradition inmodelling dailyactivities including a growing literature using digital loca-tional data (eg [57]) our ABM is not designed to representempirical sequences of actions or actual routines Instead itfocuses on trends that might emerge from the interfaces ofsimplified systems of action information and space

62 Variables and Scale Consider a unidimensional cityformed by agents and places Places form a ring with length(or perimeter) 119871 as shown in Figure 6 Our choice forrepresenting the city by a ring is justified by the minimalsufficient representation of spatial distance as a factor whose

10 Complexity

role in social organization is to be assessed This is allowedby a unidimensional model (eg [58ndash60]) The ring formallows continuous movement across a linear sequence oflocations eliminating centrality factors while taking intoaccount periodic boundary conditions in order to reduceborder effects eliminating the role of topologywhile isolatingthe problem of distance (see [61ndash63]) This stylized cityhouses 119873 agents which in each time-step 119905 select and visita specific place located at the position 119909 within the cityConsider that the position of the 119894th agent at the time 119905 isrepresented by 119909

119894(119905) The position of an agent can assume the

integer values 1 2 119871 according to the place that shehechooses to visit This means that at every time-step eachagent will choose one activity place to perform her nextaction

We considered 120590119894(119905) as the current action of the 119894-th agent

and 120590119909(119905) the activity performed in a place located at the

position 119909 = (1 2 119871) of the city both in time 119905 Wequantify action orientations weighting the three variables(latent orientation current action and activity place) Firstvariables are translated into different numeric values Let usassume that an orientation is an integer number between 0and 1000 each number represents a particular type of actionand numbers in their vicinity mean similar actions Theresulting orientation in time 119905 for each agent is the weightedaverage of those three parametersThedifference |120590

119894(119905)minus120590119909(119905)|

gives us the difference in orientation between an agent andthe activity place At every time-step each agent assesses thedifference between her orientation and every activity placein the city An agent chooses a specific place based on herorientation affinity with this place minimum difference

As we hypothesized above the urban structure has a rolein this selection In this experiment we reduced ldquourban struc-turerdquo to distance In order to investigate how distance may ornot interfere in this selection we propose two scenarios Inthe first one the agent selects her next activity based only onthe similarity in orientations between herself and the activityplace Agents can move across a space free of friction Spaceis not a constraintThis scenario considers only differences inorientation The parameter 119864

119894(119909 119905) estimates the interaction

between agent 119894 and the place located at 119909 at the time 119905 Theagent selects activity locations that minimize this quantityWe propose

119864119894 (119909 119905) =

1003816100381610038161003816120590119894 (119905) minus 1205901199091003816100381610038161003816 (1)

In the second more realistic scenario space imposesfriction to movement The agent takes into account thephysical distance between her current location in the city andthe location of her next activity and opts to minimize thisdistance along with the difference between her orientationand that of the activity place We propose

119864119894 (119909 119905) =

1003816100381610038161003816(119909119894 (119905) minus 119909) (120590119894 (119905) minus 120590119909)1003816100381610038161003816 (2)

Summing up an agent 119894 selects at time 119905 a particularactivity place located at119909 thatminimizes the function119864

119894(119909 119905)

In the first scenario she will choose activity places closerto her orientation In the second she will choose activityplaces closer in both orientation and physical distance This

brings a form of ldquoenergy costrdquo into the factors consideredSpace takes on a function role because of the energy used inldquovisitingrdquo Agents move in this linear city willing to minimizethis energy while searching for social information latentin places close to their changing orientations Changes inentropy levels are derived from this energy function In itsspatial version the model assesses effects of the effort toreduce energy in action minimizing informational differ-ences (between orientations and activity places) and spatialdistances (between agents and activity places)

63TheEvolution of ActionOrientations Orientation evolvesin time according to the following rule When an agentchooses and joins a particular place both agent and placebecome a little closer in terms of orientation This meansthat activity places are not immune to what agents performin them and that the urban activity system also changes intime To illustrate how orientations are constantly updatedconsider that agent 119894 selects an activity place located at 1199091015840Orientations will be updated according the rule

120590119894 (119905 + 1) =

1120572 + 120573 + 120574

(120572120590in119894(119905) + 120573120590119894 (119905) + 1205741205901199091015840 (119905)) (3)

Selection also depends on three additional parameterslatent orientation 120590in

119894(119905) weighted by parameter 120572 current

orientation120590119894(119905)weighted by parameter120573 and type of activity

place 1205901199091015840(119905) weighted by parameter 120574 This means that agents

will consider the three factors previously described (120590in119894(119905)

120590119894(119905) 1205901199091015840(119905)) with different weights (120572 120573 120574) to estimate which

activity she wants to perform in the next time-stepThe latent orientation evolves from an initial randomly

distributed behaviour 1205900119894 We consider that 120590in

119894(119905) is normally

distributed around 1205900119894 meaning that every agent has an

ldquoaveragerdquo latent orientation whose value will be derived ateach time-step from a normal distribution peaking at theagentrsquos average orientation If 120572 is sufficiently small in relationto 120573 and 120574 then the latent orientation does not play a rolein the agentrsquos action However if 120572 is sufficiently large (oncein relation to 120573 and 120574) then the action becomes stronglydependent on the latent orientation This would lead to apretty conservative city where inhabitants are not open tochanges in behaviour

The current action of an agent may influence a newaction with an intensity that depends on parameter 120573 If thisparameter is small in comparison to 120572 and 120574 then the agenthas no memory of an ongoing orientation However if 120573 issufficiently large then the new action is strongly dependenton a previous one (a Markovian behaviour)

The intensity of the activity placersquos influence on the agentrsquosaction depends on the value of 120574 If this parameter is smallin comparison to 120572 and 120573 then the activity place does notaffect her next actionmeaning that the place does not changebehaviours However if this parameter is sufficiently largethen the place plays a strong influence on future actions Theactivity place will be updated according to the rule

1205901199091015840 (119905 + 1) = 1205901199091015840 (119905) + 120579sum

119894isin1199091015840

120590119894 (119905) (4)

Complexity 11

where 120579 is a parameter sufficiently small That is placesare less influenced by agents than the other way roundThis means that at every time-step a place would have itstype of activity closer to the average of orientations of itsvisitors Therefore the main activity of the place will changeslightly according to visiting agents while changing theirorientations

64 Frequency of Orientations and Entropy Levels We cal-culate entropy assessing the distribution of resulting orien-tations of all agents in the social system at each time-stepusing Shannon entropy Our quantitative interpretation ofinformation removes ambiguities from the relation betweenShannon information and semantic information as differentorientations actions and types of activity places are repre-sented by numerical values This allows us to estimate socialentropy through Shannon entropy The entropy of actions iscalculated as follows consider119873(120590 119905) as the number of agentswith orientation 120590 at the time 119905 (note that the total populationis 119873 = sum

120590119873(120590 119905)) We compute the frequency (or density)

120588(120590 119905) of this orientation within a population with

120588 (120590 119905) = 1119873119873 (120590 119905) (5)

Equation (5) calculates the probability of observing anorientation 120590 at time 119905 We compute the entropy level for anydistribution of orientations with

119878 (119905) = minussum120590

120588 (120590 119905) ln (120588 (120590 119905)) (6)

Equation (6) describes how uneven is the probability of find-ing different orientations Higher values mean that differentorientations have almost the same probability to happenwhile lower values indicate a system with clear orientationtrends The reduction of entropy implies that the probabilityof certain actions increases that is actions grow in similarityIn the limit as entropy falls to zero all agents in the systemwould reach the same orientation

65Model Procedure Themodel performs the following pro-cedures

(1) The city is created by generating the following

(a) The type of activity in places that is (120590119909=1(119905 =

0) 120590119909=2(119905 = 0) 120590

119909=119871(119905 = 0)) also generated

from a random distribution This step createsactivity places in the ring-city and assign themrandomly distributed activity values 120590

1199091015840(119905)

(b) The initial position of 119873 agents that is (1199091

1199092 119909

119873) and their respective orientations

(1205901(119905 = 0) 120590

2(119905 = 0) 120590

119873(119905 = 0)) generated

from a random distribution This step createsmoving agents in random locations and assignthem latent orientations based on the initialrandomly distributed orientation 120590in

119894(119905) = 120590

119894(0)

(2) At each time-step 119905 every agent chooses a place toact This place should minimize the function 119864

119894(119909 119905)

Suppose that the chosen place by 119894 is defined as 1199091015840Then agent and place update their orientations 120590

119894and

1205901199091015840 as described above

(3) Update the distribution of orientations 120588(120590 119905)(4) Compute Shannon entropy 119878(119905)

66 Results We have developed a number of observationson the behaviour of social systems working under differentparameters

(i) Entropy reduction is only found in scenarios wherespace imposes friction to movement that is wherethe spatial distance between agents and activity placesis considered as an active factor in selection Entropyrequires an extensive space in order to be reduced Butspace cannot do it alone Surprisingly space is a nec-essary but not sufficient condition Figure 7 comparestwo scenarios In the first one distance is an issue inthe selection of activities (red line)Thedistribution oforientations changes from a homogeneous one at thestart of simulation (a) to a nearly normal distributionat the end of simulation (b) where certain kindsof action are more likely to happen Informationalcontents in extensive space help aligning contents inactions In the second scenario agents move free ofspatial friction (blue line) Orientations are initiallyrandomly distributed (a) and continue to be so at theend of simulation (b) as a similar number of agentsare distributed along different orientationsThis resultsuggests that a materially active space becomes ameans for increasing the probability of certain inter-actions easing the collective coordination of action

Nowassessing the relative influence of social and personalfactors on the selection of a next action namely the socialinformation in activity places latent orientation and the cur-rent action under the influence of spatial friction (Figure 8)we can say the following

(i) There is structure in the relation between selectionfactors and the reduction of entropy Entropy beginsto fall and fluctuate around specific values accordingto different combinations of factors For some com-binations there is a great reduction of entropy Forothers the reduction is minimal almost negligibleFluctuations derive from the random component inthe decisionmaking of the agent (under the influenceof latent orientation)

(ii) A strong latent orientation (120572 sufficiently large) leadsto increasing entropy (blue lines in Figures 8 and9) since agents cannot align their actions with otheragents and activity places Like a long-term memorysystems whose actions are dictated by latent orienta-tions are likely to preserve initial orientations whichwere randomly distributed and therefore homoge-neous This condition is responsible for limiting thereduction of entropy

(iii) Current action alone does not shape new actions inany specific way (green lines in Figures 8 and 9)

12 Complexity

Start of simulation

00000

00005

00010

00015

00020

o

f tot

al ac

tions

250 500 750 10000Type of action

Materially inactive spaceMaterially active space

t = 0

(a)

End of simulation

00000

00005

00010

00015

00020

o

f tot

al ac

tions

250 500 750 10000Type of action

Materially inactive spaceMaterially active space

t = 50

(b)

Figure 7 Histogram of probability distribution of actors performing different types of action at the start (a) and at the end (b) of simulationsin scenarios where distance is not considered (blue) and distance is considered aligning orientations in action coordination (red) Resultsare averaged for 30 runs for 125 parameter combinations

activityplace

currentaction

latentorientation

225

250

275

300

325

Entr

opy

10 20 30 40 500Time

Figure 8 Evolution of entropy colors in a RGB scale refer toparameter weights in decisions on actions in a next time-step andtheir impact on entropy levels in different simulations (under thefriction of distance) In the palette triangle each color representsa specific combination of the three parameters In each vertex aparameter reaches maximum value while the others have minimumvalue Results are averaged for 30 different runs for 125 parametercombinations

It just keeps what is already happening workingas ldquoreinforcement feedbackrdquo for either direction theagent (and the system) is going Current actions aremeans to conserve tendencies in the system

(iv) Activity places play a key role in the reduction ofentropy (red lines in Figures 8 and 9)The social infor-mation in space ldquocontaminatesrdquo agents they aligntheir actions through the social contents of placesWe have seen that space matters as extension Nowwe see that space also has a very active informationalpresence

(v) Different combinations of factors have differenteffects over entropy Figure 8 shows the transitions

between the three parameters and how nuancedweights resulting from blending factors matter inthe reduction of entropy For instance blending theweight of activity places and latent orientation on thenext action displays a strong reduction of entropy(pink lines in Figure 8) That means that when agentsdo not keep information from previous selections(ie 120573 asymp 0) the action system reduces more entropythan in other parameter combinations A strongshort-term memory leads the system into conservingitself and into a poorer capability to coordinateactions Also when current actions 120573 and activityplaces 120574 share a similar weight over the next action(orange lines on top) the social system does notexperience a great reduction of entropy as agents tendto reproduce their actions

(vi) Finally the reduction of entropy implies that theprobability of certain actions and interactionsincreases In practical terms this means morealignments between agents and more connectionsbetween actions (interactions) However if all agentsin the system reached the same orientation andentropy dropped to zero the system would lose inter-nal differentiation Agents would behave in the sameway say in a world with no personal differentiationspecialization or division of labour New orientations(therefore entropy) are necessary if the social systemis to keep differentiated agencies On the otherhand a system with maximum entropy would createdifference in actions to a point where no action coor-dination or interaction would be possible Clearly thissituation cannot be the case A social system requiresbalances neither full entropy nor total predictability

These results also suggest that we are dealing with distinctldquosocial memoriesrdquo a long-term memory is active in what we

Complexity 13

20

25

30En

trop

y

04 0602-latent orientation

04 0602-current action

20

25

30

Entr

opy

20

25

30

Entr

opy

04 0602-activity place

Figure 9 Parameter exploration isolated importance of each parameter in entropy (under the friction of distance) Each vertical set of dotsrepresents one fixed value for the highlighted parameter with the two other parameters assuming every possible combination within thetested range Each dot corresponds to the entropy value when a sufficiently large time-frame is used

call latent orientations Short-term memory is active in theinfluence of a current orientation over a new action In turnthe social information latent in activity places is a result ofsocial arrangements (say as firms a local economy and soon) which find certain stability changing much slower thenactions It is an extension of the social system projected ontourban space stabilizing the system to some extent

7 Cities and Social Interaction Conclusions

What does this approach bring to the state of the art on therelations of social interaction information and space and onsocial entropy and ABM in particular We have seen thatprevious approaches overlooked the coordination of actionas a key empirical and analytical problem Furthermorethey tend to have the thin spatiality of a passive territorialbackground (eg [16 20]) In turn although the conceptof ldquocollective actionrdquo is finally getting attention in urbanstudies [64] the problem of social organization is still largelyunderestimated The problem of entropy has been dealt withsince Wilsonrsquos [65] work on spatial interaction but has notreached the mainstream of the discipline and deals mostlywith urban form (eg [24]) and spatial distributions [66]not with social information and action Regarding agentbased models we are not aware of an approach that dealswith the problem of how cities and space are part of socialorganization In a sense our model is closer to Axelrodrsquos[67] ABM model of dissemination of culture His agentsexchange ldquoculturerdquo through direct contact with neighboursin a cellular automata model whereas our agents are mobileand actively deal with distance and social information inspace recognizing types of activities in places The intensityof exchanges between agents in Axelrodrsquos model is a functionof similarities between them In our model exchanges aremediated by the selection and interaction with space In ourunderstanding our contribution in terms of ABM lies onthe fact that a new and simple model is proposed to dealwith the material and informational dimensions of socialorganization not just as an illustration but as a proof ofconcept space can have causal presence if agents generatepreferences as detailed in a domain that coevolves with them

Now does the simulation model corroborate the theoryintroduced in this paper Our proposition cannot be empir-ically validated at this stage given that it is very hard to

assess peoplersquos choices and the whole panorama of actionsin a city Entropy may well be beyond observation In thesecircumstances simulations of the behaviour of agents underdifferent spatial conditions become useful to assess entitiesevents and entropic forces at play One expected result fromthe model is that space matters for coordinating actions butsimulations surprisingly showed that space is a necessary butnot sufficient condition for the reduction of entropy Neithermaterial cognitive resources nor individual powers of agentsare enough Information in activity places and a bit of agentsrsquopersonal history also play roles in the way social systemsdeal with their own entropy That would mean a causal butnondeterministic presence of space

Substantively this approach explores a subtle but funda-mental presence of cities in social organizationWe argue thaturban space materializes gradients of difference in potentialinteractions from less to more recognizable costly or likelyPlacing simple criteria including distance as a factor in select-ing activities our ABM showed that space becomes a meansfor producing differences in the probabilities of interactionincreasing chances of certain selections and convergences incollective action

In reality people recognize the social information ofactivity places along with uses of busy streets or local central-ities We are able to retrieve these differences from the builtenvironment and relate to them One of the key points of ourargument is that these differences help us make selectionsEvery time we move through the city and select places toperform we participate in the large-scale coordination ofaction We enact space as a ldquoreferential systemrdquo [49 50]a set of semantic bits indexing social activities informingand guiding daily decisions Space becomes inherently partof the conversion of orientations into interactions that isconnections that crisscross sequences of actions recreatingthe interaction system By means of a structure that bothmaterializes and restricts the quasi-endless combinatorial pos-sibilities of interaction this system can acquire sufficientinternal guidance to make its own reproduction possibleBy distributing sufficiently recognizable differences in theprobability of interaction cities express and release localforces of reproduction of social systems The interfaces ofaction cognitive and spatial systems transform entropiccomplexity into structured complexity constantly reorderingactions in time

14 Complexity

This materialistic viewpoint can be integrated with otherviews of social organization Further work may include therelational roles of cultures language social norms and othersteering media along with explorations of possibly similareffects of communication technologies over entropy At thisstage we attempted to integrate a social dimension (agentsengaged in coordination) a spatial dimension (an extensiveenvironment) and an informational dimension (as differ-ences in action and in the environment itself) a demandinginterdisciplinary effort The approach identifies a central rolefor coevolving agents actualizing aims as they coordinate (cf[68]) and a central role for the environment exterior butresponsive to agents leading to cumulative changes alongtheir historyThemodel also identifies roles for cognition andmemory in the ways agents deal with their own orientationsand choose their actions and activity places changing theirspatial environment with consequences in the overall levelsof entropy All this suggests that the behaviour of the modelcannot be reduced to or fully predicted from the behaviour ofindividual variables which is a key aspect of complexity

So one of the main aims of this paper was to assesswhether space as information environment had any effecton the coordination and entropy of interaction However ifphysical distance and social information in activity placesmatter what about city size and the internal spatial struc-tures of cities What about rural areas or rarefied suburbsSince our model explores an abstract circular city it onlybegins to answer such questions Our findings indicate thatshorter distances between activity places tend to reducesocial entropy Density seems to matter denser spatialities(as opposed to rarefied ones) have a role to play A pathfor further development here is Jacobsrsquos [69] idea that thediversity of activities generates positive externalities Anddiversity as spatial economics shows us has to do withthe size of cities and density of population (eg [70 71])Entropy may also be influenced by diversity Batty et al [66]have shown that information increases as cities get biggerThe remaining question is whether larger denser and moreinternally structured cities could create process and reduceentropy with more intensity than smaller cities transformingbig pools of activities into differences in the probability ofinteraction Although we proposed a unidimensional modelour agentsrsquo behaviour can be explored in more realisticrepresentations of cities testing the roles of density topologyand diversity while going beyond perfectly informed agentsBy proposing the ring model we intended to simplify acity to a minimum system where the spatial dimensionof social organization becomes intelligible without losingfundamental properties

Shouldwe conclude fromall this that entropy is a problemfor social systems Our approach suggests that it is notEntropy is a necessary force It means that novel orientationsare entering the interaction system and must be dealt withEntropy only becomes a problem if not converted into orga-nization only if numerous potential actions are not convertedinto smaller sets of actual interactions The effects of newactions over entropy would be positive once actualizedhaving to dowith the diversity of agencyConservative systemsare likely to face less entropy but also likely to produce

less novelty having homogeneity as a problematic horizonFluctuations of entropy seem vital to complex societies

Finally our intention was to explore entropy as a meansto think about the conditions of social organization ldquofromanother anglerdquo so to speak from the viewpoint of challengesinvolved in social reproduction Hence the approach focusedon the idea of uncertainty surrounding agents dealing withaims and decisions and the different probabilities of inter-action that follow an attempt to bring social cognitiveinformational and spatial theories under a single roof Ourapproach sees cities as connective systems produced to createthe delicate fabric of interaction that keeps large numbersof agents coherently living together It also suggests that thevarying states of entropy reveal deep connections betweensocial informational and spatial systems

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

Theauthors would like to thank the Ecole Polytechnique Fed-erale de Lausanne (EPFL) CAPES FAPEMIG FAPESP andCNPq for financial support Maıra Pinheiro Henrique Loreaand colleagues at ABMUS2017 for earlier discussions andMike Batty for exchanges and inspiration Finally the authorsthank Romulo Krafta maverick of the complexity studiesof cities in Brazil for his input and support throughout theyears This work is dedicated to him

References

[1] C W Mills The Sociological Imagination Oxford UniversityPress New York NY USA 1959

[2] M Castells ldquoThe information age economy society and cul-turerdquo inThe Rise of the Network Blackwell London 1996

[3] C Hidalgo Why Information Grows The Evolution of Orderfrom Atoms to Economies Basic Books New York NY USA2015

[4] D Helbing The Automation of Society Is Next ETH ZurichSwitzerland 2016

[5] N Luhmann Social Systems University Press Stanford 1995[6] I Prigogine and I Stengers Order out of Chaos Manrsquos New

Dialogue with Nature Bantam Books 1984[7] C E Shannon ldquoAmathematical theory of communicationrdquo Bell

System Technical Journal vol 27 no 4 pp 623ndash656 1948[8] W Buckley Sociology and Modern Systems Theory Prentice-

Hall Englewood Cliffs NJ USA 1967[9] D D McFarland ldquoMeasuring the Permeability of Occupational

Structures An Information-Theoretic Approachrdquo AmericanJournal of Sociology vol 75 no 1 pp 41ndash61 1969

[10] F Charvat ldquoOn philosophical aspects of the system conceptionin contemporary sociological knowledgerdquo Quality and Quan-tity vol 6 no 1 pp 3ndash16 1972

[11] F Charvat J Kucera and M Soukup ldquoToward the systemtheory of dependence further general theoretical remarksrdquoQuality and Quantity vol 7 no 1 pp 69ndash90 1973

Complexity 15

[12] P M Horan ldquoInformation-theoretic measures and the analysisof social structuresrdquo Sociological Methods and Research vol 3no 3 pp 321ndash340 1975

[13] O E Klapp ldquoOpening and closing in open systemsrdquo BehavioralScience vol 20 no 4 pp 251ndash257 1975

[14] J Galtung ldquoEntropy and the general theory of peacerdquo in Essaysin Peace Research J Galtung Ed vol L Humanities PressAtlantic Highlands NJ USA 1975

[15] T Parsons The System of Modern Societies Englewood CliffsNew Jersey NJ USA 1972

[16] K D Bailey ldquoSociological entropy theory Toward a statisticaland verbal congruencerdquo Quality amp Quantity vol 18 no 1 pp113ndash133 1983

[17] K D Bailey Social EntropyTheory State University of NewYorkPress Albany NY USA 1990

[18] J G Miller Living Systems McGraw-Hill New York NY USA1978

[19] K D Bailey Sociology and the New Systems Theory Toward aTheoretical Synthesis State University of New York PressAlbany NY USA 1994

[20] K D Bailey ldquoSociocybernetics and social entropy theoryrdquoKybernetes vol 35 no 3-4 pp 375ndash384 2006

[21] F Geyer ldquoThe challenge of sociocyberneticsrdquo Kybernetes vol24 no 4 pp 6ndash32 1995

[22] F Geyer ldquoThe march of self-referencerdquo Kybernetes vol 31 no7-8 pp 1021ndash1042 2002

[23] J HabermasTheTheory of Communicative Action Polity PressCambridge UK 1987

[24] H Haken and J Portugali Information Adaptation The Inter-play between Shannon Information and Semantic Information inCognition Springer New York NY USA 2015

[25] M Weber Economy and Society vol 1 Berkeley University ofCalifornia Press 1978

[26] L Wittgenstein Philosophical Investigations Blackwell Publish-ers Oxford London 3rd edition 2001

[27] L Vygotsky Mind in Society The Development of HigherPsychological Processes Harvard University Press CambridgeMA 1978

[28] M Wilson ldquoSix Views of Embodied Cognitionrdquo PsychonomicBulletin amp Review vol 9 no 4 pp 625ndash636 2002

[29] A Clark and D Chalmers ldquoThe extended mindrdquo Analysis vol58 no 1 pp 7ndash19 1998

[30] R PassiniWayfinding in Architecture Van Nostrand ReinholdNew York NY USA 1992

[31] E Rosch ldquoPrinciples of categorizationrdquo in Cognition and Cate-gorization E Rosch and B Lloyd Eds Hillsdale LawrenceErlbaum Ass 1978

[32] W KintschMemory and Cognition John Wiley and Sons NewYork NY USA 1970

[33] U Neisser ldquoMultiple Systems a New Approach to CognitiveTheoryrdquo European Journal of Cognitive Psychology vol 6 no 3pp 225ndash241 1994

[34] A Faria and R Krafta ldquoRepresenting urban cognitive structurethrough spatial differentiationrdquo in Proceedings of 4th Space Syn-tax International Symposium pp 531-518 UCL Press London2003

[35] J Portugali Complexity Cognition and the City Springer NewYork NY USA 2011

[36] D Kirsh and P Maglio ldquoOn distinguishing epistemic frompragmatic actionrdquo Cognitive Science vol 18 no 4 pp 513ndash5491994

[37] S M Kosslyn Image and Brain The Resolution of the ImageryDebate MIT Press Cambridge MA

[38] R Brooks Cambrian Intelligence The Early History of the NewAI MIT Press Cambridge MA 1999

[39] A Clark BeinGThere Putting Brain Body AndWorld TogetherAgain MIT Press Cambridge MA 1997

[40] R Pfeifer and C ScheierUnderstanding Intelligence MIT PressCambridge MA 1999

[41] A M Glenberg and D A Robertson ldquoIndexical understandingof instructionsrdquoDiscourse Processes vol 28 no 1 pp 1ndash26 1999

[42] JM Iverson and SGoldin-Meadow ldquoWhypeople gesturewhenthey speakrdquo Nature vol 396 no 6708 p 228 1998

[43] R M Krauss ldquoWhy do we gesture when we speakrdquo CurrentDirections in Psychological Science vol 7 no 2 pp 54ndash60 1998

[44] S Franklin Artificial Minds MIT Press Cambridge MA 1995[45] J Gibson The Ecological Approach to Visual Perception

Houghton-Mifflin Boston 1979[46] G Lakoff andM JohnsonPhilosophy in the FleshTheEmbodied

Mind and Its Challenge to Western Thought Basic Books NewYork NY USA 1999

[47] E HutchinsCognition in theWild MIT Press CambridgeMA1995

[48] E Hutchins ldquoThe cultural ecosystem of human cognitionrdquoPhilosophical Psychology vol 27 no 1 pp 34ndash49 2014

[49] V M Netto ldquoPractice space and the duality of meaningrdquoEnvironment and Planning D Society and Space vol 26 no 2pp 359ndash379 2008

[50] V M Netto The Social Fabric of Cities Routledge New YorkNY USA 2017

[51] T Parsons The Structure of Social Action The Free Press NewYork NY USA 1968

[52] T Allen Managing the Flow of Technology MIT Press Cam-bridge MA 1977

[53] F L Ribeiro J Meirelles F F Ferreira and R C Neto ldquoAmodel of urban scaling laws based on distance-dependent inter-actionsrdquo Royal Society Open Science vol 4 Article ID 1609262017

[54] W Alonso Location and Land Use Toward a General Theory ofLand Rent Harvard University Press Cambridge MA 1964

[55] WGHansen ldquoHow accessibility shapes land userdquo Journal of theAmerican Institute of Planners vol 25 no 2 pp 73ndash76 1959

[56] B Hillier Space Is the Machine Cambridge University PressCambridge UK 1996

[57] S Farber M OrsquoKelly H J Miller and T Neutens ldquoMeasuringsegregation using patterns of daily travel behavior a socialinteraction based model of exposurerdquo Journal of TransportGeography vol 49 pp 26ndash38 2015

[58] T C Schelling ldquoDynamic models of segregationrdquo Journal ofMathematical Sociology vol 1 no 2 pp 143ndash186 1971

[59] R Krafta V M Netto and L Lima ldquoUrban built form growscriticalrdquo CyberGeo vol 2011 article 565 2011

[60] N Lanchier and S Scarlatos ldquoFixation in the one-dimensionalAxelrod modelrdquo The Annals of Applied Probability vol 23 no6 pp 2538ndash2559 2013

[61] F L Ribeiro and K N Ribeiro ldquoA one dimensional modelof population growthrdquo Physica A Statistical Mechanics and itsApplications vol 434 pp 201ndash210 2015

[62] F L Ribeiro ldquoA non-phenomenological model of competitionand cooperation to explain population growth behaviorsrdquo Bul-letin of Mathematical Biology vol 77 no 3 pp 409ndash433 2015

16 Complexity

[63] R V dos Santos F L Ribeiro and A S Martinez ldquoModels forAllee effect based on physical principlesrdquo Journal of TheoreticalBiology vol 385 pp 143ndash152 2015

[64] M BattyThe New Science of Cities The MIT Press CambridgeMA 2013

[65] A G Wilson Entropy in Urban and Regional Modelling PionLondon 1970

[66] M Batty R Morphet P Masucci and K Stanilov ldquoEntropycomplexity and spatial informationrdquo Journal of GeographicalSystems vol 16 no 4 pp 363ndash385 2014

[67] R Axelrod ldquoThe dissemination of culture a model with localconvergence and global polarizationrdquo Journal of Conflict Reso-lution vol 41 no 2 pp 203ndash226 1997

[68] B H Hodges and RM Baron ldquoValues as constraints on afford-ances perceiving and acting properlyrdquo Journal for the Theory ofSocial Behaviour vol 22 no 3 pp 263ndash294 1992

[69] J JacobsThe Economy of Cities Random house New York NYUSA 1969

[70] E Glaeser H Kallal J Scheinkman and A Shleifer ldquoGrowthin Citiesrdquo Journal of Political Economy vol 100 no 6 pp 1126ndash1152 1992

[71] S Rosenthal and W Strange ldquoEvidence on the nature andsources of agglomeration economiesrdquo in Handbook of Urbanand Regional Economics J V Henderson and J-F Thisse Edsvol 4 pp 2119-2171 NorthHollandNewYork NYUSA 2004

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Complexity 7

Could cities stimulate connections If they could andgiven the fact that they are formed by space and thatspace is not fully malleable is it reasonable to supposethat certain spatial formations could ease such momentaryconnections We know that proximity increases the intensityof interaction [52 53] But what form should urbanized spacetake in order to meet such systemic expectations We mayunderstand such spatial conditions through a counterfactualscenario Imagine a fully chaotic urban formation with norecognizable concentrations or paths Visual elements couldnot convey information as spatial differences would notdevelop into structures emerging out of absolute heterogene-ity References to activities of interest would have to relyonly on our memory Selection of actions would becomedependent on very detailed mental representations of such achaotic formation Inferences about more likely places to findparticular activities would be very hard tomakeWewould beimmersed in a spatial environment without patterns to guideour choices reduce our efforts and amplify our interactionsIn a world of unstructured spaces actions would face lackof information noise and relentless entropy The possibilitythat space may have an informational presence in interactionexpresses the possibility that the physical environment mayhave informational properties and that the world is rich ininformation

Now not only our daily experience denies that cities aresuch completely chaotic structureless places but also spatialeconomics and urban studies offer plenty of theories andevidences that as complex as they may be cities are (at leastpartially) structured places From location patterns describedsince Alonso [54] and accessibility patterns described fromHansen [55] to Hillier [56] we know that urban patternsmatter they imply that activities can be arranged in ways thatare recognizable and accessible And from the point of view ofselection the realization that activities are somehow arrangedalong a spatial structure is a form of preselection Activitiestend to be distributed according to levels of accessibilityland values and practical dependencies facilitating comple-mentary actions Intentionally or not and however subject tocontingencies urban space is also structured (physically andinformationally) which allows possibilities of action to bemore easily known selected and viable In short at first theurban structure affords a range of possibilities for selectionIn a second moment the structure itself suggests gradients ofattractive possibilities

In a view of social systems constituted by connectionsof actions this means that space can be part of the constanttransition from individual acts to complexes of actions (indi-vidual act rarr places as connections rarr action systems) Ourhypothesis here to be tested in computational experimentsin the following section is that cities become a crucial part ofthis connectivity They would reduce risks such as the lack ofinformation or courses of action too costly to emerge

What does this say about the effect of cities on entropyBeginning with Luhmann interactions have structural valuebecause they represent selections from combinatorial possi-bilities If we relate this to Shannonrsquos entropy the very emer-gence of urban structures eliminates spatial scenarios whereevery possible connection would have the same probability

Another counterfactual scenario might help here Entirelyhomogeneous cities (or entirely heterogeneous for thatmatter) free from structure would present even probabilitiesof choice between actions Of course this would be a problemfor social systems connections potentially interesting tomostpeople would be as difficult to find as any If urban spacewas purely homogeneous or purely random the probabilityof finding an activity would be distributed homogeneouslyin space That is a scenario of maximum entropy The spatialenvironment would be useless in stimulating certain sought-after connections which could otherwise be there say whenwe concentrate activities like final supply or input-outputexchanges in production Effort time spent and economiccosts involved in finding activities of interest would rise

On the other hand when urban space finds differentiatedcontents and structures say location patterns along acces-sible paths it naturally creates differences in the probabilityof interaction and decreases in entropy Location patterns arein fact material and informational expressions of interactionsystems The differences engendered by such patterns gener-ate differences in the chances of finding an agent a service orcommerce If a city did not have recognizable internal forma-tions such as a centralities or high streets agents would havemore difficulties in knowingwhere they aremost likely to findwhat they are after A partially structured environment allowsus to build inferences about actions and agencies available in asocial systemWhen these agencies are also related temporallythrough proximity and functional complementarity there isan even greater potential for differences in probability toemerge suggesting new connections and sequences Whatall this means is that cities internally differentiated yet farfrom absolute heterogeneity have the effect of reducing socialentropy

Before experimenting with these relationships in silicolet us attempt an introductory description of what thesedifferences in probability entail in theway social systems han-dle entropy (Figure 3) Linearizing for the sake of simplicityprocesses that in fact occur simultaneously the reductionof social entropy would take the following form (a) In aninitial state free of space think of the actions as lines movingin time Agents can do anything and we cannot foreseewhat they will do a potential state of high entropy Colorsof the lines in Figure 4 represent different informationalcontents in personal orientations (b) Then agents startselecting actions and converging into different positions in aspatial system arranged as places or buildings (representedby colors in the vertical strip) in order to connect withothers Action lines might have subtle differences in relationto the places they approach so they converge into placesby informational proximity Connections can only happenif all other possibilities of connection are excluded (c)As these convergences happen the initially chaotic mazebecomes a momentarily coordinated system where agentscooperate The overall entropy of actions is reduced (d)After each spatially held event actions may change againaccording to changing orientations Accordingly the colorsof lines change Entropy increases as actions go into anothermomentary state of unpredictability as new possibilities arepresented to agents and somemust be selected if actions are to

8 Complexity

Momentarilycoherent systems

State of unpredictabledecisions

Action systems

Reduction of entropy

Coordination of actions

Selectionof actions

High entropy state

Information growth(Hidalgo 2015)

Possibilities of action(Luhmann 1995)

Differences in theprobability ofinteractions

(Shannon 1948)

Agents interact Orientations align inactivity places

Actions mediated byinformation

(Luhmann 1995)

City as informationenvironment(Haken and Portugali 2015)

Figure 3 Recursive process that conducts interaction systems from chaotic states (high entropy) into coherence (low entropy)

(a) (b) (c)

(e) (d)

Figure 4 The cycle of social entropy and space clockwise from the high unpredictability of potential actions to a momentarily orderedsystem as actions converge into places only to dive in entropy again

Complexity 9

Actionagent x

Actionagent y

Activity place

Cooperation

Change insubjectiveorientation

Figure 5 Aligning different actions in the cooperation process

be actualized (e) And then actions ldquoreferrdquo again to meaningsand positions in space starting a new cycle of entropy(Figure 4)

These distinct moments are theoretical of course Inreality reminding Prigogine and Stengers [6 page 124] thisldquomultitude of eventsrdquo takes place simultaneously and mayrender any clearly emergent cycle invisible But that neithermeans that individual cycles do not happen or aggregate inbunches nor they compensate one another like peaks andtroughs in colliding sound waves In fact routinization indaily life leads to rhythms and cycles unfolding from orfolding over the seemingly chaotic maze of actions But evenif actions were far from synchronic the fabric of interactionstill faces the challenge of entropy that is the challenge ofproducing differences in the probability of connections as a wayto ease organization whenever agents feel compelled to createnew interactions and this process is remarkably recursiveWe converge to places to act with others and do so on adaily basis Once engaged in an activity place interactionrequires levels of cooperationWemay bring our own distincttendencies or latent orientations in the form of long-termmemories of how to act along with short-term memoriesof a previous action both shaping our next actions As weact socially we bring our orientations into contact in away to open them in the communication process regulatingcollective action A result of this coordination is to reduceinitial differences Our actions ldquoalignrdquowithin an activity place(Figure 5)

All this suggests that social life involves a continuousalthough unconscious handling of entropy a vital oscillationbetween unpredictability and predictability As this processengages an enormous number of agents and events andtranscends places and contexts and therefore cannot beobserved as a whole entropy can only be fully grasped intheoretical representations A useful way to examine space asan information environment for action is through a compu-tational model a means to examine a counterfactual worldwhere other steering media such as language or social ruleswere ldquoswitched offrdquo keeping only properties of space activealong with agentsrsquo cognitive abilities to align orientationswith activities Once we abstract from other media wemay assess whether space could have any presence in theself-organization of action Let us examine this possibilitythrough some experiments

LBuildingactivity type

Figure 6The theoretical one-dimensional citywith a set of differentactivity places Here there are 119871 buildings distributed in a ringconstituting the city space Colors represent activity types

6 Digital ExperimentsA Unidimensional Model

61 Design Concepts An agent based model (ABM) is pro-posed Agents are defined to perform actions at each time-step and places by corresponding activity types The decisionon which action to perform next may be influenced bythree different conditions namely (i) latent orientation120590in

119894(119905)

representing the tendency of a single agent to act arounda particular type of action initially randomly distributedthis condition remains over time (ii) current action 120590

119894(119905)

performed by an agent as heshe selects an activity placein order to perform a new action this means an influenceof the current action while allowing gradual changes oforientation in time influenced by other agents and activitiesand (iii) types of activity places 120590

1199091015840(119905) where agents perform

supporting specific types of action Agents follow simplerules selecting their activity places closer to their latentorientations and current actions while being influencedby those activities Activity places are also influenced byvisiting agents but they change at a slower rate In shortagents coevolve with their spatial and social environmentThe model simulates different situations under this basicstructure Differences in the weight of these factors overthe next action may lead to quite different levels of socialentropy suggesting new possibilities for understanding thephenomenonAlthough there is a tradition inmodelling dailyactivities including a growing literature using digital loca-tional data (eg [57]) our ABM is not designed to representempirical sequences of actions or actual routines Instead itfocuses on trends that might emerge from the interfaces ofsimplified systems of action information and space

62 Variables and Scale Consider a unidimensional cityformed by agents and places Places form a ring with length(or perimeter) 119871 as shown in Figure 6 Our choice forrepresenting the city by a ring is justified by the minimalsufficient representation of spatial distance as a factor whose

10 Complexity

role in social organization is to be assessed This is allowedby a unidimensional model (eg [58ndash60]) The ring formallows continuous movement across a linear sequence oflocations eliminating centrality factors while taking intoaccount periodic boundary conditions in order to reduceborder effects eliminating the role of topologywhile isolatingthe problem of distance (see [61ndash63]) This stylized cityhouses 119873 agents which in each time-step 119905 select and visita specific place located at the position 119909 within the cityConsider that the position of the 119894th agent at the time 119905 isrepresented by 119909

119894(119905) The position of an agent can assume the

integer values 1 2 119871 according to the place that shehechooses to visit This means that at every time-step eachagent will choose one activity place to perform her nextaction

We considered 120590119894(119905) as the current action of the 119894-th agent

and 120590119909(119905) the activity performed in a place located at the

position 119909 = (1 2 119871) of the city both in time 119905 Wequantify action orientations weighting the three variables(latent orientation current action and activity place) Firstvariables are translated into different numeric values Let usassume that an orientation is an integer number between 0and 1000 each number represents a particular type of actionand numbers in their vicinity mean similar actions Theresulting orientation in time 119905 for each agent is the weightedaverage of those three parametersThedifference |120590

119894(119905)minus120590119909(119905)|

gives us the difference in orientation between an agent andthe activity place At every time-step each agent assesses thedifference between her orientation and every activity placein the city An agent chooses a specific place based on herorientation affinity with this place minimum difference

As we hypothesized above the urban structure has a rolein this selection In this experiment we reduced ldquourban struc-turerdquo to distance In order to investigate how distance may ornot interfere in this selection we propose two scenarios Inthe first one the agent selects her next activity based only onthe similarity in orientations between herself and the activityplace Agents can move across a space free of friction Spaceis not a constraintThis scenario considers only differences inorientation The parameter 119864

119894(119909 119905) estimates the interaction

between agent 119894 and the place located at 119909 at the time 119905 Theagent selects activity locations that minimize this quantityWe propose

119864119894 (119909 119905) =

1003816100381610038161003816120590119894 (119905) minus 1205901199091003816100381610038161003816 (1)

In the second more realistic scenario space imposesfriction to movement The agent takes into account thephysical distance between her current location in the city andthe location of her next activity and opts to minimize thisdistance along with the difference between her orientationand that of the activity place We propose

119864119894 (119909 119905) =

1003816100381610038161003816(119909119894 (119905) minus 119909) (120590119894 (119905) minus 120590119909)1003816100381610038161003816 (2)

Summing up an agent 119894 selects at time 119905 a particularactivity place located at119909 thatminimizes the function119864

119894(119909 119905)

In the first scenario she will choose activity places closerto her orientation In the second she will choose activityplaces closer in both orientation and physical distance This

brings a form of ldquoenergy costrdquo into the factors consideredSpace takes on a function role because of the energy used inldquovisitingrdquo Agents move in this linear city willing to minimizethis energy while searching for social information latentin places close to their changing orientations Changes inentropy levels are derived from this energy function In itsspatial version the model assesses effects of the effort toreduce energy in action minimizing informational differ-ences (between orientations and activity places) and spatialdistances (between agents and activity places)

63TheEvolution of ActionOrientations Orientation evolvesin time according to the following rule When an agentchooses and joins a particular place both agent and placebecome a little closer in terms of orientation This meansthat activity places are not immune to what agents performin them and that the urban activity system also changes intime To illustrate how orientations are constantly updatedconsider that agent 119894 selects an activity place located at 1199091015840Orientations will be updated according the rule

120590119894 (119905 + 1) =

1120572 + 120573 + 120574

(120572120590in119894(119905) + 120573120590119894 (119905) + 1205741205901199091015840 (119905)) (3)

Selection also depends on three additional parameterslatent orientation 120590in

119894(119905) weighted by parameter 120572 current

orientation120590119894(119905)weighted by parameter120573 and type of activity

place 1205901199091015840(119905) weighted by parameter 120574 This means that agents

will consider the three factors previously described (120590in119894(119905)

120590119894(119905) 1205901199091015840(119905)) with different weights (120572 120573 120574) to estimate which

activity she wants to perform in the next time-stepThe latent orientation evolves from an initial randomly

distributed behaviour 1205900119894 We consider that 120590in

119894(119905) is normally

distributed around 1205900119894 meaning that every agent has an

ldquoaveragerdquo latent orientation whose value will be derived ateach time-step from a normal distribution peaking at theagentrsquos average orientation If 120572 is sufficiently small in relationto 120573 and 120574 then the latent orientation does not play a rolein the agentrsquos action However if 120572 is sufficiently large (oncein relation to 120573 and 120574) then the action becomes stronglydependent on the latent orientation This would lead to apretty conservative city where inhabitants are not open tochanges in behaviour

The current action of an agent may influence a newaction with an intensity that depends on parameter 120573 If thisparameter is small in comparison to 120572 and 120574 then the agenthas no memory of an ongoing orientation However if 120573 issufficiently large then the new action is strongly dependenton a previous one (a Markovian behaviour)

The intensity of the activity placersquos influence on the agentrsquosaction depends on the value of 120574 If this parameter is smallin comparison to 120572 and 120573 then the activity place does notaffect her next actionmeaning that the place does not changebehaviours However if this parameter is sufficiently largethen the place plays a strong influence on future actions Theactivity place will be updated according to the rule

1205901199091015840 (119905 + 1) = 1205901199091015840 (119905) + 120579sum

119894isin1199091015840

120590119894 (119905) (4)

Complexity 11

where 120579 is a parameter sufficiently small That is placesare less influenced by agents than the other way roundThis means that at every time-step a place would have itstype of activity closer to the average of orientations of itsvisitors Therefore the main activity of the place will changeslightly according to visiting agents while changing theirorientations

64 Frequency of Orientations and Entropy Levels We cal-culate entropy assessing the distribution of resulting orien-tations of all agents in the social system at each time-stepusing Shannon entropy Our quantitative interpretation ofinformation removes ambiguities from the relation betweenShannon information and semantic information as differentorientations actions and types of activity places are repre-sented by numerical values This allows us to estimate socialentropy through Shannon entropy The entropy of actions iscalculated as follows consider119873(120590 119905) as the number of agentswith orientation 120590 at the time 119905 (note that the total populationis 119873 = sum

120590119873(120590 119905)) We compute the frequency (or density)

120588(120590 119905) of this orientation within a population with

120588 (120590 119905) = 1119873119873 (120590 119905) (5)

Equation (5) calculates the probability of observing anorientation 120590 at time 119905 We compute the entropy level for anydistribution of orientations with

119878 (119905) = minussum120590

120588 (120590 119905) ln (120588 (120590 119905)) (6)

Equation (6) describes how uneven is the probability of find-ing different orientations Higher values mean that differentorientations have almost the same probability to happenwhile lower values indicate a system with clear orientationtrends The reduction of entropy implies that the probabilityof certain actions increases that is actions grow in similarityIn the limit as entropy falls to zero all agents in the systemwould reach the same orientation

65Model Procedure Themodel performs the following pro-cedures

(1) The city is created by generating the following

(a) The type of activity in places that is (120590119909=1(119905 =

0) 120590119909=2(119905 = 0) 120590

119909=119871(119905 = 0)) also generated

from a random distribution This step createsactivity places in the ring-city and assign themrandomly distributed activity values 120590

1199091015840(119905)

(b) The initial position of 119873 agents that is (1199091

1199092 119909

119873) and their respective orientations

(1205901(119905 = 0) 120590

2(119905 = 0) 120590

119873(119905 = 0)) generated

from a random distribution This step createsmoving agents in random locations and assignthem latent orientations based on the initialrandomly distributed orientation 120590in

119894(119905) = 120590

119894(0)

(2) At each time-step 119905 every agent chooses a place toact This place should minimize the function 119864

119894(119909 119905)

Suppose that the chosen place by 119894 is defined as 1199091015840Then agent and place update their orientations 120590

119894and

1205901199091015840 as described above

(3) Update the distribution of orientations 120588(120590 119905)(4) Compute Shannon entropy 119878(119905)

66 Results We have developed a number of observationson the behaviour of social systems working under differentparameters

(i) Entropy reduction is only found in scenarios wherespace imposes friction to movement that is wherethe spatial distance between agents and activity placesis considered as an active factor in selection Entropyrequires an extensive space in order to be reduced Butspace cannot do it alone Surprisingly space is a nec-essary but not sufficient condition Figure 7 comparestwo scenarios In the first one distance is an issue inthe selection of activities (red line)Thedistribution oforientations changes from a homogeneous one at thestart of simulation (a) to a nearly normal distributionat the end of simulation (b) where certain kindsof action are more likely to happen Informationalcontents in extensive space help aligning contents inactions In the second scenario agents move free ofspatial friction (blue line) Orientations are initiallyrandomly distributed (a) and continue to be so at theend of simulation (b) as a similar number of agentsare distributed along different orientationsThis resultsuggests that a materially active space becomes ameans for increasing the probability of certain inter-actions easing the collective coordination of action

Nowassessing the relative influence of social and personalfactors on the selection of a next action namely the socialinformation in activity places latent orientation and the cur-rent action under the influence of spatial friction (Figure 8)we can say the following

(i) There is structure in the relation between selectionfactors and the reduction of entropy Entropy beginsto fall and fluctuate around specific values accordingto different combinations of factors For some com-binations there is a great reduction of entropy Forothers the reduction is minimal almost negligibleFluctuations derive from the random component inthe decisionmaking of the agent (under the influenceof latent orientation)

(ii) A strong latent orientation (120572 sufficiently large) leadsto increasing entropy (blue lines in Figures 8 and9) since agents cannot align their actions with otheragents and activity places Like a long-term memorysystems whose actions are dictated by latent orienta-tions are likely to preserve initial orientations whichwere randomly distributed and therefore homoge-neous This condition is responsible for limiting thereduction of entropy

(iii) Current action alone does not shape new actions inany specific way (green lines in Figures 8 and 9)

12 Complexity

Start of simulation

00000

00005

00010

00015

00020

o

f tot

al ac

tions

250 500 750 10000Type of action

Materially inactive spaceMaterially active space

t = 0

(a)

End of simulation

00000

00005

00010

00015

00020

o

f tot

al ac

tions

250 500 750 10000Type of action

Materially inactive spaceMaterially active space

t = 50

(b)

Figure 7 Histogram of probability distribution of actors performing different types of action at the start (a) and at the end (b) of simulationsin scenarios where distance is not considered (blue) and distance is considered aligning orientations in action coordination (red) Resultsare averaged for 30 runs for 125 parameter combinations

activityplace

currentaction

latentorientation

225

250

275

300

325

Entr

opy

10 20 30 40 500Time

Figure 8 Evolution of entropy colors in a RGB scale refer toparameter weights in decisions on actions in a next time-step andtheir impact on entropy levels in different simulations (under thefriction of distance) In the palette triangle each color representsa specific combination of the three parameters In each vertex aparameter reaches maximum value while the others have minimumvalue Results are averaged for 30 different runs for 125 parametercombinations

It just keeps what is already happening workingas ldquoreinforcement feedbackrdquo for either direction theagent (and the system) is going Current actions aremeans to conserve tendencies in the system

(iv) Activity places play a key role in the reduction ofentropy (red lines in Figures 8 and 9)The social infor-mation in space ldquocontaminatesrdquo agents they aligntheir actions through the social contents of placesWe have seen that space matters as extension Nowwe see that space also has a very active informationalpresence

(v) Different combinations of factors have differenteffects over entropy Figure 8 shows the transitions

between the three parameters and how nuancedweights resulting from blending factors matter inthe reduction of entropy For instance blending theweight of activity places and latent orientation on thenext action displays a strong reduction of entropy(pink lines in Figure 8) That means that when agentsdo not keep information from previous selections(ie 120573 asymp 0) the action system reduces more entropythan in other parameter combinations A strongshort-term memory leads the system into conservingitself and into a poorer capability to coordinateactions Also when current actions 120573 and activityplaces 120574 share a similar weight over the next action(orange lines on top) the social system does notexperience a great reduction of entropy as agents tendto reproduce their actions

(vi) Finally the reduction of entropy implies that theprobability of certain actions and interactionsincreases In practical terms this means morealignments between agents and more connectionsbetween actions (interactions) However if all agentsin the system reached the same orientation andentropy dropped to zero the system would lose inter-nal differentiation Agents would behave in the sameway say in a world with no personal differentiationspecialization or division of labour New orientations(therefore entropy) are necessary if the social systemis to keep differentiated agencies On the otherhand a system with maximum entropy would createdifference in actions to a point where no action coor-dination or interaction would be possible Clearly thissituation cannot be the case A social system requiresbalances neither full entropy nor total predictability

These results also suggest that we are dealing with distinctldquosocial memoriesrdquo a long-term memory is active in what we

Complexity 13

20

25

30En

trop

y

04 0602-latent orientation

04 0602-current action

20

25

30

Entr

opy

20

25

30

Entr

opy

04 0602-activity place

Figure 9 Parameter exploration isolated importance of each parameter in entropy (under the friction of distance) Each vertical set of dotsrepresents one fixed value for the highlighted parameter with the two other parameters assuming every possible combination within thetested range Each dot corresponds to the entropy value when a sufficiently large time-frame is used

call latent orientations Short-term memory is active in theinfluence of a current orientation over a new action In turnthe social information latent in activity places is a result ofsocial arrangements (say as firms a local economy and soon) which find certain stability changing much slower thenactions It is an extension of the social system projected ontourban space stabilizing the system to some extent

7 Cities and Social Interaction Conclusions

What does this approach bring to the state of the art on therelations of social interaction information and space and onsocial entropy and ABM in particular We have seen thatprevious approaches overlooked the coordination of actionas a key empirical and analytical problem Furthermorethey tend to have the thin spatiality of a passive territorialbackground (eg [16 20]) In turn although the conceptof ldquocollective actionrdquo is finally getting attention in urbanstudies [64] the problem of social organization is still largelyunderestimated The problem of entropy has been dealt withsince Wilsonrsquos [65] work on spatial interaction but has notreached the mainstream of the discipline and deals mostlywith urban form (eg [24]) and spatial distributions [66]not with social information and action Regarding agentbased models we are not aware of an approach that dealswith the problem of how cities and space are part of socialorganization In a sense our model is closer to Axelrodrsquos[67] ABM model of dissemination of culture His agentsexchange ldquoculturerdquo through direct contact with neighboursin a cellular automata model whereas our agents are mobileand actively deal with distance and social information inspace recognizing types of activities in places The intensityof exchanges between agents in Axelrodrsquos model is a functionof similarities between them In our model exchanges aremediated by the selection and interaction with space In ourunderstanding our contribution in terms of ABM lies onthe fact that a new and simple model is proposed to dealwith the material and informational dimensions of socialorganization not just as an illustration but as a proof ofconcept space can have causal presence if agents generatepreferences as detailed in a domain that coevolves with them

Now does the simulation model corroborate the theoryintroduced in this paper Our proposition cannot be empir-ically validated at this stage given that it is very hard to

assess peoplersquos choices and the whole panorama of actionsin a city Entropy may well be beyond observation In thesecircumstances simulations of the behaviour of agents underdifferent spatial conditions become useful to assess entitiesevents and entropic forces at play One expected result fromthe model is that space matters for coordinating actions butsimulations surprisingly showed that space is a necessary butnot sufficient condition for the reduction of entropy Neithermaterial cognitive resources nor individual powers of agentsare enough Information in activity places and a bit of agentsrsquopersonal history also play roles in the way social systemsdeal with their own entropy That would mean a causal butnondeterministic presence of space

Substantively this approach explores a subtle but funda-mental presence of cities in social organizationWe argue thaturban space materializes gradients of difference in potentialinteractions from less to more recognizable costly or likelyPlacing simple criteria including distance as a factor in select-ing activities our ABM showed that space becomes a meansfor producing differences in the probabilities of interactionincreasing chances of certain selections and convergences incollective action

In reality people recognize the social information ofactivity places along with uses of busy streets or local central-ities We are able to retrieve these differences from the builtenvironment and relate to them One of the key points of ourargument is that these differences help us make selectionsEvery time we move through the city and select places toperform we participate in the large-scale coordination ofaction We enact space as a ldquoreferential systemrdquo [49 50]a set of semantic bits indexing social activities informingand guiding daily decisions Space becomes inherently partof the conversion of orientations into interactions that isconnections that crisscross sequences of actions recreatingthe interaction system By means of a structure that bothmaterializes and restricts the quasi-endless combinatorial pos-sibilities of interaction this system can acquire sufficientinternal guidance to make its own reproduction possibleBy distributing sufficiently recognizable differences in theprobability of interaction cities express and release localforces of reproduction of social systems The interfaces ofaction cognitive and spatial systems transform entropiccomplexity into structured complexity constantly reorderingactions in time

14 Complexity

This materialistic viewpoint can be integrated with otherviews of social organization Further work may include therelational roles of cultures language social norms and othersteering media along with explorations of possibly similareffects of communication technologies over entropy At thisstage we attempted to integrate a social dimension (agentsengaged in coordination) a spatial dimension (an extensiveenvironment) and an informational dimension (as differ-ences in action and in the environment itself) a demandinginterdisciplinary effort The approach identifies a central rolefor coevolving agents actualizing aims as they coordinate (cf[68]) and a central role for the environment exterior butresponsive to agents leading to cumulative changes alongtheir historyThemodel also identifies roles for cognition andmemory in the ways agents deal with their own orientationsand choose their actions and activity places changing theirspatial environment with consequences in the overall levelsof entropy All this suggests that the behaviour of the modelcannot be reduced to or fully predicted from the behaviour ofindividual variables which is a key aspect of complexity

So one of the main aims of this paper was to assesswhether space as information environment had any effecton the coordination and entropy of interaction However ifphysical distance and social information in activity placesmatter what about city size and the internal spatial struc-tures of cities What about rural areas or rarefied suburbsSince our model explores an abstract circular city it onlybegins to answer such questions Our findings indicate thatshorter distances between activity places tend to reducesocial entropy Density seems to matter denser spatialities(as opposed to rarefied ones) have a role to play A pathfor further development here is Jacobsrsquos [69] idea that thediversity of activities generates positive externalities Anddiversity as spatial economics shows us has to do withthe size of cities and density of population (eg [70 71])Entropy may also be influenced by diversity Batty et al [66]have shown that information increases as cities get biggerThe remaining question is whether larger denser and moreinternally structured cities could create process and reduceentropy with more intensity than smaller cities transformingbig pools of activities into differences in the probability ofinteraction Although we proposed a unidimensional modelour agentsrsquo behaviour can be explored in more realisticrepresentations of cities testing the roles of density topologyand diversity while going beyond perfectly informed agentsBy proposing the ring model we intended to simplify acity to a minimum system where the spatial dimensionof social organization becomes intelligible without losingfundamental properties

Shouldwe conclude fromall this that entropy is a problemfor social systems Our approach suggests that it is notEntropy is a necessary force It means that novel orientationsare entering the interaction system and must be dealt withEntropy only becomes a problem if not converted into orga-nization only if numerous potential actions are not convertedinto smaller sets of actual interactions The effects of newactions over entropy would be positive once actualizedhaving to dowith the diversity of agencyConservative systemsare likely to face less entropy but also likely to produce

less novelty having homogeneity as a problematic horizonFluctuations of entropy seem vital to complex societies

Finally our intention was to explore entropy as a meansto think about the conditions of social organization ldquofromanother anglerdquo so to speak from the viewpoint of challengesinvolved in social reproduction Hence the approach focusedon the idea of uncertainty surrounding agents dealing withaims and decisions and the different probabilities of inter-action that follow an attempt to bring social cognitiveinformational and spatial theories under a single roof Ourapproach sees cities as connective systems produced to createthe delicate fabric of interaction that keeps large numbersof agents coherently living together It also suggests that thevarying states of entropy reveal deep connections betweensocial informational and spatial systems

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

Theauthors would like to thank the Ecole Polytechnique Fed-erale de Lausanne (EPFL) CAPES FAPEMIG FAPESP andCNPq for financial support Maıra Pinheiro Henrique Loreaand colleagues at ABMUS2017 for earlier discussions andMike Batty for exchanges and inspiration Finally the authorsthank Romulo Krafta maverick of the complexity studiesof cities in Brazil for his input and support throughout theyears This work is dedicated to him

References

[1] C W Mills The Sociological Imagination Oxford UniversityPress New York NY USA 1959

[2] M Castells ldquoThe information age economy society and cul-turerdquo inThe Rise of the Network Blackwell London 1996

[3] C Hidalgo Why Information Grows The Evolution of Orderfrom Atoms to Economies Basic Books New York NY USA2015

[4] D Helbing The Automation of Society Is Next ETH ZurichSwitzerland 2016

[5] N Luhmann Social Systems University Press Stanford 1995[6] I Prigogine and I Stengers Order out of Chaos Manrsquos New

Dialogue with Nature Bantam Books 1984[7] C E Shannon ldquoAmathematical theory of communicationrdquo Bell

System Technical Journal vol 27 no 4 pp 623ndash656 1948[8] W Buckley Sociology and Modern Systems Theory Prentice-

Hall Englewood Cliffs NJ USA 1967[9] D D McFarland ldquoMeasuring the Permeability of Occupational

Structures An Information-Theoretic Approachrdquo AmericanJournal of Sociology vol 75 no 1 pp 41ndash61 1969

[10] F Charvat ldquoOn philosophical aspects of the system conceptionin contemporary sociological knowledgerdquo Quality and Quan-tity vol 6 no 1 pp 3ndash16 1972

[11] F Charvat J Kucera and M Soukup ldquoToward the systemtheory of dependence further general theoretical remarksrdquoQuality and Quantity vol 7 no 1 pp 69ndash90 1973

Complexity 15

[12] P M Horan ldquoInformation-theoretic measures and the analysisof social structuresrdquo Sociological Methods and Research vol 3no 3 pp 321ndash340 1975

[13] O E Klapp ldquoOpening and closing in open systemsrdquo BehavioralScience vol 20 no 4 pp 251ndash257 1975

[14] J Galtung ldquoEntropy and the general theory of peacerdquo in Essaysin Peace Research J Galtung Ed vol L Humanities PressAtlantic Highlands NJ USA 1975

[15] T Parsons The System of Modern Societies Englewood CliffsNew Jersey NJ USA 1972

[16] K D Bailey ldquoSociological entropy theory Toward a statisticaland verbal congruencerdquo Quality amp Quantity vol 18 no 1 pp113ndash133 1983

[17] K D Bailey Social EntropyTheory State University of NewYorkPress Albany NY USA 1990

[18] J G Miller Living Systems McGraw-Hill New York NY USA1978

[19] K D Bailey Sociology and the New Systems Theory Toward aTheoretical Synthesis State University of New York PressAlbany NY USA 1994

[20] K D Bailey ldquoSociocybernetics and social entropy theoryrdquoKybernetes vol 35 no 3-4 pp 375ndash384 2006

[21] F Geyer ldquoThe challenge of sociocyberneticsrdquo Kybernetes vol24 no 4 pp 6ndash32 1995

[22] F Geyer ldquoThe march of self-referencerdquo Kybernetes vol 31 no7-8 pp 1021ndash1042 2002

[23] J HabermasTheTheory of Communicative Action Polity PressCambridge UK 1987

[24] H Haken and J Portugali Information Adaptation The Inter-play between Shannon Information and Semantic Information inCognition Springer New York NY USA 2015

[25] M Weber Economy and Society vol 1 Berkeley University ofCalifornia Press 1978

[26] L Wittgenstein Philosophical Investigations Blackwell Publish-ers Oxford London 3rd edition 2001

[27] L Vygotsky Mind in Society The Development of HigherPsychological Processes Harvard University Press CambridgeMA 1978

[28] M Wilson ldquoSix Views of Embodied Cognitionrdquo PsychonomicBulletin amp Review vol 9 no 4 pp 625ndash636 2002

[29] A Clark and D Chalmers ldquoThe extended mindrdquo Analysis vol58 no 1 pp 7ndash19 1998

[30] R PassiniWayfinding in Architecture Van Nostrand ReinholdNew York NY USA 1992

[31] E Rosch ldquoPrinciples of categorizationrdquo in Cognition and Cate-gorization E Rosch and B Lloyd Eds Hillsdale LawrenceErlbaum Ass 1978

[32] W KintschMemory and Cognition John Wiley and Sons NewYork NY USA 1970

[33] U Neisser ldquoMultiple Systems a New Approach to CognitiveTheoryrdquo European Journal of Cognitive Psychology vol 6 no 3pp 225ndash241 1994

[34] A Faria and R Krafta ldquoRepresenting urban cognitive structurethrough spatial differentiationrdquo in Proceedings of 4th Space Syn-tax International Symposium pp 531-518 UCL Press London2003

[35] J Portugali Complexity Cognition and the City Springer NewYork NY USA 2011

[36] D Kirsh and P Maglio ldquoOn distinguishing epistemic frompragmatic actionrdquo Cognitive Science vol 18 no 4 pp 513ndash5491994

[37] S M Kosslyn Image and Brain The Resolution of the ImageryDebate MIT Press Cambridge MA

[38] R Brooks Cambrian Intelligence The Early History of the NewAI MIT Press Cambridge MA 1999

[39] A Clark BeinGThere Putting Brain Body AndWorld TogetherAgain MIT Press Cambridge MA 1997

[40] R Pfeifer and C ScheierUnderstanding Intelligence MIT PressCambridge MA 1999

[41] A M Glenberg and D A Robertson ldquoIndexical understandingof instructionsrdquoDiscourse Processes vol 28 no 1 pp 1ndash26 1999

[42] JM Iverson and SGoldin-Meadow ldquoWhypeople gesturewhenthey speakrdquo Nature vol 396 no 6708 p 228 1998

[43] R M Krauss ldquoWhy do we gesture when we speakrdquo CurrentDirections in Psychological Science vol 7 no 2 pp 54ndash60 1998

[44] S Franklin Artificial Minds MIT Press Cambridge MA 1995[45] J Gibson The Ecological Approach to Visual Perception

Houghton-Mifflin Boston 1979[46] G Lakoff andM JohnsonPhilosophy in the FleshTheEmbodied

Mind and Its Challenge to Western Thought Basic Books NewYork NY USA 1999

[47] E HutchinsCognition in theWild MIT Press CambridgeMA1995

[48] E Hutchins ldquoThe cultural ecosystem of human cognitionrdquoPhilosophical Psychology vol 27 no 1 pp 34ndash49 2014

[49] V M Netto ldquoPractice space and the duality of meaningrdquoEnvironment and Planning D Society and Space vol 26 no 2pp 359ndash379 2008

[50] V M Netto The Social Fabric of Cities Routledge New YorkNY USA 2017

[51] T Parsons The Structure of Social Action The Free Press NewYork NY USA 1968

[52] T Allen Managing the Flow of Technology MIT Press Cam-bridge MA 1977

[53] F L Ribeiro J Meirelles F F Ferreira and R C Neto ldquoAmodel of urban scaling laws based on distance-dependent inter-actionsrdquo Royal Society Open Science vol 4 Article ID 1609262017

[54] W Alonso Location and Land Use Toward a General Theory ofLand Rent Harvard University Press Cambridge MA 1964

[55] WGHansen ldquoHow accessibility shapes land userdquo Journal of theAmerican Institute of Planners vol 25 no 2 pp 73ndash76 1959

[56] B Hillier Space Is the Machine Cambridge University PressCambridge UK 1996

[57] S Farber M OrsquoKelly H J Miller and T Neutens ldquoMeasuringsegregation using patterns of daily travel behavior a socialinteraction based model of exposurerdquo Journal of TransportGeography vol 49 pp 26ndash38 2015

[58] T C Schelling ldquoDynamic models of segregationrdquo Journal ofMathematical Sociology vol 1 no 2 pp 143ndash186 1971

[59] R Krafta V M Netto and L Lima ldquoUrban built form growscriticalrdquo CyberGeo vol 2011 article 565 2011

[60] N Lanchier and S Scarlatos ldquoFixation in the one-dimensionalAxelrod modelrdquo The Annals of Applied Probability vol 23 no6 pp 2538ndash2559 2013

[61] F L Ribeiro and K N Ribeiro ldquoA one dimensional modelof population growthrdquo Physica A Statistical Mechanics and itsApplications vol 434 pp 201ndash210 2015

[62] F L Ribeiro ldquoA non-phenomenological model of competitionand cooperation to explain population growth behaviorsrdquo Bul-letin of Mathematical Biology vol 77 no 3 pp 409ndash433 2015

16 Complexity

[63] R V dos Santos F L Ribeiro and A S Martinez ldquoModels forAllee effect based on physical principlesrdquo Journal of TheoreticalBiology vol 385 pp 143ndash152 2015

[64] M BattyThe New Science of Cities The MIT Press CambridgeMA 2013

[65] A G Wilson Entropy in Urban and Regional Modelling PionLondon 1970

[66] M Batty R Morphet P Masucci and K Stanilov ldquoEntropycomplexity and spatial informationrdquo Journal of GeographicalSystems vol 16 no 4 pp 363ndash385 2014

[67] R Axelrod ldquoThe dissemination of culture a model with localconvergence and global polarizationrdquo Journal of Conflict Reso-lution vol 41 no 2 pp 203ndash226 1997

[68] B H Hodges and RM Baron ldquoValues as constraints on afford-ances perceiving and acting properlyrdquo Journal for the Theory ofSocial Behaviour vol 22 no 3 pp 263ndash294 1992

[69] J JacobsThe Economy of Cities Random house New York NYUSA 1969

[70] E Glaeser H Kallal J Scheinkman and A Shleifer ldquoGrowthin Citiesrdquo Journal of Political Economy vol 100 no 6 pp 1126ndash1152 1992

[71] S Rosenthal and W Strange ldquoEvidence on the nature andsources of agglomeration economiesrdquo in Handbook of Urbanand Regional Economics J V Henderson and J-F Thisse Edsvol 4 pp 2119-2171 NorthHollandNewYork NYUSA 2004

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Stochastic AnalysisInternational Journal of

8 Complexity

Momentarilycoherent systems

State of unpredictabledecisions

Action systems

Reduction of entropy

Coordination of actions

Selectionof actions

High entropy state

Information growth(Hidalgo 2015)

Possibilities of action(Luhmann 1995)

Differences in theprobability ofinteractions

(Shannon 1948)

Agents interact Orientations align inactivity places

Actions mediated byinformation

(Luhmann 1995)

City as informationenvironment(Haken and Portugali 2015)

Figure 3 Recursive process that conducts interaction systems from chaotic states (high entropy) into coherence (low entropy)

(a) (b) (c)

(e) (d)

Figure 4 The cycle of social entropy and space clockwise from the high unpredictability of potential actions to a momentarily orderedsystem as actions converge into places only to dive in entropy again

Complexity 9

Actionagent x

Actionagent y

Activity place

Cooperation

Change insubjectiveorientation

Figure 5 Aligning different actions in the cooperation process

be actualized (e) And then actions ldquoreferrdquo again to meaningsand positions in space starting a new cycle of entropy(Figure 4)

These distinct moments are theoretical of course Inreality reminding Prigogine and Stengers [6 page 124] thisldquomultitude of eventsrdquo takes place simultaneously and mayrender any clearly emergent cycle invisible But that neithermeans that individual cycles do not happen or aggregate inbunches nor they compensate one another like peaks andtroughs in colliding sound waves In fact routinization indaily life leads to rhythms and cycles unfolding from orfolding over the seemingly chaotic maze of actions But evenif actions were far from synchronic the fabric of interactionstill faces the challenge of entropy that is the challenge ofproducing differences in the probability of connections as a wayto ease organization whenever agents feel compelled to createnew interactions and this process is remarkably recursiveWe converge to places to act with others and do so on adaily basis Once engaged in an activity place interactionrequires levels of cooperationWemay bring our own distincttendencies or latent orientations in the form of long-termmemories of how to act along with short-term memoriesof a previous action both shaping our next actions As weact socially we bring our orientations into contact in away to open them in the communication process regulatingcollective action A result of this coordination is to reduceinitial differences Our actions ldquoalignrdquowithin an activity place(Figure 5)

All this suggests that social life involves a continuousalthough unconscious handling of entropy a vital oscillationbetween unpredictability and predictability As this processengages an enormous number of agents and events andtranscends places and contexts and therefore cannot beobserved as a whole entropy can only be fully grasped intheoretical representations A useful way to examine space asan information environment for action is through a compu-tational model a means to examine a counterfactual worldwhere other steering media such as language or social ruleswere ldquoswitched offrdquo keeping only properties of space activealong with agentsrsquo cognitive abilities to align orientationswith activities Once we abstract from other media wemay assess whether space could have any presence in theself-organization of action Let us examine this possibilitythrough some experiments

LBuildingactivity type

Figure 6The theoretical one-dimensional citywith a set of differentactivity places Here there are 119871 buildings distributed in a ringconstituting the city space Colors represent activity types

6 Digital ExperimentsA Unidimensional Model

61 Design Concepts An agent based model (ABM) is pro-posed Agents are defined to perform actions at each time-step and places by corresponding activity types The decisionon which action to perform next may be influenced bythree different conditions namely (i) latent orientation120590in

119894(119905)

representing the tendency of a single agent to act arounda particular type of action initially randomly distributedthis condition remains over time (ii) current action 120590

119894(119905)

performed by an agent as heshe selects an activity placein order to perform a new action this means an influenceof the current action while allowing gradual changes oforientation in time influenced by other agents and activitiesand (iii) types of activity places 120590

1199091015840(119905) where agents perform

supporting specific types of action Agents follow simplerules selecting their activity places closer to their latentorientations and current actions while being influencedby those activities Activity places are also influenced byvisiting agents but they change at a slower rate In shortagents coevolve with their spatial and social environmentThe model simulates different situations under this basicstructure Differences in the weight of these factors overthe next action may lead to quite different levels of socialentropy suggesting new possibilities for understanding thephenomenonAlthough there is a tradition inmodelling dailyactivities including a growing literature using digital loca-tional data (eg [57]) our ABM is not designed to representempirical sequences of actions or actual routines Instead itfocuses on trends that might emerge from the interfaces ofsimplified systems of action information and space

62 Variables and Scale Consider a unidimensional cityformed by agents and places Places form a ring with length(or perimeter) 119871 as shown in Figure 6 Our choice forrepresenting the city by a ring is justified by the minimalsufficient representation of spatial distance as a factor whose

10 Complexity

role in social organization is to be assessed This is allowedby a unidimensional model (eg [58ndash60]) The ring formallows continuous movement across a linear sequence oflocations eliminating centrality factors while taking intoaccount periodic boundary conditions in order to reduceborder effects eliminating the role of topologywhile isolatingthe problem of distance (see [61ndash63]) This stylized cityhouses 119873 agents which in each time-step 119905 select and visita specific place located at the position 119909 within the cityConsider that the position of the 119894th agent at the time 119905 isrepresented by 119909

119894(119905) The position of an agent can assume the

integer values 1 2 119871 according to the place that shehechooses to visit This means that at every time-step eachagent will choose one activity place to perform her nextaction

We considered 120590119894(119905) as the current action of the 119894-th agent

and 120590119909(119905) the activity performed in a place located at the

position 119909 = (1 2 119871) of the city both in time 119905 Wequantify action orientations weighting the three variables(latent orientation current action and activity place) Firstvariables are translated into different numeric values Let usassume that an orientation is an integer number between 0and 1000 each number represents a particular type of actionand numbers in their vicinity mean similar actions Theresulting orientation in time 119905 for each agent is the weightedaverage of those three parametersThedifference |120590

119894(119905)minus120590119909(119905)|

gives us the difference in orientation between an agent andthe activity place At every time-step each agent assesses thedifference between her orientation and every activity placein the city An agent chooses a specific place based on herorientation affinity with this place minimum difference

As we hypothesized above the urban structure has a rolein this selection In this experiment we reduced ldquourban struc-turerdquo to distance In order to investigate how distance may ornot interfere in this selection we propose two scenarios Inthe first one the agent selects her next activity based only onthe similarity in orientations between herself and the activityplace Agents can move across a space free of friction Spaceis not a constraintThis scenario considers only differences inorientation The parameter 119864

119894(119909 119905) estimates the interaction

between agent 119894 and the place located at 119909 at the time 119905 Theagent selects activity locations that minimize this quantityWe propose

119864119894 (119909 119905) =

1003816100381610038161003816120590119894 (119905) minus 1205901199091003816100381610038161003816 (1)

In the second more realistic scenario space imposesfriction to movement The agent takes into account thephysical distance between her current location in the city andthe location of her next activity and opts to minimize thisdistance along with the difference between her orientationand that of the activity place We propose

119864119894 (119909 119905) =

1003816100381610038161003816(119909119894 (119905) minus 119909) (120590119894 (119905) minus 120590119909)1003816100381610038161003816 (2)

Summing up an agent 119894 selects at time 119905 a particularactivity place located at119909 thatminimizes the function119864

119894(119909 119905)

In the first scenario she will choose activity places closerto her orientation In the second she will choose activityplaces closer in both orientation and physical distance This

brings a form of ldquoenergy costrdquo into the factors consideredSpace takes on a function role because of the energy used inldquovisitingrdquo Agents move in this linear city willing to minimizethis energy while searching for social information latentin places close to their changing orientations Changes inentropy levels are derived from this energy function In itsspatial version the model assesses effects of the effort toreduce energy in action minimizing informational differ-ences (between orientations and activity places) and spatialdistances (between agents and activity places)

63TheEvolution of ActionOrientations Orientation evolvesin time according to the following rule When an agentchooses and joins a particular place both agent and placebecome a little closer in terms of orientation This meansthat activity places are not immune to what agents performin them and that the urban activity system also changes intime To illustrate how orientations are constantly updatedconsider that agent 119894 selects an activity place located at 1199091015840Orientations will be updated according the rule

120590119894 (119905 + 1) =

1120572 + 120573 + 120574

(120572120590in119894(119905) + 120573120590119894 (119905) + 1205741205901199091015840 (119905)) (3)

Selection also depends on three additional parameterslatent orientation 120590in

119894(119905) weighted by parameter 120572 current

orientation120590119894(119905)weighted by parameter120573 and type of activity

place 1205901199091015840(119905) weighted by parameter 120574 This means that agents

will consider the three factors previously described (120590in119894(119905)

120590119894(119905) 1205901199091015840(119905)) with different weights (120572 120573 120574) to estimate which

activity she wants to perform in the next time-stepThe latent orientation evolves from an initial randomly

distributed behaviour 1205900119894 We consider that 120590in

119894(119905) is normally

distributed around 1205900119894 meaning that every agent has an

ldquoaveragerdquo latent orientation whose value will be derived ateach time-step from a normal distribution peaking at theagentrsquos average orientation If 120572 is sufficiently small in relationto 120573 and 120574 then the latent orientation does not play a rolein the agentrsquos action However if 120572 is sufficiently large (oncein relation to 120573 and 120574) then the action becomes stronglydependent on the latent orientation This would lead to apretty conservative city where inhabitants are not open tochanges in behaviour

The current action of an agent may influence a newaction with an intensity that depends on parameter 120573 If thisparameter is small in comparison to 120572 and 120574 then the agenthas no memory of an ongoing orientation However if 120573 issufficiently large then the new action is strongly dependenton a previous one (a Markovian behaviour)

The intensity of the activity placersquos influence on the agentrsquosaction depends on the value of 120574 If this parameter is smallin comparison to 120572 and 120573 then the activity place does notaffect her next actionmeaning that the place does not changebehaviours However if this parameter is sufficiently largethen the place plays a strong influence on future actions Theactivity place will be updated according to the rule

1205901199091015840 (119905 + 1) = 1205901199091015840 (119905) + 120579sum

119894isin1199091015840

120590119894 (119905) (4)

Complexity 11

where 120579 is a parameter sufficiently small That is placesare less influenced by agents than the other way roundThis means that at every time-step a place would have itstype of activity closer to the average of orientations of itsvisitors Therefore the main activity of the place will changeslightly according to visiting agents while changing theirorientations

64 Frequency of Orientations and Entropy Levels We cal-culate entropy assessing the distribution of resulting orien-tations of all agents in the social system at each time-stepusing Shannon entropy Our quantitative interpretation ofinformation removes ambiguities from the relation betweenShannon information and semantic information as differentorientations actions and types of activity places are repre-sented by numerical values This allows us to estimate socialentropy through Shannon entropy The entropy of actions iscalculated as follows consider119873(120590 119905) as the number of agentswith orientation 120590 at the time 119905 (note that the total populationis 119873 = sum

120590119873(120590 119905)) We compute the frequency (or density)

120588(120590 119905) of this orientation within a population with

120588 (120590 119905) = 1119873119873 (120590 119905) (5)

Equation (5) calculates the probability of observing anorientation 120590 at time 119905 We compute the entropy level for anydistribution of orientations with

119878 (119905) = minussum120590

120588 (120590 119905) ln (120588 (120590 119905)) (6)

Equation (6) describes how uneven is the probability of find-ing different orientations Higher values mean that differentorientations have almost the same probability to happenwhile lower values indicate a system with clear orientationtrends The reduction of entropy implies that the probabilityof certain actions increases that is actions grow in similarityIn the limit as entropy falls to zero all agents in the systemwould reach the same orientation

65Model Procedure Themodel performs the following pro-cedures

(1) The city is created by generating the following

(a) The type of activity in places that is (120590119909=1(119905 =

0) 120590119909=2(119905 = 0) 120590

119909=119871(119905 = 0)) also generated

from a random distribution This step createsactivity places in the ring-city and assign themrandomly distributed activity values 120590

1199091015840(119905)

(b) The initial position of 119873 agents that is (1199091

1199092 119909

119873) and their respective orientations

(1205901(119905 = 0) 120590

2(119905 = 0) 120590

119873(119905 = 0)) generated

from a random distribution This step createsmoving agents in random locations and assignthem latent orientations based on the initialrandomly distributed orientation 120590in

119894(119905) = 120590

119894(0)

(2) At each time-step 119905 every agent chooses a place toact This place should minimize the function 119864

119894(119909 119905)

Suppose that the chosen place by 119894 is defined as 1199091015840Then agent and place update their orientations 120590

119894and

1205901199091015840 as described above

(3) Update the distribution of orientations 120588(120590 119905)(4) Compute Shannon entropy 119878(119905)

66 Results We have developed a number of observationson the behaviour of social systems working under differentparameters

(i) Entropy reduction is only found in scenarios wherespace imposes friction to movement that is wherethe spatial distance between agents and activity placesis considered as an active factor in selection Entropyrequires an extensive space in order to be reduced Butspace cannot do it alone Surprisingly space is a nec-essary but not sufficient condition Figure 7 comparestwo scenarios In the first one distance is an issue inthe selection of activities (red line)Thedistribution oforientations changes from a homogeneous one at thestart of simulation (a) to a nearly normal distributionat the end of simulation (b) where certain kindsof action are more likely to happen Informationalcontents in extensive space help aligning contents inactions In the second scenario agents move free ofspatial friction (blue line) Orientations are initiallyrandomly distributed (a) and continue to be so at theend of simulation (b) as a similar number of agentsare distributed along different orientationsThis resultsuggests that a materially active space becomes ameans for increasing the probability of certain inter-actions easing the collective coordination of action

Nowassessing the relative influence of social and personalfactors on the selection of a next action namely the socialinformation in activity places latent orientation and the cur-rent action under the influence of spatial friction (Figure 8)we can say the following

(i) There is structure in the relation between selectionfactors and the reduction of entropy Entropy beginsto fall and fluctuate around specific values accordingto different combinations of factors For some com-binations there is a great reduction of entropy Forothers the reduction is minimal almost negligibleFluctuations derive from the random component inthe decisionmaking of the agent (under the influenceof latent orientation)

(ii) A strong latent orientation (120572 sufficiently large) leadsto increasing entropy (blue lines in Figures 8 and9) since agents cannot align their actions with otheragents and activity places Like a long-term memorysystems whose actions are dictated by latent orienta-tions are likely to preserve initial orientations whichwere randomly distributed and therefore homoge-neous This condition is responsible for limiting thereduction of entropy

(iii) Current action alone does not shape new actions inany specific way (green lines in Figures 8 and 9)

12 Complexity

Start of simulation

00000

00005

00010

00015

00020

o

f tot

al ac

tions

250 500 750 10000Type of action

Materially inactive spaceMaterially active space

t = 0

(a)

End of simulation

00000

00005

00010

00015

00020

o

f tot

al ac

tions

250 500 750 10000Type of action

Materially inactive spaceMaterially active space

t = 50

(b)

Figure 7 Histogram of probability distribution of actors performing different types of action at the start (a) and at the end (b) of simulationsin scenarios where distance is not considered (blue) and distance is considered aligning orientations in action coordination (red) Resultsare averaged for 30 runs for 125 parameter combinations

activityplace

currentaction

latentorientation

225

250

275

300

325

Entr

opy

10 20 30 40 500Time

Figure 8 Evolution of entropy colors in a RGB scale refer toparameter weights in decisions on actions in a next time-step andtheir impact on entropy levels in different simulations (under thefriction of distance) In the palette triangle each color representsa specific combination of the three parameters In each vertex aparameter reaches maximum value while the others have minimumvalue Results are averaged for 30 different runs for 125 parametercombinations

It just keeps what is already happening workingas ldquoreinforcement feedbackrdquo for either direction theagent (and the system) is going Current actions aremeans to conserve tendencies in the system

(iv) Activity places play a key role in the reduction ofentropy (red lines in Figures 8 and 9)The social infor-mation in space ldquocontaminatesrdquo agents they aligntheir actions through the social contents of placesWe have seen that space matters as extension Nowwe see that space also has a very active informationalpresence

(v) Different combinations of factors have differenteffects over entropy Figure 8 shows the transitions

between the three parameters and how nuancedweights resulting from blending factors matter inthe reduction of entropy For instance blending theweight of activity places and latent orientation on thenext action displays a strong reduction of entropy(pink lines in Figure 8) That means that when agentsdo not keep information from previous selections(ie 120573 asymp 0) the action system reduces more entropythan in other parameter combinations A strongshort-term memory leads the system into conservingitself and into a poorer capability to coordinateactions Also when current actions 120573 and activityplaces 120574 share a similar weight over the next action(orange lines on top) the social system does notexperience a great reduction of entropy as agents tendto reproduce their actions

(vi) Finally the reduction of entropy implies that theprobability of certain actions and interactionsincreases In practical terms this means morealignments between agents and more connectionsbetween actions (interactions) However if all agentsin the system reached the same orientation andentropy dropped to zero the system would lose inter-nal differentiation Agents would behave in the sameway say in a world with no personal differentiationspecialization or division of labour New orientations(therefore entropy) are necessary if the social systemis to keep differentiated agencies On the otherhand a system with maximum entropy would createdifference in actions to a point where no action coor-dination or interaction would be possible Clearly thissituation cannot be the case A social system requiresbalances neither full entropy nor total predictability

These results also suggest that we are dealing with distinctldquosocial memoriesrdquo a long-term memory is active in what we

Complexity 13

20

25

30En

trop

y

04 0602-latent orientation

04 0602-current action

20

25

30

Entr

opy

20

25

30

Entr

opy

04 0602-activity place

Figure 9 Parameter exploration isolated importance of each parameter in entropy (under the friction of distance) Each vertical set of dotsrepresents one fixed value for the highlighted parameter with the two other parameters assuming every possible combination within thetested range Each dot corresponds to the entropy value when a sufficiently large time-frame is used

call latent orientations Short-term memory is active in theinfluence of a current orientation over a new action In turnthe social information latent in activity places is a result ofsocial arrangements (say as firms a local economy and soon) which find certain stability changing much slower thenactions It is an extension of the social system projected ontourban space stabilizing the system to some extent

7 Cities and Social Interaction Conclusions

What does this approach bring to the state of the art on therelations of social interaction information and space and onsocial entropy and ABM in particular We have seen thatprevious approaches overlooked the coordination of actionas a key empirical and analytical problem Furthermorethey tend to have the thin spatiality of a passive territorialbackground (eg [16 20]) In turn although the conceptof ldquocollective actionrdquo is finally getting attention in urbanstudies [64] the problem of social organization is still largelyunderestimated The problem of entropy has been dealt withsince Wilsonrsquos [65] work on spatial interaction but has notreached the mainstream of the discipline and deals mostlywith urban form (eg [24]) and spatial distributions [66]not with social information and action Regarding agentbased models we are not aware of an approach that dealswith the problem of how cities and space are part of socialorganization In a sense our model is closer to Axelrodrsquos[67] ABM model of dissemination of culture His agentsexchange ldquoculturerdquo through direct contact with neighboursin a cellular automata model whereas our agents are mobileand actively deal with distance and social information inspace recognizing types of activities in places The intensityof exchanges between agents in Axelrodrsquos model is a functionof similarities between them In our model exchanges aremediated by the selection and interaction with space In ourunderstanding our contribution in terms of ABM lies onthe fact that a new and simple model is proposed to dealwith the material and informational dimensions of socialorganization not just as an illustration but as a proof ofconcept space can have causal presence if agents generatepreferences as detailed in a domain that coevolves with them

Now does the simulation model corroborate the theoryintroduced in this paper Our proposition cannot be empir-ically validated at this stage given that it is very hard to

assess peoplersquos choices and the whole panorama of actionsin a city Entropy may well be beyond observation In thesecircumstances simulations of the behaviour of agents underdifferent spatial conditions become useful to assess entitiesevents and entropic forces at play One expected result fromthe model is that space matters for coordinating actions butsimulations surprisingly showed that space is a necessary butnot sufficient condition for the reduction of entropy Neithermaterial cognitive resources nor individual powers of agentsare enough Information in activity places and a bit of agentsrsquopersonal history also play roles in the way social systemsdeal with their own entropy That would mean a causal butnondeterministic presence of space

Substantively this approach explores a subtle but funda-mental presence of cities in social organizationWe argue thaturban space materializes gradients of difference in potentialinteractions from less to more recognizable costly or likelyPlacing simple criteria including distance as a factor in select-ing activities our ABM showed that space becomes a meansfor producing differences in the probabilities of interactionincreasing chances of certain selections and convergences incollective action

In reality people recognize the social information ofactivity places along with uses of busy streets or local central-ities We are able to retrieve these differences from the builtenvironment and relate to them One of the key points of ourargument is that these differences help us make selectionsEvery time we move through the city and select places toperform we participate in the large-scale coordination ofaction We enact space as a ldquoreferential systemrdquo [49 50]a set of semantic bits indexing social activities informingand guiding daily decisions Space becomes inherently partof the conversion of orientations into interactions that isconnections that crisscross sequences of actions recreatingthe interaction system By means of a structure that bothmaterializes and restricts the quasi-endless combinatorial pos-sibilities of interaction this system can acquire sufficientinternal guidance to make its own reproduction possibleBy distributing sufficiently recognizable differences in theprobability of interaction cities express and release localforces of reproduction of social systems The interfaces ofaction cognitive and spatial systems transform entropiccomplexity into structured complexity constantly reorderingactions in time

14 Complexity

This materialistic viewpoint can be integrated with otherviews of social organization Further work may include therelational roles of cultures language social norms and othersteering media along with explorations of possibly similareffects of communication technologies over entropy At thisstage we attempted to integrate a social dimension (agentsengaged in coordination) a spatial dimension (an extensiveenvironment) and an informational dimension (as differ-ences in action and in the environment itself) a demandinginterdisciplinary effort The approach identifies a central rolefor coevolving agents actualizing aims as they coordinate (cf[68]) and a central role for the environment exterior butresponsive to agents leading to cumulative changes alongtheir historyThemodel also identifies roles for cognition andmemory in the ways agents deal with their own orientationsand choose their actions and activity places changing theirspatial environment with consequences in the overall levelsof entropy All this suggests that the behaviour of the modelcannot be reduced to or fully predicted from the behaviour ofindividual variables which is a key aspect of complexity

So one of the main aims of this paper was to assesswhether space as information environment had any effecton the coordination and entropy of interaction However ifphysical distance and social information in activity placesmatter what about city size and the internal spatial struc-tures of cities What about rural areas or rarefied suburbsSince our model explores an abstract circular city it onlybegins to answer such questions Our findings indicate thatshorter distances between activity places tend to reducesocial entropy Density seems to matter denser spatialities(as opposed to rarefied ones) have a role to play A pathfor further development here is Jacobsrsquos [69] idea that thediversity of activities generates positive externalities Anddiversity as spatial economics shows us has to do withthe size of cities and density of population (eg [70 71])Entropy may also be influenced by diversity Batty et al [66]have shown that information increases as cities get biggerThe remaining question is whether larger denser and moreinternally structured cities could create process and reduceentropy with more intensity than smaller cities transformingbig pools of activities into differences in the probability ofinteraction Although we proposed a unidimensional modelour agentsrsquo behaviour can be explored in more realisticrepresentations of cities testing the roles of density topologyand diversity while going beyond perfectly informed agentsBy proposing the ring model we intended to simplify acity to a minimum system where the spatial dimensionof social organization becomes intelligible without losingfundamental properties

Shouldwe conclude fromall this that entropy is a problemfor social systems Our approach suggests that it is notEntropy is a necessary force It means that novel orientationsare entering the interaction system and must be dealt withEntropy only becomes a problem if not converted into orga-nization only if numerous potential actions are not convertedinto smaller sets of actual interactions The effects of newactions over entropy would be positive once actualizedhaving to dowith the diversity of agencyConservative systemsare likely to face less entropy but also likely to produce

less novelty having homogeneity as a problematic horizonFluctuations of entropy seem vital to complex societies

Finally our intention was to explore entropy as a meansto think about the conditions of social organization ldquofromanother anglerdquo so to speak from the viewpoint of challengesinvolved in social reproduction Hence the approach focusedon the idea of uncertainty surrounding agents dealing withaims and decisions and the different probabilities of inter-action that follow an attempt to bring social cognitiveinformational and spatial theories under a single roof Ourapproach sees cities as connective systems produced to createthe delicate fabric of interaction that keeps large numbersof agents coherently living together It also suggests that thevarying states of entropy reveal deep connections betweensocial informational and spatial systems

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

Theauthors would like to thank the Ecole Polytechnique Fed-erale de Lausanne (EPFL) CAPES FAPEMIG FAPESP andCNPq for financial support Maıra Pinheiro Henrique Loreaand colleagues at ABMUS2017 for earlier discussions andMike Batty for exchanges and inspiration Finally the authorsthank Romulo Krafta maverick of the complexity studiesof cities in Brazil for his input and support throughout theyears This work is dedicated to him

References

[1] C W Mills The Sociological Imagination Oxford UniversityPress New York NY USA 1959

[2] M Castells ldquoThe information age economy society and cul-turerdquo inThe Rise of the Network Blackwell London 1996

[3] C Hidalgo Why Information Grows The Evolution of Orderfrom Atoms to Economies Basic Books New York NY USA2015

[4] D Helbing The Automation of Society Is Next ETH ZurichSwitzerland 2016

[5] N Luhmann Social Systems University Press Stanford 1995[6] I Prigogine and I Stengers Order out of Chaos Manrsquos New

Dialogue with Nature Bantam Books 1984[7] C E Shannon ldquoAmathematical theory of communicationrdquo Bell

System Technical Journal vol 27 no 4 pp 623ndash656 1948[8] W Buckley Sociology and Modern Systems Theory Prentice-

Hall Englewood Cliffs NJ USA 1967[9] D D McFarland ldquoMeasuring the Permeability of Occupational

Structures An Information-Theoretic Approachrdquo AmericanJournal of Sociology vol 75 no 1 pp 41ndash61 1969

[10] F Charvat ldquoOn philosophical aspects of the system conceptionin contemporary sociological knowledgerdquo Quality and Quan-tity vol 6 no 1 pp 3ndash16 1972

[11] F Charvat J Kucera and M Soukup ldquoToward the systemtheory of dependence further general theoretical remarksrdquoQuality and Quantity vol 7 no 1 pp 69ndash90 1973

Complexity 15

[12] P M Horan ldquoInformation-theoretic measures and the analysisof social structuresrdquo Sociological Methods and Research vol 3no 3 pp 321ndash340 1975

[13] O E Klapp ldquoOpening and closing in open systemsrdquo BehavioralScience vol 20 no 4 pp 251ndash257 1975

[14] J Galtung ldquoEntropy and the general theory of peacerdquo in Essaysin Peace Research J Galtung Ed vol L Humanities PressAtlantic Highlands NJ USA 1975

[15] T Parsons The System of Modern Societies Englewood CliffsNew Jersey NJ USA 1972

[16] K D Bailey ldquoSociological entropy theory Toward a statisticaland verbal congruencerdquo Quality amp Quantity vol 18 no 1 pp113ndash133 1983

[17] K D Bailey Social EntropyTheory State University of NewYorkPress Albany NY USA 1990

[18] J G Miller Living Systems McGraw-Hill New York NY USA1978

[19] K D Bailey Sociology and the New Systems Theory Toward aTheoretical Synthesis State University of New York PressAlbany NY USA 1994

[20] K D Bailey ldquoSociocybernetics and social entropy theoryrdquoKybernetes vol 35 no 3-4 pp 375ndash384 2006

[21] F Geyer ldquoThe challenge of sociocyberneticsrdquo Kybernetes vol24 no 4 pp 6ndash32 1995

[22] F Geyer ldquoThe march of self-referencerdquo Kybernetes vol 31 no7-8 pp 1021ndash1042 2002

[23] J HabermasTheTheory of Communicative Action Polity PressCambridge UK 1987

[24] H Haken and J Portugali Information Adaptation The Inter-play between Shannon Information and Semantic Information inCognition Springer New York NY USA 2015

[25] M Weber Economy and Society vol 1 Berkeley University ofCalifornia Press 1978

[26] L Wittgenstein Philosophical Investigations Blackwell Publish-ers Oxford London 3rd edition 2001

[27] L Vygotsky Mind in Society The Development of HigherPsychological Processes Harvard University Press CambridgeMA 1978

[28] M Wilson ldquoSix Views of Embodied Cognitionrdquo PsychonomicBulletin amp Review vol 9 no 4 pp 625ndash636 2002

[29] A Clark and D Chalmers ldquoThe extended mindrdquo Analysis vol58 no 1 pp 7ndash19 1998

[30] R PassiniWayfinding in Architecture Van Nostrand ReinholdNew York NY USA 1992

[31] E Rosch ldquoPrinciples of categorizationrdquo in Cognition and Cate-gorization E Rosch and B Lloyd Eds Hillsdale LawrenceErlbaum Ass 1978

[32] W KintschMemory and Cognition John Wiley and Sons NewYork NY USA 1970

[33] U Neisser ldquoMultiple Systems a New Approach to CognitiveTheoryrdquo European Journal of Cognitive Psychology vol 6 no 3pp 225ndash241 1994

[34] A Faria and R Krafta ldquoRepresenting urban cognitive structurethrough spatial differentiationrdquo in Proceedings of 4th Space Syn-tax International Symposium pp 531-518 UCL Press London2003

[35] J Portugali Complexity Cognition and the City Springer NewYork NY USA 2011

[36] D Kirsh and P Maglio ldquoOn distinguishing epistemic frompragmatic actionrdquo Cognitive Science vol 18 no 4 pp 513ndash5491994

[37] S M Kosslyn Image and Brain The Resolution of the ImageryDebate MIT Press Cambridge MA

[38] R Brooks Cambrian Intelligence The Early History of the NewAI MIT Press Cambridge MA 1999

[39] A Clark BeinGThere Putting Brain Body AndWorld TogetherAgain MIT Press Cambridge MA 1997

[40] R Pfeifer and C ScheierUnderstanding Intelligence MIT PressCambridge MA 1999

[41] A M Glenberg and D A Robertson ldquoIndexical understandingof instructionsrdquoDiscourse Processes vol 28 no 1 pp 1ndash26 1999

[42] JM Iverson and SGoldin-Meadow ldquoWhypeople gesturewhenthey speakrdquo Nature vol 396 no 6708 p 228 1998

[43] R M Krauss ldquoWhy do we gesture when we speakrdquo CurrentDirections in Psychological Science vol 7 no 2 pp 54ndash60 1998

[44] S Franklin Artificial Minds MIT Press Cambridge MA 1995[45] J Gibson The Ecological Approach to Visual Perception

Houghton-Mifflin Boston 1979[46] G Lakoff andM JohnsonPhilosophy in the FleshTheEmbodied

Mind and Its Challenge to Western Thought Basic Books NewYork NY USA 1999

[47] E HutchinsCognition in theWild MIT Press CambridgeMA1995

[48] E Hutchins ldquoThe cultural ecosystem of human cognitionrdquoPhilosophical Psychology vol 27 no 1 pp 34ndash49 2014

[49] V M Netto ldquoPractice space and the duality of meaningrdquoEnvironment and Planning D Society and Space vol 26 no 2pp 359ndash379 2008

[50] V M Netto The Social Fabric of Cities Routledge New YorkNY USA 2017

[51] T Parsons The Structure of Social Action The Free Press NewYork NY USA 1968

[52] T Allen Managing the Flow of Technology MIT Press Cam-bridge MA 1977

[53] F L Ribeiro J Meirelles F F Ferreira and R C Neto ldquoAmodel of urban scaling laws based on distance-dependent inter-actionsrdquo Royal Society Open Science vol 4 Article ID 1609262017

[54] W Alonso Location and Land Use Toward a General Theory ofLand Rent Harvard University Press Cambridge MA 1964

[55] WGHansen ldquoHow accessibility shapes land userdquo Journal of theAmerican Institute of Planners vol 25 no 2 pp 73ndash76 1959

[56] B Hillier Space Is the Machine Cambridge University PressCambridge UK 1996

[57] S Farber M OrsquoKelly H J Miller and T Neutens ldquoMeasuringsegregation using patterns of daily travel behavior a socialinteraction based model of exposurerdquo Journal of TransportGeography vol 49 pp 26ndash38 2015

[58] T C Schelling ldquoDynamic models of segregationrdquo Journal ofMathematical Sociology vol 1 no 2 pp 143ndash186 1971

[59] R Krafta V M Netto and L Lima ldquoUrban built form growscriticalrdquo CyberGeo vol 2011 article 565 2011

[60] N Lanchier and S Scarlatos ldquoFixation in the one-dimensionalAxelrod modelrdquo The Annals of Applied Probability vol 23 no6 pp 2538ndash2559 2013

[61] F L Ribeiro and K N Ribeiro ldquoA one dimensional modelof population growthrdquo Physica A Statistical Mechanics and itsApplications vol 434 pp 201ndash210 2015

[62] F L Ribeiro ldquoA non-phenomenological model of competitionand cooperation to explain population growth behaviorsrdquo Bul-letin of Mathematical Biology vol 77 no 3 pp 409ndash433 2015

16 Complexity

[63] R V dos Santos F L Ribeiro and A S Martinez ldquoModels forAllee effect based on physical principlesrdquo Journal of TheoreticalBiology vol 385 pp 143ndash152 2015

[64] M BattyThe New Science of Cities The MIT Press CambridgeMA 2013

[65] A G Wilson Entropy in Urban and Regional Modelling PionLondon 1970

[66] M Batty R Morphet P Masucci and K Stanilov ldquoEntropycomplexity and spatial informationrdquo Journal of GeographicalSystems vol 16 no 4 pp 363ndash385 2014

[67] R Axelrod ldquoThe dissemination of culture a model with localconvergence and global polarizationrdquo Journal of Conflict Reso-lution vol 41 no 2 pp 203ndash226 1997

[68] B H Hodges and RM Baron ldquoValues as constraints on afford-ances perceiving and acting properlyrdquo Journal for the Theory ofSocial Behaviour vol 22 no 3 pp 263ndash294 1992

[69] J JacobsThe Economy of Cities Random house New York NYUSA 1969

[70] E Glaeser H Kallal J Scheinkman and A Shleifer ldquoGrowthin Citiesrdquo Journal of Political Economy vol 100 no 6 pp 1126ndash1152 1992

[71] S Rosenthal and W Strange ldquoEvidence on the nature andsources of agglomeration economiesrdquo in Handbook of Urbanand Regional Economics J V Henderson and J-F Thisse Edsvol 4 pp 2119-2171 NorthHollandNewYork NYUSA 2004

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

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Mathematical PhysicsAdvances in

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Operations ResearchAdvances in

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

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Algebra

Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Complexity 9

Actionagent x

Actionagent y

Activity place

Cooperation

Change insubjectiveorientation

Figure 5 Aligning different actions in the cooperation process

be actualized (e) And then actions ldquoreferrdquo again to meaningsand positions in space starting a new cycle of entropy(Figure 4)

These distinct moments are theoretical of course Inreality reminding Prigogine and Stengers [6 page 124] thisldquomultitude of eventsrdquo takes place simultaneously and mayrender any clearly emergent cycle invisible But that neithermeans that individual cycles do not happen or aggregate inbunches nor they compensate one another like peaks andtroughs in colliding sound waves In fact routinization indaily life leads to rhythms and cycles unfolding from orfolding over the seemingly chaotic maze of actions But evenif actions were far from synchronic the fabric of interactionstill faces the challenge of entropy that is the challenge ofproducing differences in the probability of connections as a wayto ease organization whenever agents feel compelled to createnew interactions and this process is remarkably recursiveWe converge to places to act with others and do so on adaily basis Once engaged in an activity place interactionrequires levels of cooperationWemay bring our own distincttendencies or latent orientations in the form of long-termmemories of how to act along with short-term memoriesof a previous action both shaping our next actions As weact socially we bring our orientations into contact in away to open them in the communication process regulatingcollective action A result of this coordination is to reduceinitial differences Our actions ldquoalignrdquowithin an activity place(Figure 5)

All this suggests that social life involves a continuousalthough unconscious handling of entropy a vital oscillationbetween unpredictability and predictability As this processengages an enormous number of agents and events andtranscends places and contexts and therefore cannot beobserved as a whole entropy can only be fully grasped intheoretical representations A useful way to examine space asan information environment for action is through a compu-tational model a means to examine a counterfactual worldwhere other steering media such as language or social ruleswere ldquoswitched offrdquo keeping only properties of space activealong with agentsrsquo cognitive abilities to align orientationswith activities Once we abstract from other media wemay assess whether space could have any presence in theself-organization of action Let us examine this possibilitythrough some experiments

LBuildingactivity type

Figure 6The theoretical one-dimensional citywith a set of differentactivity places Here there are 119871 buildings distributed in a ringconstituting the city space Colors represent activity types

6 Digital ExperimentsA Unidimensional Model

61 Design Concepts An agent based model (ABM) is pro-posed Agents are defined to perform actions at each time-step and places by corresponding activity types The decisionon which action to perform next may be influenced bythree different conditions namely (i) latent orientation120590in

119894(119905)

representing the tendency of a single agent to act arounda particular type of action initially randomly distributedthis condition remains over time (ii) current action 120590

119894(119905)

performed by an agent as heshe selects an activity placein order to perform a new action this means an influenceof the current action while allowing gradual changes oforientation in time influenced by other agents and activitiesand (iii) types of activity places 120590

1199091015840(119905) where agents perform

supporting specific types of action Agents follow simplerules selecting their activity places closer to their latentorientations and current actions while being influencedby those activities Activity places are also influenced byvisiting agents but they change at a slower rate In shortagents coevolve with their spatial and social environmentThe model simulates different situations under this basicstructure Differences in the weight of these factors overthe next action may lead to quite different levels of socialentropy suggesting new possibilities for understanding thephenomenonAlthough there is a tradition inmodelling dailyactivities including a growing literature using digital loca-tional data (eg [57]) our ABM is not designed to representempirical sequences of actions or actual routines Instead itfocuses on trends that might emerge from the interfaces ofsimplified systems of action information and space

62 Variables and Scale Consider a unidimensional cityformed by agents and places Places form a ring with length(or perimeter) 119871 as shown in Figure 6 Our choice forrepresenting the city by a ring is justified by the minimalsufficient representation of spatial distance as a factor whose

10 Complexity

role in social organization is to be assessed This is allowedby a unidimensional model (eg [58ndash60]) The ring formallows continuous movement across a linear sequence oflocations eliminating centrality factors while taking intoaccount periodic boundary conditions in order to reduceborder effects eliminating the role of topologywhile isolatingthe problem of distance (see [61ndash63]) This stylized cityhouses 119873 agents which in each time-step 119905 select and visita specific place located at the position 119909 within the cityConsider that the position of the 119894th agent at the time 119905 isrepresented by 119909

119894(119905) The position of an agent can assume the

integer values 1 2 119871 according to the place that shehechooses to visit This means that at every time-step eachagent will choose one activity place to perform her nextaction

We considered 120590119894(119905) as the current action of the 119894-th agent

and 120590119909(119905) the activity performed in a place located at the

position 119909 = (1 2 119871) of the city both in time 119905 Wequantify action orientations weighting the three variables(latent orientation current action and activity place) Firstvariables are translated into different numeric values Let usassume that an orientation is an integer number between 0and 1000 each number represents a particular type of actionand numbers in their vicinity mean similar actions Theresulting orientation in time 119905 for each agent is the weightedaverage of those three parametersThedifference |120590

119894(119905)minus120590119909(119905)|

gives us the difference in orientation between an agent andthe activity place At every time-step each agent assesses thedifference between her orientation and every activity placein the city An agent chooses a specific place based on herorientation affinity with this place minimum difference

As we hypothesized above the urban structure has a rolein this selection In this experiment we reduced ldquourban struc-turerdquo to distance In order to investigate how distance may ornot interfere in this selection we propose two scenarios Inthe first one the agent selects her next activity based only onthe similarity in orientations between herself and the activityplace Agents can move across a space free of friction Spaceis not a constraintThis scenario considers only differences inorientation The parameter 119864

119894(119909 119905) estimates the interaction

between agent 119894 and the place located at 119909 at the time 119905 Theagent selects activity locations that minimize this quantityWe propose

119864119894 (119909 119905) =

1003816100381610038161003816120590119894 (119905) minus 1205901199091003816100381610038161003816 (1)

In the second more realistic scenario space imposesfriction to movement The agent takes into account thephysical distance between her current location in the city andthe location of her next activity and opts to minimize thisdistance along with the difference between her orientationand that of the activity place We propose

119864119894 (119909 119905) =

1003816100381610038161003816(119909119894 (119905) minus 119909) (120590119894 (119905) minus 120590119909)1003816100381610038161003816 (2)

Summing up an agent 119894 selects at time 119905 a particularactivity place located at119909 thatminimizes the function119864

119894(119909 119905)

In the first scenario she will choose activity places closerto her orientation In the second she will choose activityplaces closer in both orientation and physical distance This

brings a form of ldquoenergy costrdquo into the factors consideredSpace takes on a function role because of the energy used inldquovisitingrdquo Agents move in this linear city willing to minimizethis energy while searching for social information latentin places close to their changing orientations Changes inentropy levels are derived from this energy function In itsspatial version the model assesses effects of the effort toreduce energy in action minimizing informational differ-ences (between orientations and activity places) and spatialdistances (between agents and activity places)

63TheEvolution of ActionOrientations Orientation evolvesin time according to the following rule When an agentchooses and joins a particular place both agent and placebecome a little closer in terms of orientation This meansthat activity places are not immune to what agents performin them and that the urban activity system also changes intime To illustrate how orientations are constantly updatedconsider that agent 119894 selects an activity place located at 1199091015840Orientations will be updated according the rule

120590119894 (119905 + 1) =

1120572 + 120573 + 120574

(120572120590in119894(119905) + 120573120590119894 (119905) + 1205741205901199091015840 (119905)) (3)

Selection also depends on three additional parameterslatent orientation 120590in

119894(119905) weighted by parameter 120572 current

orientation120590119894(119905)weighted by parameter120573 and type of activity

place 1205901199091015840(119905) weighted by parameter 120574 This means that agents

will consider the three factors previously described (120590in119894(119905)

120590119894(119905) 1205901199091015840(119905)) with different weights (120572 120573 120574) to estimate which

activity she wants to perform in the next time-stepThe latent orientation evolves from an initial randomly

distributed behaviour 1205900119894 We consider that 120590in

119894(119905) is normally

distributed around 1205900119894 meaning that every agent has an

ldquoaveragerdquo latent orientation whose value will be derived ateach time-step from a normal distribution peaking at theagentrsquos average orientation If 120572 is sufficiently small in relationto 120573 and 120574 then the latent orientation does not play a rolein the agentrsquos action However if 120572 is sufficiently large (oncein relation to 120573 and 120574) then the action becomes stronglydependent on the latent orientation This would lead to apretty conservative city where inhabitants are not open tochanges in behaviour

The current action of an agent may influence a newaction with an intensity that depends on parameter 120573 If thisparameter is small in comparison to 120572 and 120574 then the agenthas no memory of an ongoing orientation However if 120573 issufficiently large then the new action is strongly dependenton a previous one (a Markovian behaviour)

The intensity of the activity placersquos influence on the agentrsquosaction depends on the value of 120574 If this parameter is smallin comparison to 120572 and 120573 then the activity place does notaffect her next actionmeaning that the place does not changebehaviours However if this parameter is sufficiently largethen the place plays a strong influence on future actions Theactivity place will be updated according to the rule

1205901199091015840 (119905 + 1) = 1205901199091015840 (119905) + 120579sum

119894isin1199091015840

120590119894 (119905) (4)

Complexity 11

where 120579 is a parameter sufficiently small That is placesare less influenced by agents than the other way roundThis means that at every time-step a place would have itstype of activity closer to the average of orientations of itsvisitors Therefore the main activity of the place will changeslightly according to visiting agents while changing theirorientations

64 Frequency of Orientations and Entropy Levels We cal-culate entropy assessing the distribution of resulting orien-tations of all agents in the social system at each time-stepusing Shannon entropy Our quantitative interpretation ofinformation removes ambiguities from the relation betweenShannon information and semantic information as differentorientations actions and types of activity places are repre-sented by numerical values This allows us to estimate socialentropy through Shannon entropy The entropy of actions iscalculated as follows consider119873(120590 119905) as the number of agentswith orientation 120590 at the time 119905 (note that the total populationis 119873 = sum

120590119873(120590 119905)) We compute the frequency (or density)

120588(120590 119905) of this orientation within a population with

120588 (120590 119905) = 1119873119873 (120590 119905) (5)

Equation (5) calculates the probability of observing anorientation 120590 at time 119905 We compute the entropy level for anydistribution of orientations with

119878 (119905) = minussum120590

120588 (120590 119905) ln (120588 (120590 119905)) (6)

Equation (6) describes how uneven is the probability of find-ing different orientations Higher values mean that differentorientations have almost the same probability to happenwhile lower values indicate a system with clear orientationtrends The reduction of entropy implies that the probabilityof certain actions increases that is actions grow in similarityIn the limit as entropy falls to zero all agents in the systemwould reach the same orientation

65Model Procedure Themodel performs the following pro-cedures

(1) The city is created by generating the following

(a) The type of activity in places that is (120590119909=1(119905 =

0) 120590119909=2(119905 = 0) 120590

119909=119871(119905 = 0)) also generated

from a random distribution This step createsactivity places in the ring-city and assign themrandomly distributed activity values 120590

1199091015840(119905)

(b) The initial position of 119873 agents that is (1199091

1199092 119909

119873) and their respective orientations

(1205901(119905 = 0) 120590

2(119905 = 0) 120590

119873(119905 = 0)) generated

from a random distribution This step createsmoving agents in random locations and assignthem latent orientations based on the initialrandomly distributed orientation 120590in

119894(119905) = 120590

119894(0)

(2) At each time-step 119905 every agent chooses a place toact This place should minimize the function 119864

119894(119909 119905)

Suppose that the chosen place by 119894 is defined as 1199091015840Then agent and place update their orientations 120590

119894and

1205901199091015840 as described above

(3) Update the distribution of orientations 120588(120590 119905)(4) Compute Shannon entropy 119878(119905)

66 Results We have developed a number of observationson the behaviour of social systems working under differentparameters

(i) Entropy reduction is only found in scenarios wherespace imposes friction to movement that is wherethe spatial distance between agents and activity placesis considered as an active factor in selection Entropyrequires an extensive space in order to be reduced Butspace cannot do it alone Surprisingly space is a nec-essary but not sufficient condition Figure 7 comparestwo scenarios In the first one distance is an issue inthe selection of activities (red line)Thedistribution oforientations changes from a homogeneous one at thestart of simulation (a) to a nearly normal distributionat the end of simulation (b) where certain kindsof action are more likely to happen Informationalcontents in extensive space help aligning contents inactions In the second scenario agents move free ofspatial friction (blue line) Orientations are initiallyrandomly distributed (a) and continue to be so at theend of simulation (b) as a similar number of agentsare distributed along different orientationsThis resultsuggests that a materially active space becomes ameans for increasing the probability of certain inter-actions easing the collective coordination of action

Nowassessing the relative influence of social and personalfactors on the selection of a next action namely the socialinformation in activity places latent orientation and the cur-rent action under the influence of spatial friction (Figure 8)we can say the following

(i) There is structure in the relation between selectionfactors and the reduction of entropy Entropy beginsto fall and fluctuate around specific values accordingto different combinations of factors For some com-binations there is a great reduction of entropy Forothers the reduction is minimal almost negligibleFluctuations derive from the random component inthe decisionmaking of the agent (under the influenceof latent orientation)

(ii) A strong latent orientation (120572 sufficiently large) leadsto increasing entropy (blue lines in Figures 8 and9) since agents cannot align their actions with otheragents and activity places Like a long-term memorysystems whose actions are dictated by latent orienta-tions are likely to preserve initial orientations whichwere randomly distributed and therefore homoge-neous This condition is responsible for limiting thereduction of entropy

(iii) Current action alone does not shape new actions inany specific way (green lines in Figures 8 and 9)

12 Complexity

Start of simulation

00000

00005

00010

00015

00020

o

f tot

al ac

tions

250 500 750 10000Type of action

Materially inactive spaceMaterially active space

t = 0

(a)

End of simulation

00000

00005

00010

00015

00020

o

f tot

al ac

tions

250 500 750 10000Type of action

Materially inactive spaceMaterially active space

t = 50

(b)

Figure 7 Histogram of probability distribution of actors performing different types of action at the start (a) and at the end (b) of simulationsin scenarios where distance is not considered (blue) and distance is considered aligning orientations in action coordination (red) Resultsare averaged for 30 runs for 125 parameter combinations

activityplace

currentaction

latentorientation

225

250

275

300

325

Entr

opy

10 20 30 40 500Time

Figure 8 Evolution of entropy colors in a RGB scale refer toparameter weights in decisions on actions in a next time-step andtheir impact on entropy levels in different simulations (under thefriction of distance) In the palette triangle each color representsa specific combination of the three parameters In each vertex aparameter reaches maximum value while the others have minimumvalue Results are averaged for 30 different runs for 125 parametercombinations

It just keeps what is already happening workingas ldquoreinforcement feedbackrdquo for either direction theagent (and the system) is going Current actions aremeans to conserve tendencies in the system

(iv) Activity places play a key role in the reduction ofentropy (red lines in Figures 8 and 9)The social infor-mation in space ldquocontaminatesrdquo agents they aligntheir actions through the social contents of placesWe have seen that space matters as extension Nowwe see that space also has a very active informationalpresence

(v) Different combinations of factors have differenteffects over entropy Figure 8 shows the transitions

between the three parameters and how nuancedweights resulting from blending factors matter inthe reduction of entropy For instance blending theweight of activity places and latent orientation on thenext action displays a strong reduction of entropy(pink lines in Figure 8) That means that when agentsdo not keep information from previous selections(ie 120573 asymp 0) the action system reduces more entropythan in other parameter combinations A strongshort-term memory leads the system into conservingitself and into a poorer capability to coordinateactions Also when current actions 120573 and activityplaces 120574 share a similar weight over the next action(orange lines on top) the social system does notexperience a great reduction of entropy as agents tendto reproduce their actions

(vi) Finally the reduction of entropy implies that theprobability of certain actions and interactionsincreases In practical terms this means morealignments between agents and more connectionsbetween actions (interactions) However if all agentsin the system reached the same orientation andentropy dropped to zero the system would lose inter-nal differentiation Agents would behave in the sameway say in a world with no personal differentiationspecialization or division of labour New orientations(therefore entropy) are necessary if the social systemis to keep differentiated agencies On the otherhand a system with maximum entropy would createdifference in actions to a point where no action coor-dination or interaction would be possible Clearly thissituation cannot be the case A social system requiresbalances neither full entropy nor total predictability

These results also suggest that we are dealing with distinctldquosocial memoriesrdquo a long-term memory is active in what we

Complexity 13

20

25

30En

trop

y

04 0602-latent orientation

04 0602-current action

20

25

30

Entr

opy

20

25

30

Entr

opy

04 0602-activity place

Figure 9 Parameter exploration isolated importance of each parameter in entropy (under the friction of distance) Each vertical set of dotsrepresents one fixed value for the highlighted parameter with the two other parameters assuming every possible combination within thetested range Each dot corresponds to the entropy value when a sufficiently large time-frame is used

call latent orientations Short-term memory is active in theinfluence of a current orientation over a new action In turnthe social information latent in activity places is a result ofsocial arrangements (say as firms a local economy and soon) which find certain stability changing much slower thenactions It is an extension of the social system projected ontourban space stabilizing the system to some extent

7 Cities and Social Interaction Conclusions

What does this approach bring to the state of the art on therelations of social interaction information and space and onsocial entropy and ABM in particular We have seen thatprevious approaches overlooked the coordination of actionas a key empirical and analytical problem Furthermorethey tend to have the thin spatiality of a passive territorialbackground (eg [16 20]) In turn although the conceptof ldquocollective actionrdquo is finally getting attention in urbanstudies [64] the problem of social organization is still largelyunderestimated The problem of entropy has been dealt withsince Wilsonrsquos [65] work on spatial interaction but has notreached the mainstream of the discipline and deals mostlywith urban form (eg [24]) and spatial distributions [66]not with social information and action Regarding agentbased models we are not aware of an approach that dealswith the problem of how cities and space are part of socialorganization In a sense our model is closer to Axelrodrsquos[67] ABM model of dissemination of culture His agentsexchange ldquoculturerdquo through direct contact with neighboursin a cellular automata model whereas our agents are mobileand actively deal with distance and social information inspace recognizing types of activities in places The intensityof exchanges between agents in Axelrodrsquos model is a functionof similarities between them In our model exchanges aremediated by the selection and interaction with space In ourunderstanding our contribution in terms of ABM lies onthe fact that a new and simple model is proposed to dealwith the material and informational dimensions of socialorganization not just as an illustration but as a proof ofconcept space can have causal presence if agents generatepreferences as detailed in a domain that coevolves with them

Now does the simulation model corroborate the theoryintroduced in this paper Our proposition cannot be empir-ically validated at this stage given that it is very hard to

assess peoplersquos choices and the whole panorama of actionsin a city Entropy may well be beyond observation In thesecircumstances simulations of the behaviour of agents underdifferent spatial conditions become useful to assess entitiesevents and entropic forces at play One expected result fromthe model is that space matters for coordinating actions butsimulations surprisingly showed that space is a necessary butnot sufficient condition for the reduction of entropy Neithermaterial cognitive resources nor individual powers of agentsare enough Information in activity places and a bit of agentsrsquopersonal history also play roles in the way social systemsdeal with their own entropy That would mean a causal butnondeterministic presence of space

Substantively this approach explores a subtle but funda-mental presence of cities in social organizationWe argue thaturban space materializes gradients of difference in potentialinteractions from less to more recognizable costly or likelyPlacing simple criteria including distance as a factor in select-ing activities our ABM showed that space becomes a meansfor producing differences in the probabilities of interactionincreasing chances of certain selections and convergences incollective action

In reality people recognize the social information ofactivity places along with uses of busy streets or local central-ities We are able to retrieve these differences from the builtenvironment and relate to them One of the key points of ourargument is that these differences help us make selectionsEvery time we move through the city and select places toperform we participate in the large-scale coordination ofaction We enact space as a ldquoreferential systemrdquo [49 50]a set of semantic bits indexing social activities informingand guiding daily decisions Space becomes inherently partof the conversion of orientations into interactions that isconnections that crisscross sequences of actions recreatingthe interaction system By means of a structure that bothmaterializes and restricts the quasi-endless combinatorial pos-sibilities of interaction this system can acquire sufficientinternal guidance to make its own reproduction possibleBy distributing sufficiently recognizable differences in theprobability of interaction cities express and release localforces of reproduction of social systems The interfaces ofaction cognitive and spatial systems transform entropiccomplexity into structured complexity constantly reorderingactions in time

14 Complexity

This materialistic viewpoint can be integrated with otherviews of social organization Further work may include therelational roles of cultures language social norms and othersteering media along with explorations of possibly similareffects of communication technologies over entropy At thisstage we attempted to integrate a social dimension (agentsengaged in coordination) a spatial dimension (an extensiveenvironment) and an informational dimension (as differ-ences in action and in the environment itself) a demandinginterdisciplinary effort The approach identifies a central rolefor coevolving agents actualizing aims as they coordinate (cf[68]) and a central role for the environment exterior butresponsive to agents leading to cumulative changes alongtheir historyThemodel also identifies roles for cognition andmemory in the ways agents deal with their own orientationsand choose their actions and activity places changing theirspatial environment with consequences in the overall levelsof entropy All this suggests that the behaviour of the modelcannot be reduced to or fully predicted from the behaviour ofindividual variables which is a key aspect of complexity

So one of the main aims of this paper was to assesswhether space as information environment had any effecton the coordination and entropy of interaction However ifphysical distance and social information in activity placesmatter what about city size and the internal spatial struc-tures of cities What about rural areas or rarefied suburbsSince our model explores an abstract circular city it onlybegins to answer such questions Our findings indicate thatshorter distances between activity places tend to reducesocial entropy Density seems to matter denser spatialities(as opposed to rarefied ones) have a role to play A pathfor further development here is Jacobsrsquos [69] idea that thediversity of activities generates positive externalities Anddiversity as spatial economics shows us has to do withthe size of cities and density of population (eg [70 71])Entropy may also be influenced by diversity Batty et al [66]have shown that information increases as cities get biggerThe remaining question is whether larger denser and moreinternally structured cities could create process and reduceentropy with more intensity than smaller cities transformingbig pools of activities into differences in the probability ofinteraction Although we proposed a unidimensional modelour agentsrsquo behaviour can be explored in more realisticrepresentations of cities testing the roles of density topologyand diversity while going beyond perfectly informed agentsBy proposing the ring model we intended to simplify acity to a minimum system where the spatial dimensionof social organization becomes intelligible without losingfundamental properties

Shouldwe conclude fromall this that entropy is a problemfor social systems Our approach suggests that it is notEntropy is a necessary force It means that novel orientationsare entering the interaction system and must be dealt withEntropy only becomes a problem if not converted into orga-nization only if numerous potential actions are not convertedinto smaller sets of actual interactions The effects of newactions over entropy would be positive once actualizedhaving to dowith the diversity of agencyConservative systemsare likely to face less entropy but also likely to produce

less novelty having homogeneity as a problematic horizonFluctuations of entropy seem vital to complex societies

Finally our intention was to explore entropy as a meansto think about the conditions of social organization ldquofromanother anglerdquo so to speak from the viewpoint of challengesinvolved in social reproduction Hence the approach focusedon the idea of uncertainty surrounding agents dealing withaims and decisions and the different probabilities of inter-action that follow an attempt to bring social cognitiveinformational and spatial theories under a single roof Ourapproach sees cities as connective systems produced to createthe delicate fabric of interaction that keeps large numbersof agents coherently living together It also suggests that thevarying states of entropy reveal deep connections betweensocial informational and spatial systems

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

Theauthors would like to thank the Ecole Polytechnique Fed-erale de Lausanne (EPFL) CAPES FAPEMIG FAPESP andCNPq for financial support Maıra Pinheiro Henrique Loreaand colleagues at ABMUS2017 for earlier discussions andMike Batty for exchanges and inspiration Finally the authorsthank Romulo Krafta maverick of the complexity studiesof cities in Brazil for his input and support throughout theyears This work is dedicated to him

References

[1] C W Mills The Sociological Imagination Oxford UniversityPress New York NY USA 1959

[2] M Castells ldquoThe information age economy society and cul-turerdquo inThe Rise of the Network Blackwell London 1996

[3] C Hidalgo Why Information Grows The Evolution of Orderfrom Atoms to Economies Basic Books New York NY USA2015

[4] D Helbing The Automation of Society Is Next ETH ZurichSwitzerland 2016

[5] N Luhmann Social Systems University Press Stanford 1995[6] I Prigogine and I Stengers Order out of Chaos Manrsquos New

Dialogue with Nature Bantam Books 1984[7] C E Shannon ldquoAmathematical theory of communicationrdquo Bell

System Technical Journal vol 27 no 4 pp 623ndash656 1948[8] W Buckley Sociology and Modern Systems Theory Prentice-

Hall Englewood Cliffs NJ USA 1967[9] D D McFarland ldquoMeasuring the Permeability of Occupational

Structures An Information-Theoretic Approachrdquo AmericanJournal of Sociology vol 75 no 1 pp 41ndash61 1969

[10] F Charvat ldquoOn philosophical aspects of the system conceptionin contemporary sociological knowledgerdquo Quality and Quan-tity vol 6 no 1 pp 3ndash16 1972

[11] F Charvat J Kucera and M Soukup ldquoToward the systemtheory of dependence further general theoretical remarksrdquoQuality and Quantity vol 7 no 1 pp 69ndash90 1973

Complexity 15

[12] P M Horan ldquoInformation-theoretic measures and the analysisof social structuresrdquo Sociological Methods and Research vol 3no 3 pp 321ndash340 1975

[13] O E Klapp ldquoOpening and closing in open systemsrdquo BehavioralScience vol 20 no 4 pp 251ndash257 1975

[14] J Galtung ldquoEntropy and the general theory of peacerdquo in Essaysin Peace Research J Galtung Ed vol L Humanities PressAtlantic Highlands NJ USA 1975

[15] T Parsons The System of Modern Societies Englewood CliffsNew Jersey NJ USA 1972

[16] K D Bailey ldquoSociological entropy theory Toward a statisticaland verbal congruencerdquo Quality amp Quantity vol 18 no 1 pp113ndash133 1983

[17] K D Bailey Social EntropyTheory State University of NewYorkPress Albany NY USA 1990

[18] J G Miller Living Systems McGraw-Hill New York NY USA1978

[19] K D Bailey Sociology and the New Systems Theory Toward aTheoretical Synthesis State University of New York PressAlbany NY USA 1994

[20] K D Bailey ldquoSociocybernetics and social entropy theoryrdquoKybernetes vol 35 no 3-4 pp 375ndash384 2006

[21] F Geyer ldquoThe challenge of sociocyberneticsrdquo Kybernetes vol24 no 4 pp 6ndash32 1995

[22] F Geyer ldquoThe march of self-referencerdquo Kybernetes vol 31 no7-8 pp 1021ndash1042 2002

[23] J HabermasTheTheory of Communicative Action Polity PressCambridge UK 1987

[24] H Haken and J Portugali Information Adaptation The Inter-play between Shannon Information and Semantic Information inCognition Springer New York NY USA 2015

[25] M Weber Economy and Society vol 1 Berkeley University ofCalifornia Press 1978

[26] L Wittgenstein Philosophical Investigations Blackwell Publish-ers Oxford London 3rd edition 2001

[27] L Vygotsky Mind in Society The Development of HigherPsychological Processes Harvard University Press CambridgeMA 1978

[28] M Wilson ldquoSix Views of Embodied Cognitionrdquo PsychonomicBulletin amp Review vol 9 no 4 pp 625ndash636 2002

[29] A Clark and D Chalmers ldquoThe extended mindrdquo Analysis vol58 no 1 pp 7ndash19 1998

[30] R PassiniWayfinding in Architecture Van Nostrand ReinholdNew York NY USA 1992

[31] E Rosch ldquoPrinciples of categorizationrdquo in Cognition and Cate-gorization E Rosch and B Lloyd Eds Hillsdale LawrenceErlbaum Ass 1978

[32] W KintschMemory and Cognition John Wiley and Sons NewYork NY USA 1970

[33] U Neisser ldquoMultiple Systems a New Approach to CognitiveTheoryrdquo European Journal of Cognitive Psychology vol 6 no 3pp 225ndash241 1994

[34] A Faria and R Krafta ldquoRepresenting urban cognitive structurethrough spatial differentiationrdquo in Proceedings of 4th Space Syn-tax International Symposium pp 531-518 UCL Press London2003

[35] J Portugali Complexity Cognition and the City Springer NewYork NY USA 2011

[36] D Kirsh and P Maglio ldquoOn distinguishing epistemic frompragmatic actionrdquo Cognitive Science vol 18 no 4 pp 513ndash5491994

[37] S M Kosslyn Image and Brain The Resolution of the ImageryDebate MIT Press Cambridge MA

[38] R Brooks Cambrian Intelligence The Early History of the NewAI MIT Press Cambridge MA 1999

[39] A Clark BeinGThere Putting Brain Body AndWorld TogetherAgain MIT Press Cambridge MA 1997

[40] R Pfeifer and C ScheierUnderstanding Intelligence MIT PressCambridge MA 1999

[41] A M Glenberg and D A Robertson ldquoIndexical understandingof instructionsrdquoDiscourse Processes vol 28 no 1 pp 1ndash26 1999

[42] JM Iverson and SGoldin-Meadow ldquoWhypeople gesturewhenthey speakrdquo Nature vol 396 no 6708 p 228 1998

[43] R M Krauss ldquoWhy do we gesture when we speakrdquo CurrentDirections in Psychological Science vol 7 no 2 pp 54ndash60 1998

[44] S Franklin Artificial Minds MIT Press Cambridge MA 1995[45] J Gibson The Ecological Approach to Visual Perception

Houghton-Mifflin Boston 1979[46] G Lakoff andM JohnsonPhilosophy in the FleshTheEmbodied

Mind and Its Challenge to Western Thought Basic Books NewYork NY USA 1999

[47] E HutchinsCognition in theWild MIT Press CambridgeMA1995

[48] E Hutchins ldquoThe cultural ecosystem of human cognitionrdquoPhilosophical Psychology vol 27 no 1 pp 34ndash49 2014

[49] V M Netto ldquoPractice space and the duality of meaningrdquoEnvironment and Planning D Society and Space vol 26 no 2pp 359ndash379 2008

[50] V M Netto The Social Fabric of Cities Routledge New YorkNY USA 2017

[51] T Parsons The Structure of Social Action The Free Press NewYork NY USA 1968

[52] T Allen Managing the Flow of Technology MIT Press Cam-bridge MA 1977

[53] F L Ribeiro J Meirelles F F Ferreira and R C Neto ldquoAmodel of urban scaling laws based on distance-dependent inter-actionsrdquo Royal Society Open Science vol 4 Article ID 1609262017

[54] W Alonso Location and Land Use Toward a General Theory ofLand Rent Harvard University Press Cambridge MA 1964

[55] WGHansen ldquoHow accessibility shapes land userdquo Journal of theAmerican Institute of Planners vol 25 no 2 pp 73ndash76 1959

[56] B Hillier Space Is the Machine Cambridge University PressCambridge UK 1996

[57] S Farber M OrsquoKelly H J Miller and T Neutens ldquoMeasuringsegregation using patterns of daily travel behavior a socialinteraction based model of exposurerdquo Journal of TransportGeography vol 49 pp 26ndash38 2015

[58] T C Schelling ldquoDynamic models of segregationrdquo Journal ofMathematical Sociology vol 1 no 2 pp 143ndash186 1971

[59] R Krafta V M Netto and L Lima ldquoUrban built form growscriticalrdquo CyberGeo vol 2011 article 565 2011

[60] N Lanchier and S Scarlatos ldquoFixation in the one-dimensionalAxelrod modelrdquo The Annals of Applied Probability vol 23 no6 pp 2538ndash2559 2013

[61] F L Ribeiro and K N Ribeiro ldquoA one dimensional modelof population growthrdquo Physica A Statistical Mechanics and itsApplications vol 434 pp 201ndash210 2015

[62] F L Ribeiro ldquoA non-phenomenological model of competitionand cooperation to explain population growth behaviorsrdquo Bul-letin of Mathematical Biology vol 77 no 3 pp 409ndash433 2015

16 Complexity

[63] R V dos Santos F L Ribeiro and A S Martinez ldquoModels forAllee effect based on physical principlesrdquo Journal of TheoreticalBiology vol 385 pp 143ndash152 2015

[64] M BattyThe New Science of Cities The MIT Press CambridgeMA 2013

[65] A G Wilson Entropy in Urban and Regional Modelling PionLondon 1970

[66] M Batty R Morphet P Masucci and K Stanilov ldquoEntropycomplexity and spatial informationrdquo Journal of GeographicalSystems vol 16 no 4 pp 363ndash385 2014

[67] R Axelrod ldquoThe dissemination of culture a model with localconvergence and global polarizationrdquo Journal of Conflict Reso-lution vol 41 no 2 pp 203ndash226 1997

[68] B H Hodges and RM Baron ldquoValues as constraints on afford-ances perceiving and acting properlyrdquo Journal for the Theory ofSocial Behaviour vol 22 no 3 pp 263ndash294 1992

[69] J JacobsThe Economy of Cities Random house New York NYUSA 1969

[70] E Glaeser H Kallal J Scheinkman and A Shleifer ldquoGrowthin Citiesrdquo Journal of Political Economy vol 100 no 6 pp 1126ndash1152 1992

[71] S Rosenthal and W Strange ldquoEvidence on the nature andsources of agglomeration economiesrdquo in Handbook of Urbanand Regional Economics J V Henderson and J-F Thisse Edsvol 4 pp 2119-2171 NorthHollandNewYork NYUSA 2004

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Complexity

role in social organization is to be assessed This is allowedby a unidimensional model (eg [58ndash60]) The ring formallows continuous movement across a linear sequence oflocations eliminating centrality factors while taking intoaccount periodic boundary conditions in order to reduceborder effects eliminating the role of topologywhile isolatingthe problem of distance (see [61ndash63]) This stylized cityhouses 119873 agents which in each time-step 119905 select and visita specific place located at the position 119909 within the cityConsider that the position of the 119894th agent at the time 119905 isrepresented by 119909

119894(119905) The position of an agent can assume the

integer values 1 2 119871 according to the place that shehechooses to visit This means that at every time-step eachagent will choose one activity place to perform her nextaction

We considered 120590119894(119905) as the current action of the 119894-th agent

and 120590119909(119905) the activity performed in a place located at the

position 119909 = (1 2 119871) of the city both in time 119905 Wequantify action orientations weighting the three variables(latent orientation current action and activity place) Firstvariables are translated into different numeric values Let usassume that an orientation is an integer number between 0and 1000 each number represents a particular type of actionand numbers in their vicinity mean similar actions Theresulting orientation in time 119905 for each agent is the weightedaverage of those three parametersThedifference |120590

119894(119905)minus120590119909(119905)|

gives us the difference in orientation between an agent andthe activity place At every time-step each agent assesses thedifference between her orientation and every activity placein the city An agent chooses a specific place based on herorientation affinity with this place minimum difference

As we hypothesized above the urban structure has a rolein this selection In this experiment we reduced ldquourban struc-turerdquo to distance In order to investigate how distance may ornot interfere in this selection we propose two scenarios Inthe first one the agent selects her next activity based only onthe similarity in orientations between herself and the activityplace Agents can move across a space free of friction Spaceis not a constraintThis scenario considers only differences inorientation The parameter 119864

119894(119909 119905) estimates the interaction

between agent 119894 and the place located at 119909 at the time 119905 Theagent selects activity locations that minimize this quantityWe propose

119864119894 (119909 119905) =

1003816100381610038161003816120590119894 (119905) minus 1205901199091003816100381610038161003816 (1)

In the second more realistic scenario space imposesfriction to movement The agent takes into account thephysical distance between her current location in the city andthe location of her next activity and opts to minimize thisdistance along with the difference between her orientationand that of the activity place We propose

119864119894 (119909 119905) =

1003816100381610038161003816(119909119894 (119905) minus 119909) (120590119894 (119905) minus 120590119909)1003816100381610038161003816 (2)

Summing up an agent 119894 selects at time 119905 a particularactivity place located at119909 thatminimizes the function119864

119894(119909 119905)

In the first scenario she will choose activity places closerto her orientation In the second she will choose activityplaces closer in both orientation and physical distance This

brings a form of ldquoenergy costrdquo into the factors consideredSpace takes on a function role because of the energy used inldquovisitingrdquo Agents move in this linear city willing to minimizethis energy while searching for social information latentin places close to their changing orientations Changes inentropy levels are derived from this energy function In itsspatial version the model assesses effects of the effort toreduce energy in action minimizing informational differ-ences (between orientations and activity places) and spatialdistances (between agents and activity places)

63TheEvolution of ActionOrientations Orientation evolvesin time according to the following rule When an agentchooses and joins a particular place both agent and placebecome a little closer in terms of orientation This meansthat activity places are not immune to what agents performin them and that the urban activity system also changes intime To illustrate how orientations are constantly updatedconsider that agent 119894 selects an activity place located at 1199091015840Orientations will be updated according the rule

120590119894 (119905 + 1) =

1120572 + 120573 + 120574

(120572120590in119894(119905) + 120573120590119894 (119905) + 1205741205901199091015840 (119905)) (3)

Selection also depends on three additional parameterslatent orientation 120590in

119894(119905) weighted by parameter 120572 current

orientation120590119894(119905)weighted by parameter120573 and type of activity

place 1205901199091015840(119905) weighted by parameter 120574 This means that agents

will consider the three factors previously described (120590in119894(119905)

120590119894(119905) 1205901199091015840(119905)) with different weights (120572 120573 120574) to estimate which

activity she wants to perform in the next time-stepThe latent orientation evolves from an initial randomly

distributed behaviour 1205900119894 We consider that 120590in

119894(119905) is normally

distributed around 1205900119894 meaning that every agent has an

ldquoaveragerdquo latent orientation whose value will be derived ateach time-step from a normal distribution peaking at theagentrsquos average orientation If 120572 is sufficiently small in relationto 120573 and 120574 then the latent orientation does not play a rolein the agentrsquos action However if 120572 is sufficiently large (oncein relation to 120573 and 120574) then the action becomes stronglydependent on the latent orientation This would lead to apretty conservative city where inhabitants are not open tochanges in behaviour

The current action of an agent may influence a newaction with an intensity that depends on parameter 120573 If thisparameter is small in comparison to 120572 and 120574 then the agenthas no memory of an ongoing orientation However if 120573 issufficiently large then the new action is strongly dependenton a previous one (a Markovian behaviour)

The intensity of the activity placersquos influence on the agentrsquosaction depends on the value of 120574 If this parameter is smallin comparison to 120572 and 120573 then the activity place does notaffect her next actionmeaning that the place does not changebehaviours However if this parameter is sufficiently largethen the place plays a strong influence on future actions Theactivity place will be updated according to the rule

1205901199091015840 (119905 + 1) = 1205901199091015840 (119905) + 120579sum

119894isin1199091015840

120590119894 (119905) (4)

Complexity 11

where 120579 is a parameter sufficiently small That is placesare less influenced by agents than the other way roundThis means that at every time-step a place would have itstype of activity closer to the average of orientations of itsvisitors Therefore the main activity of the place will changeslightly according to visiting agents while changing theirorientations

64 Frequency of Orientations and Entropy Levels We cal-culate entropy assessing the distribution of resulting orien-tations of all agents in the social system at each time-stepusing Shannon entropy Our quantitative interpretation ofinformation removes ambiguities from the relation betweenShannon information and semantic information as differentorientations actions and types of activity places are repre-sented by numerical values This allows us to estimate socialentropy through Shannon entropy The entropy of actions iscalculated as follows consider119873(120590 119905) as the number of agentswith orientation 120590 at the time 119905 (note that the total populationis 119873 = sum

120590119873(120590 119905)) We compute the frequency (or density)

120588(120590 119905) of this orientation within a population with

120588 (120590 119905) = 1119873119873 (120590 119905) (5)

Equation (5) calculates the probability of observing anorientation 120590 at time 119905 We compute the entropy level for anydistribution of orientations with

119878 (119905) = minussum120590

120588 (120590 119905) ln (120588 (120590 119905)) (6)

Equation (6) describes how uneven is the probability of find-ing different orientations Higher values mean that differentorientations have almost the same probability to happenwhile lower values indicate a system with clear orientationtrends The reduction of entropy implies that the probabilityof certain actions increases that is actions grow in similarityIn the limit as entropy falls to zero all agents in the systemwould reach the same orientation

65Model Procedure Themodel performs the following pro-cedures

(1) The city is created by generating the following

(a) The type of activity in places that is (120590119909=1(119905 =

0) 120590119909=2(119905 = 0) 120590

119909=119871(119905 = 0)) also generated

from a random distribution This step createsactivity places in the ring-city and assign themrandomly distributed activity values 120590

1199091015840(119905)

(b) The initial position of 119873 agents that is (1199091

1199092 119909

119873) and their respective orientations

(1205901(119905 = 0) 120590

2(119905 = 0) 120590

119873(119905 = 0)) generated

from a random distribution This step createsmoving agents in random locations and assignthem latent orientations based on the initialrandomly distributed orientation 120590in

119894(119905) = 120590

119894(0)

(2) At each time-step 119905 every agent chooses a place toact This place should minimize the function 119864

119894(119909 119905)

Suppose that the chosen place by 119894 is defined as 1199091015840Then agent and place update their orientations 120590

119894and

1205901199091015840 as described above

(3) Update the distribution of orientations 120588(120590 119905)(4) Compute Shannon entropy 119878(119905)

66 Results We have developed a number of observationson the behaviour of social systems working under differentparameters

(i) Entropy reduction is only found in scenarios wherespace imposes friction to movement that is wherethe spatial distance between agents and activity placesis considered as an active factor in selection Entropyrequires an extensive space in order to be reduced Butspace cannot do it alone Surprisingly space is a nec-essary but not sufficient condition Figure 7 comparestwo scenarios In the first one distance is an issue inthe selection of activities (red line)Thedistribution oforientations changes from a homogeneous one at thestart of simulation (a) to a nearly normal distributionat the end of simulation (b) where certain kindsof action are more likely to happen Informationalcontents in extensive space help aligning contents inactions In the second scenario agents move free ofspatial friction (blue line) Orientations are initiallyrandomly distributed (a) and continue to be so at theend of simulation (b) as a similar number of agentsare distributed along different orientationsThis resultsuggests that a materially active space becomes ameans for increasing the probability of certain inter-actions easing the collective coordination of action

Nowassessing the relative influence of social and personalfactors on the selection of a next action namely the socialinformation in activity places latent orientation and the cur-rent action under the influence of spatial friction (Figure 8)we can say the following

(i) There is structure in the relation between selectionfactors and the reduction of entropy Entropy beginsto fall and fluctuate around specific values accordingto different combinations of factors For some com-binations there is a great reduction of entropy Forothers the reduction is minimal almost negligibleFluctuations derive from the random component inthe decisionmaking of the agent (under the influenceof latent orientation)

(ii) A strong latent orientation (120572 sufficiently large) leadsto increasing entropy (blue lines in Figures 8 and9) since agents cannot align their actions with otheragents and activity places Like a long-term memorysystems whose actions are dictated by latent orienta-tions are likely to preserve initial orientations whichwere randomly distributed and therefore homoge-neous This condition is responsible for limiting thereduction of entropy

(iii) Current action alone does not shape new actions inany specific way (green lines in Figures 8 and 9)

12 Complexity

Start of simulation

00000

00005

00010

00015

00020

o

f tot

al ac

tions

250 500 750 10000Type of action

Materially inactive spaceMaterially active space

t = 0

(a)

End of simulation

00000

00005

00010

00015

00020

o

f tot

al ac

tions

250 500 750 10000Type of action

Materially inactive spaceMaterially active space

t = 50

(b)

Figure 7 Histogram of probability distribution of actors performing different types of action at the start (a) and at the end (b) of simulationsin scenarios where distance is not considered (blue) and distance is considered aligning orientations in action coordination (red) Resultsare averaged for 30 runs for 125 parameter combinations

activityplace

currentaction

latentorientation

225

250

275

300

325

Entr

opy

10 20 30 40 500Time

Figure 8 Evolution of entropy colors in a RGB scale refer toparameter weights in decisions on actions in a next time-step andtheir impact on entropy levels in different simulations (under thefriction of distance) In the palette triangle each color representsa specific combination of the three parameters In each vertex aparameter reaches maximum value while the others have minimumvalue Results are averaged for 30 different runs for 125 parametercombinations

It just keeps what is already happening workingas ldquoreinforcement feedbackrdquo for either direction theagent (and the system) is going Current actions aremeans to conserve tendencies in the system

(iv) Activity places play a key role in the reduction ofentropy (red lines in Figures 8 and 9)The social infor-mation in space ldquocontaminatesrdquo agents they aligntheir actions through the social contents of placesWe have seen that space matters as extension Nowwe see that space also has a very active informationalpresence

(v) Different combinations of factors have differenteffects over entropy Figure 8 shows the transitions

between the three parameters and how nuancedweights resulting from blending factors matter inthe reduction of entropy For instance blending theweight of activity places and latent orientation on thenext action displays a strong reduction of entropy(pink lines in Figure 8) That means that when agentsdo not keep information from previous selections(ie 120573 asymp 0) the action system reduces more entropythan in other parameter combinations A strongshort-term memory leads the system into conservingitself and into a poorer capability to coordinateactions Also when current actions 120573 and activityplaces 120574 share a similar weight over the next action(orange lines on top) the social system does notexperience a great reduction of entropy as agents tendto reproduce their actions

(vi) Finally the reduction of entropy implies that theprobability of certain actions and interactionsincreases In practical terms this means morealignments between agents and more connectionsbetween actions (interactions) However if all agentsin the system reached the same orientation andentropy dropped to zero the system would lose inter-nal differentiation Agents would behave in the sameway say in a world with no personal differentiationspecialization or division of labour New orientations(therefore entropy) are necessary if the social systemis to keep differentiated agencies On the otherhand a system with maximum entropy would createdifference in actions to a point where no action coor-dination or interaction would be possible Clearly thissituation cannot be the case A social system requiresbalances neither full entropy nor total predictability

These results also suggest that we are dealing with distinctldquosocial memoriesrdquo a long-term memory is active in what we

Complexity 13

20

25

30En

trop

y

04 0602-latent orientation

04 0602-current action

20

25

30

Entr

opy

20

25

30

Entr

opy

04 0602-activity place

Figure 9 Parameter exploration isolated importance of each parameter in entropy (under the friction of distance) Each vertical set of dotsrepresents one fixed value for the highlighted parameter with the two other parameters assuming every possible combination within thetested range Each dot corresponds to the entropy value when a sufficiently large time-frame is used

call latent orientations Short-term memory is active in theinfluence of a current orientation over a new action In turnthe social information latent in activity places is a result ofsocial arrangements (say as firms a local economy and soon) which find certain stability changing much slower thenactions It is an extension of the social system projected ontourban space stabilizing the system to some extent

7 Cities and Social Interaction Conclusions

What does this approach bring to the state of the art on therelations of social interaction information and space and onsocial entropy and ABM in particular We have seen thatprevious approaches overlooked the coordination of actionas a key empirical and analytical problem Furthermorethey tend to have the thin spatiality of a passive territorialbackground (eg [16 20]) In turn although the conceptof ldquocollective actionrdquo is finally getting attention in urbanstudies [64] the problem of social organization is still largelyunderestimated The problem of entropy has been dealt withsince Wilsonrsquos [65] work on spatial interaction but has notreached the mainstream of the discipline and deals mostlywith urban form (eg [24]) and spatial distributions [66]not with social information and action Regarding agentbased models we are not aware of an approach that dealswith the problem of how cities and space are part of socialorganization In a sense our model is closer to Axelrodrsquos[67] ABM model of dissemination of culture His agentsexchange ldquoculturerdquo through direct contact with neighboursin a cellular automata model whereas our agents are mobileand actively deal with distance and social information inspace recognizing types of activities in places The intensityof exchanges between agents in Axelrodrsquos model is a functionof similarities between them In our model exchanges aremediated by the selection and interaction with space In ourunderstanding our contribution in terms of ABM lies onthe fact that a new and simple model is proposed to dealwith the material and informational dimensions of socialorganization not just as an illustration but as a proof ofconcept space can have causal presence if agents generatepreferences as detailed in a domain that coevolves with them

Now does the simulation model corroborate the theoryintroduced in this paper Our proposition cannot be empir-ically validated at this stage given that it is very hard to

assess peoplersquos choices and the whole panorama of actionsin a city Entropy may well be beyond observation In thesecircumstances simulations of the behaviour of agents underdifferent spatial conditions become useful to assess entitiesevents and entropic forces at play One expected result fromthe model is that space matters for coordinating actions butsimulations surprisingly showed that space is a necessary butnot sufficient condition for the reduction of entropy Neithermaterial cognitive resources nor individual powers of agentsare enough Information in activity places and a bit of agentsrsquopersonal history also play roles in the way social systemsdeal with their own entropy That would mean a causal butnondeterministic presence of space

Substantively this approach explores a subtle but funda-mental presence of cities in social organizationWe argue thaturban space materializes gradients of difference in potentialinteractions from less to more recognizable costly or likelyPlacing simple criteria including distance as a factor in select-ing activities our ABM showed that space becomes a meansfor producing differences in the probabilities of interactionincreasing chances of certain selections and convergences incollective action

In reality people recognize the social information ofactivity places along with uses of busy streets or local central-ities We are able to retrieve these differences from the builtenvironment and relate to them One of the key points of ourargument is that these differences help us make selectionsEvery time we move through the city and select places toperform we participate in the large-scale coordination ofaction We enact space as a ldquoreferential systemrdquo [49 50]a set of semantic bits indexing social activities informingand guiding daily decisions Space becomes inherently partof the conversion of orientations into interactions that isconnections that crisscross sequences of actions recreatingthe interaction system By means of a structure that bothmaterializes and restricts the quasi-endless combinatorial pos-sibilities of interaction this system can acquire sufficientinternal guidance to make its own reproduction possibleBy distributing sufficiently recognizable differences in theprobability of interaction cities express and release localforces of reproduction of social systems The interfaces ofaction cognitive and spatial systems transform entropiccomplexity into structured complexity constantly reorderingactions in time

14 Complexity

This materialistic viewpoint can be integrated with otherviews of social organization Further work may include therelational roles of cultures language social norms and othersteering media along with explorations of possibly similareffects of communication technologies over entropy At thisstage we attempted to integrate a social dimension (agentsengaged in coordination) a spatial dimension (an extensiveenvironment) and an informational dimension (as differ-ences in action and in the environment itself) a demandinginterdisciplinary effort The approach identifies a central rolefor coevolving agents actualizing aims as they coordinate (cf[68]) and a central role for the environment exterior butresponsive to agents leading to cumulative changes alongtheir historyThemodel also identifies roles for cognition andmemory in the ways agents deal with their own orientationsand choose their actions and activity places changing theirspatial environment with consequences in the overall levelsof entropy All this suggests that the behaviour of the modelcannot be reduced to or fully predicted from the behaviour ofindividual variables which is a key aspect of complexity

So one of the main aims of this paper was to assesswhether space as information environment had any effecton the coordination and entropy of interaction However ifphysical distance and social information in activity placesmatter what about city size and the internal spatial struc-tures of cities What about rural areas or rarefied suburbsSince our model explores an abstract circular city it onlybegins to answer such questions Our findings indicate thatshorter distances between activity places tend to reducesocial entropy Density seems to matter denser spatialities(as opposed to rarefied ones) have a role to play A pathfor further development here is Jacobsrsquos [69] idea that thediversity of activities generates positive externalities Anddiversity as spatial economics shows us has to do withthe size of cities and density of population (eg [70 71])Entropy may also be influenced by diversity Batty et al [66]have shown that information increases as cities get biggerThe remaining question is whether larger denser and moreinternally structured cities could create process and reduceentropy with more intensity than smaller cities transformingbig pools of activities into differences in the probability ofinteraction Although we proposed a unidimensional modelour agentsrsquo behaviour can be explored in more realisticrepresentations of cities testing the roles of density topologyand diversity while going beyond perfectly informed agentsBy proposing the ring model we intended to simplify acity to a minimum system where the spatial dimensionof social organization becomes intelligible without losingfundamental properties

Shouldwe conclude fromall this that entropy is a problemfor social systems Our approach suggests that it is notEntropy is a necessary force It means that novel orientationsare entering the interaction system and must be dealt withEntropy only becomes a problem if not converted into orga-nization only if numerous potential actions are not convertedinto smaller sets of actual interactions The effects of newactions over entropy would be positive once actualizedhaving to dowith the diversity of agencyConservative systemsare likely to face less entropy but also likely to produce

less novelty having homogeneity as a problematic horizonFluctuations of entropy seem vital to complex societies

Finally our intention was to explore entropy as a meansto think about the conditions of social organization ldquofromanother anglerdquo so to speak from the viewpoint of challengesinvolved in social reproduction Hence the approach focusedon the idea of uncertainty surrounding agents dealing withaims and decisions and the different probabilities of inter-action that follow an attempt to bring social cognitiveinformational and spatial theories under a single roof Ourapproach sees cities as connective systems produced to createthe delicate fabric of interaction that keeps large numbersof agents coherently living together It also suggests that thevarying states of entropy reveal deep connections betweensocial informational and spatial systems

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

Theauthors would like to thank the Ecole Polytechnique Fed-erale de Lausanne (EPFL) CAPES FAPEMIG FAPESP andCNPq for financial support Maıra Pinheiro Henrique Loreaand colleagues at ABMUS2017 for earlier discussions andMike Batty for exchanges and inspiration Finally the authorsthank Romulo Krafta maverick of the complexity studiesof cities in Brazil for his input and support throughout theyears This work is dedicated to him

References

[1] C W Mills The Sociological Imagination Oxford UniversityPress New York NY USA 1959

[2] M Castells ldquoThe information age economy society and cul-turerdquo inThe Rise of the Network Blackwell London 1996

[3] C Hidalgo Why Information Grows The Evolution of Orderfrom Atoms to Economies Basic Books New York NY USA2015

[4] D Helbing The Automation of Society Is Next ETH ZurichSwitzerland 2016

[5] N Luhmann Social Systems University Press Stanford 1995[6] I Prigogine and I Stengers Order out of Chaos Manrsquos New

Dialogue with Nature Bantam Books 1984[7] C E Shannon ldquoAmathematical theory of communicationrdquo Bell

System Technical Journal vol 27 no 4 pp 623ndash656 1948[8] W Buckley Sociology and Modern Systems Theory Prentice-

Hall Englewood Cliffs NJ USA 1967[9] D D McFarland ldquoMeasuring the Permeability of Occupational

Structures An Information-Theoretic Approachrdquo AmericanJournal of Sociology vol 75 no 1 pp 41ndash61 1969

[10] F Charvat ldquoOn philosophical aspects of the system conceptionin contemporary sociological knowledgerdquo Quality and Quan-tity vol 6 no 1 pp 3ndash16 1972

[11] F Charvat J Kucera and M Soukup ldquoToward the systemtheory of dependence further general theoretical remarksrdquoQuality and Quantity vol 7 no 1 pp 69ndash90 1973

Complexity 15

[12] P M Horan ldquoInformation-theoretic measures and the analysisof social structuresrdquo Sociological Methods and Research vol 3no 3 pp 321ndash340 1975

[13] O E Klapp ldquoOpening and closing in open systemsrdquo BehavioralScience vol 20 no 4 pp 251ndash257 1975

[14] J Galtung ldquoEntropy and the general theory of peacerdquo in Essaysin Peace Research J Galtung Ed vol L Humanities PressAtlantic Highlands NJ USA 1975

[15] T Parsons The System of Modern Societies Englewood CliffsNew Jersey NJ USA 1972

[16] K D Bailey ldquoSociological entropy theory Toward a statisticaland verbal congruencerdquo Quality amp Quantity vol 18 no 1 pp113ndash133 1983

[17] K D Bailey Social EntropyTheory State University of NewYorkPress Albany NY USA 1990

[18] J G Miller Living Systems McGraw-Hill New York NY USA1978

[19] K D Bailey Sociology and the New Systems Theory Toward aTheoretical Synthesis State University of New York PressAlbany NY USA 1994

[20] K D Bailey ldquoSociocybernetics and social entropy theoryrdquoKybernetes vol 35 no 3-4 pp 375ndash384 2006

[21] F Geyer ldquoThe challenge of sociocyberneticsrdquo Kybernetes vol24 no 4 pp 6ndash32 1995

[22] F Geyer ldquoThe march of self-referencerdquo Kybernetes vol 31 no7-8 pp 1021ndash1042 2002

[23] J HabermasTheTheory of Communicative Action Polity PressCambridge UK 1987

[24] H Haken and J Portugali Information Adaptation The Inter-play between Shannon Information and Semantic Information inCognition Springer New York NY USA 2015

[25] M Weber Economy and Society vol 1 Berkeley University ofCalifornia Press 1978

[26] L Wittgenstein Philosophical Investigations Blackwell Publish-ers Oxford London 3rd edition 2001

[27] L Vygotsky Mind in Society The Development of HigherPsychological Processes Harvard University Press CambridgeMA 1978

[28] M Wilson ldquoSix Views of Embodied Cognitionrdquo PsychonomicBulletin amp Review vol 9 no 4 pp 625ndash636 2002

[29] A Clark and D Chalmers ldquoThe extended mindrdquo Analysis vol58 no 1 pp 7ndash19 1998

[30] R PassiniWayfinding in Architecture Van Nostrand ReinholdNew York NY USA 1992

[31] E Rosch ldquoPrinciples of categorizationrdquo in Cognition and Cate-gorization E Rosch and B Lloyd Eds Hillsdale LawrenceErlbaum Ass 1978

[32] W KintschMemory and Cognition John Wiley and Sons NewYork NY USA 1970

[33] U Neisser ldquoMultiple Systems a New Approach to CognitiveTheoryrdquo European Journal of Cognitive Psychology vol 6 no 3pp 225ndash241 1994

[34] A Faria and R Krafta ldquoRepresenting urban cognitive structurethrough spatial differentiationrdquo in Proceedings of 4th Space Syn-tax International Symposium pp 531-518 UCL Press London2003

[35] J Portugali Complexity Cognition and the City Springer NewYork NY USA 2011

[36] D Kirsh and P Maglio ldquoOn distinguishing epistemic frompragmatic actionrdquo Cognitive Science vol 18 no 4 pp 513ndash5491994

[37] S M Kosslyn Image and Brain The Resolution of the ImageryDebate MIT Press Cambridge MA

[38] R Brooks Cambrian Intelligence The Early History of the NewAI MIT Press Cambridge MA 1999

[39] A Clark BeinGThere Putting Brain Body AndWorld TogetherAgain MIT Press Cambridge MA 1997

[40] R Pfeifer and C ScheierUnderstanding Intelligence MIT PressCambridge MA 1999

[41] A M Glenberg and D A Robertson ldquoIndexical understandingof instructionsrdquoDiscourse Processes vol 28 no 1 pp 1ndash26 1999

[42] JM Iverson and SGoldin-Meadow ldquoWhypeople gesturewhenthey speakrdquo Nature vol 396 no 6708 p 228 1998

[43] R M Krauss ldquoWhy do we gesture when we speakrdquo CurrentDirections in Psychological Science vol 7 no 2 pp 54ndash60 1998

[44] S Franklin Artificial Minds MIT Press Cambridge MA 1995[45] J Gibson The Ecological Approach to Visual Perception

Houghton-Mifflin Boston 1979[46] G Lakoff andM JohnsonPhilosophy in the FleshTheEmbodied

Mind and Its Challenge to Western Thought Basic Books NewYork NY USA 1999

[47] E HutchinsCognition in theWild MIT Press CambridgeMA1995

[48] E Hutchins ldquoThe cultural ecosystem of human cognitionrdquoPhilosophical Psychology vol 27 no 1 pp 34ndash49 2014

[49] V M Netto ldquoPractice space and the duality of meaningrdquoEnvironment and Planning D Society and Space vol 26 no 2pp 359ndash379 2008

[50] V M Netto The Social Fabric of Cities Routledge New YorkNY USA 2017

[51] T Parsons The Structure of Social Action The Free Press NewYork NY USA 1968

[52] T Allen Managing the Flow of Technology MIT Press Cam-bridge MA 1977

[53] F L Ribeiro J Meirelles F F Ferreira and R C Neto ldquoAmodel of urban scaling laws based on distance-dependent inter-actionsrdquo Royal Society Open Science vol 4 Article ID 1609262017

[54] W Alonso Location and Land Use Toward a General Theory ofLand Rent Harvard University Press Cambridge MA 1964

[55] WGHansen ldquoHow accessibility shapes land userdquo Journal of theAmerican Institute of Planners vol 25 no 2 pp 73ndash76 1959

[56] B Hillier Space Is the Machine Cambridge University PressCambridge UK 1996

[57] S Farber M OrsquoKelly H J Miller and T Neutens ldquoMeasuringsegregation using patterns of daily travel behavior a socialinteraction based model of exposurerdquo Journal of TransportGeography vol 49 pp 26ndash38 2015

[58] T C Schelling ldquoDynamic models of segregationrdquo Journal ofMathematical Sociology vol 1 no 2 pp 143ndash186 1971

[59] R Krafta V M Netto and L Lima ldquoUrban built form growscriticalrdquo CyberGeo vol 2011 article 565 2011

[60] N Lanchier and S Scarlatos ldquoFixation in the one-dimensionalAxelrod modelrdquo The Annals of Applied Probability vol 23 no6 pp 2538ndash2559 2013

[61] F L Ribeiro and K N Ribeiro ldquoA one dimensional modelof population growthrdquo Physica A Statistical Mechanics and itsApplications vol 434 pp 201ndash210 2015

[62] F L Ribeiro ldquoA non-phenomenological model of competitionand cooperation to explain population growth behaviorsrdquo Bul-letin of Mathematical Biology vol 77 no 3 pp 409ndash433 2015

16 Complexity

[63] R V dos Santos F L Ribeiro and A S Martinez ldquoModels forAllee effect based on physical principlesrdquo Journal of TheoreticalBiology vol 385 pp 143ndash152 2015

[64] M BattyThe New Science of Cities The MIT Press CambridgeMA 2013

[65] A G Wilson Entropy in Urban and Regional Modelling PionLondon 1970

[66] M Batty R Morphet P Masucci and K Stanilov ldquoEntropycomplexity and spatial informationrdquo Journal of GeographicalSystems vol 16 no 4 pp 363ndash385 2014

[67] R Axelrod ldquoThe dissemination of culture a model with localconvergence and global polarizationrdquo Journal of Conflict Reso-lution vol 41 no 2 pp 203ndash226 1997

[68] B H Hodges and RM Baron ldquoValues as constraints on afford-ances perceiving and acting properlyrdquo Journal for the Theory ofSocial Behaviour vol 22 no 3 pp 263ndash294 1992

[69] J JacobsThe Economy of Cities Random house New York NYUSA 1969

[70] E Glaeser H Kallal J Scheinkman and A Shleifer ldquoGrowthin Citiesrdquo Journal of Political Economy vol 100 no 6 pp 1126ndash1152 1992

[71] S Rosenthal and W Strange ldquoEvidence on the nature andsources of agglomeration economiesrdquo in Handbook of Urbanand Regional Economics J V Henderson and J-F Thisse Edsvol 4 pp 2119-2171 NorthHollandNewYork NYUSA 2004

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Complexity 11

where 120579 is a parameter sufficiently small That is placesare less influenced by agents than the other way roundThis means that at every time-step a place would have itstype of activity closer to the average of orientations of itsvisitors Therefore the main activity of the place will changeslightly according to visiting agents while changing theirorientations

64 Frequency of Orientations and Entropy Levels We cal-culate entropy assessing the distribution of resulting orien-tations of all agents in the social system at each time-stepusing Shannon entropy Our quantitative interpretation ofinformation removes ambiguities from the relation betweenShannon information and semantic information as differentorientations actions and types of activity places are repre-sented by numerical values This allows us to estimate socialentropy through Shannon entropy The entropy of actions iscalculated as follows consider119873(120590 119905) as the number of agentswith orientation 120590 at the time 119905 (note that the total populationis 119873 = sum

120590119873(120590 119905)) We compute the frequency (or density)

120588(120590 119905) of this orientation within a population with

120588 (120590 119905) = 1119873119873 (120590 119905) (5)

Equation (5) calculates the probability of observing anorientation 120590 at time 119905 We compute the entropy level for anydistribution of orientations with

119878 (119905) = minussum120590

120588 (120590 119905) ln (120588 (120590 119905)) (6)

Equation (6) describes how uneven is the probability of find-ing different orientations Higher values mean that differentorientations have almost the same probability to happenwhile lower values indicate a system with clear orientationtrends The reduction of entropy implies that the probabilityof certain actions increases that is actions grow in similarityIn the limit as entropy falls to zero all agents in the systemwould reach the same orientation

65Model Procedure Themodel performs the following pro-cedures

(1) The city is created by generating the following

(a) The type of activity in places that is (120590119909=1(119905 =

0) 120590119909=2(119905 = 0) 120590

119909=119871(119905 = 0)) also generated

from a random distribution This step createsactivity places in the ring-city and assign themrandomly distributed activity values 120590

1199091015840(119905)

(b) The initial position of 119873 agents that is (1199091

1199092 119909

119873) and their respective orientations

(1205901(119905 = 0) 120590

2(119905 = 0) 120590

119873(119905 = 0)) generated

from a random distribution This step createsmoving agents in random locations and assignthem latent orientations based on the initialrandomly distributed orientation 120590in

119894(119905) = 120590

119894(0)

(2) At each time-step 119905 every agent chooses a place toact This place should minimize the function 119864

119894(119909 119905)

Suppose that the chosen place by 119894 is defined as 1199091015840Then agent and place update their orientations 120590

119894and

1205901199091015840 as described above

(3) Update the distribution of orientations 120588(120590 119905)(4) Compute Shannon entropy 119878(119905)

66 Results We have developed a number of observationson the behaviour of social systems working under differentparameters

(i) Entropy reduction is only found in scenarios wherespace imposes friction to movement that is wherethe spatial distance between agents and activity placesis considered as an active factor in selection Entropyrequires an extensive space in order to be reduced Butspace cannot do it alone Surprisingly space is a nec-essary but not sufficient condition Figure 7 comparestwo scenarios In the first one distance is an issue inthe selection of activities (red line)Thedistribution oforientations changes from a homogeneous one at thestart of simulation (a) to a nearly normal distributionat the end of simulation (b) where certain kindsof action are more likely to happen Informationalcontents in extensive space help aligning contents inactions In the second scenario agents move free ofspatial friction (blue line) Orientations are initiallyrandomly distributed (a) and continue to be so at theend of simulation (b) as a similar number of agentsare distributed along different orientationsThis resultsuggests that a materially active space becomes ameans for increasing the probability of certain inter-actions easing the collective coordination of action

Nowassessing the relative influence of social and personalfactors on the selection of a next action namely the socialinformation in activity places latent orientation and the cur-rent action under the influence of spatial friction (Figure 8)we can say the following

(i) There is structure in the relation between selectionfactors and the reduction of entropy Entropy beginsto fall and fluctuate around specific values accordingto different combinations of factors For some com-binations there is a great reduction of entropy Forothers the reduction is minimal almost negligibleFluctuations derive from the random component inthe decisionmaking of the agent (under the influenceof latent orientation)

(ii) A strong latent orientation (120572 sufficiently large) leadsto increasing entropy (blue lines in Figures 8 and9) since agents cannot align their actions with otheragents and activity places Like a long-term memorysystems whose actions are dictated by latent orienta-tions are likely to preserve initial orientations whichwere randomly distributed and therefore homoge-neous This condition is responsible for limiting thereduction of entropy

(iii) Current action alone does not shape new actions inany specific way (green lines in Figures 8 and 9)

12 Complexity

Start of simulation

00000

00005

00010

00015

00020

o

f tot

al ac

tions

250 500 750 10000Type of action

Materially inactive spaceMaterially active space

t = 0

(a)

End of simulation

00000

00005

00010

00015

00020

o

f tot

al ac

tions

250 500 750 10000Type of action

Materially inactive spaceMaterially active space

t = 50

(b)

Figure 7 Histogram of probability distribution of actors performing different types of action at the start (a) and at the end (b) of simulationsin scenarios where distance is not considered (blue) and distance is considered aligning orientations in action coordination (red) Resultsare averaged for 30 runs for 125 parameter combinations

activityplace

currentaction

latentorientation

225

250

275

300

325

Entr

opy

10 20 30 40 500Time

Figure 8 Evolution of entropy colors in a RGB scale refer toparameter weights in decisions on actions in a next time-step andtheir impact on entropy levels in different simulations (under thefriction of distance) In the palette triangle each color representsa specific combination of the three parameters In each vertex aparameter reaches maximum value while the others have minimumvalue Results are averaged for 30 different runs for 125 parametercombinations

It just keeps what is already happening workingas ldquoreinforcement feedbackrdquo for either direction theagent (and the system) is going Current actions aremeans to conserve tendencies in the system

(iv) Activity places play a key role in the reduction ofentropy (red lines in Figures 8 and 9)The social infor-mation in space ldquocontaminatesrdquo agents they aligntheir actions through the social contents of placesWe have seen that space matters as extension Nowwe see that space also has a very active informationalpresence

(v) Different combinations of factors have differenteffects over entropy Figure 8 shows the transitions

between the three parameters and how nuancedweights resulting from blending factors matter inthe reduction of entropy For instance blending theweight of activity places and latent orientation on thenext action displays a strong reduction of entropy(pink lines in Figure 8) That means that when agentsdo not keep information from previous selections(ie 120573 asymp 0) the action system reduces more entropythan in other parameter combinations A strongshort-term memory leads the system into conservingitself and into a poorer capability to coordinateactions Also when current actions 120573 and activityplaces 120574 share a similar weight over the next action(orange lines on top) the social system does notexperience a great reduction of entropy as agents tendto reproduce their actions

(vi) Finally the reduction of entropy implies that theprobability of certain actions and interactionsincreases In practical terms this means morealignments between agents and more connectionsbetween actions (interactions) However if all agentsin the system reached the same orientation andentropy dropped to zero the system would lose inter-nal differentiation Agents would behave in the sameway say in a world with no personal differentiationspecialization or division of labour New orientations(therefore entropy) are necessary if the social systemis to keep differentiated agencies On the otherhand a system with maximum entropy would createdifference in actions to a point where no action coor-dination or interaction would be possible Clearly thissituation cannot be the case A social system requiresbalances neither full entropy nor total predictability

These results also suggest that we are dealing with distinctldquosocial memoriesrdquo a long-term memory is active in what we

Complexity 13

20

25

30En

trop

y

04 0602-latent orientation

04 0602-current action

20

25

30

Entr

opy

20

25

30

Entr

opy

04 0602-activity place

Figure 9 Parameter exploration isolated importance of each parameter in entropy (under the friction of distance) Each vertical set of dotsrepresents one fixed value for the highlighted parameter with the two other parameters assuming every possible combination within thetested range Each dot corresponds to the entropy value when a sufficiently large time-frame is used

call latent orientations Short-term memory is active in theinfluence of a current orientation over a new action In turnthe social information latent in activity places is a result ofsocial arrangements (say as firms a local economy and soon) which find certain stability changing much slower thenactions It is an extension of the social system projected ontourban space stabilizing the system to some extent

7 Cities and Social Interaction Conclusions

What does this approach bring to the state of the art on therelations of social interaction information and space and onsocial entropy and ABM in particular We have seen thatprevious approaches overlooked the coordination of actionas a key empirical and analytical problem Furthermorethey tend to have the thin spatiality of a passive territorialbackground (eg [16 20]) In turn although the conceptof ldquocollective actionrdquo is finally getting attention in urbanstudies [64] the problem of social organization is still largelyunderestimated The problem of entropy has been dealt withsince Wilsonrsquos [65] work on spatial interaction but has notreached the mainstream of the discipline and deals mostlywith urban form (eg [24]) and spatial distributions [66]not with social information and action Regarding agentbased models we are not aware of an approach that dealswith the problem of how cities and space are part of socialorganization In a sense our model is closer to Axelrodrsquos[67] ABM model of dissemination of culture His agentsexchange ldquoculturerdquo through direct contact with neighboursin a cellular automata model whereas our agents are mobileand actively deal with distance and social information inspace recognizing types of activities in places The intensityof exchanges between agents in Axelrodrsquos model is a functionof similarities between them In our model exchanges aremediated by the selection and interaction with space In ourunderstanding our contribution in terms of ABM lies onthe fact that a new and simple model is proposed to dealwith the material and informational dimensions of socialorganization not just as an illustration but as a proof ofconcept space can have causal presence if agents generatepreferences as detailed in a domain that coevolves with them

Now does the simulation model corroborate the theoryintroduced in this paper Our proposition cannot be empir-ically validated at this stage given that it is very hard to

assess peoplersquos choices and the whole panorama of actionsin a city Entropy may well be beyond observation In thesecircumstances simulations of the behaviour of agents underdifferent spatial conditions become useful to assess entitiesevents and entropic forces at play One expected result fromthe model is that space matters for coordinating actions butsimulations surprisingly showed that space is a necessary butnot sufficient condition for the reduction of entropy Neithermaterial cognitive resources nor individual powers of agentsare enough Information in activity places and a bit of agentsrsquopersonal history also play roles in the way social systemsdeal with their own entropy That would mean a causal butnondeterministic presence of space

Substantively this approach explores a subtle but funda-mental presence of cities in social organizationWe argue thaturban space materializes gradients of difference in potentialinteractions from less to more recognizable costly or likelyPlacing simple criteria including distance as a factor in select-ing activities our ABM showed that space becomes a meansfor producing differences in the probabilities of interactionincreasing chances of certain selections and convergences incollective action

In reality people recognize the social information ofactivity places along with uses of busy streets or local central-ities We are able to retrieve these differences from the builtenvironment and relate to them One of the key points of ourargument is that these differences help us make selectionsEvery time we move through the city and select places toperform we participate in the large-scale coordination ofaction We enact space as a ldquoreferential systemrdquo [49 50]a set of semantic bits indexing social activities informingand guiding daily decisions Space becomes inherently partof the conversion of orientations into interactions that isconnections that crisscross sequences of actions recreatingthe interaction system By means of a structure that bothmaterializes and restricts the quasi-endless combinatorial pos-sibilities of interaction this system can acquire sufficientinternal guidance to make its own reproduction possibleBy distributing sufficiently recognizable differences in theprobability of interaction cities express and release localforces of reproduction of social systems The interfaces ofaction cognitive and spatial systems transform entropiccomplexity into structured complexity constantly reorderingactions in time

14 Complexity

This materialistic viewpoint can be integrated with otherviews of social organization Further work may include therelational roles of cultures language social norms and othersteering media along with explorations of possibly similareffects of communication technologies over entropy At thisstage we attempted to integrate a social dimension (agentsengaged in coordination) a spatial dimension (an extensiveenvironment) and an informational dimension (as differ-ences in action and in the environment itself) a demandinginterdisciplinary effort The approach identifies a central rolefor coevolving agents actualizing aims as they coordinate (cf[68]) and a central role for the environment exterior butresponsive to agents leading to cumulative changes alongtheir historyThemodel also identifies roles for cognition andmemory in the ways agents deal with their own orientationsand choose their actions and activity places changing theirspatial environment with consequences in the overall levelsof entropy All this suggests that the behaviour of the modelcannot be reduced to or fully predicted from the behaviour ofindividual variables which is a key aspect of complexity

So one of the main aims of this paper was to assesswhether space as information environment had any effecton the coordination and entropy of interaction However ifphysical distance and social information in activity placesmatter what about city size and the internal spatial struc-tures of cities What about rural areas or rarefied suburbsSince our model explores an abstract circular city it onlybegins to answer such questions Our findings indicate thatshorter distances between activity places tend to reducesocial entropy Density seems to matter denser spatialities(as opposed to rarefied ones) have a role to play A pathfor further development here is Jacobsrsquos [69] idea that thediversity of activities generates positive externalities Anddiversity as spatial economics shows us has to do withthe size of cities and density of population (eg [70 71])Entropy may also be influenced by diversity Batty et al [66]have shown that information increases as cities get biggerThe remaining question is whether larger denser and moreinternally structured cities could create process and reduceentropy with more intensity than smaller cities transformingbig pools of activities into differences in the probability ofinteraction Although we proposed a unidimensional modelour agentsrsquo behaviour can be explored in more realisticrepresentations of cities testing the roles of density topologyand diversity while going beyond perfectly informed agentsBy proposing the ring model we intended to simplify acity to a minimum system where the spatial dimensionof social organization becomes intelligible without losingfundamental properties

Shouldwe conclude fromall this that entropy is a problemfor social systems Our approach suggests that it is notEntropy is a necessary force It means that novel orientationsare entering the interaction system and must be dealt withEntropy only becomes a problem if not converted into orga-nization only if numerous potential actions are not convertedinto smaller sets of actual interactions The effects of newactions over entropy would be positive once actualizedhaving to dowith the diversity of agencyConservative systemsare likely to face less entropy but also likely to produce

less novelty having homogeneity as a problematic horizonFluctuations of entropy seem vital to complex societies

Finally our intention was to explore entropy as a meansto think about the conditions of social organization ldquofromanother anglerdquo so to speak from the viewpoint of challengesinvolved in social reproduction Hence the approach focusedon the idea of uncertainty surrounding agents dealing withaims and decisions and the different probabilities of inter-action that follow an attempt to bring social cognitiveinformational and spatial theories under a single roof Ourapproach sees cities as connective systems produced to createthe delicate fabric of interaction that keeps large numbersof agents coherently living together It also suggests that thevarying states of entropy reveal deep connections betweensocial informational and spatial systems

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

Theauthors would like to thank the Ecole Polytechnique Fed-erale de Lausanne (EPFL) CAPES FAPEMIG FAPESP andCNPq for financial support Maıra Pinheiro Henrique Loreaand colleagues at ABMUS2017 for earlier discussions andMike Batty for exchanges and inspiration Finally the authorsthank Romulo Krafta maverick of the complexity studiesof cities in Brazil for his input and support throughout theyears This work is dedicated to him

References

[1] C W Mills The Sociological Imagination Oxford UniversityPress New York NY USA 1959

[2] M Castells ldquoThe information age economy society and cul-turerdquo inThe Rise of the Network Blackwell London 1996

[3] C Hidalgo Why Information Grows The Evolution of Orderfrom Atoms to Economies Basic Books New York NY USA2015

[4] D Helbing The Automation of Society Is Next ETH ZurichSwitzerland 2016

[5] N Luhmann Social Systems University Press Stanford 1995[6] I Prigogine and I Stengers Order out of Chaos Manrsquos New

Dialogue with Nature Bantam Books 1984[7] C E Shannon ldquoAmathematical theory of communicationrdquo Bell

System Technical Journal vol 27 no 4 pp 623ndash656 1948[8] W Buckley Sociology and Modern Systems Theory Prentice-

Hall Englewood Cliffs NJ USA 1967[9] D D McFarland ldquoMeasuring the Permeability of Occupational

Structures An Information-Theoretic Approachrdquo AmericanJournal of Sociology vol 75 no 1 pp 41ndash61 1969

[10] F Charvat ldquoOn philosophical aspects of the system conceptionin contemporary sociological knowledgerdquo Quality and Quan-tity vol 6 no 1 pp 3ndash16 1972

[11] F Charvat J Kucera and M Soukup ldquoToward the systemtheory of dependence further general theoretical remarksrdquoQuality and Quantity vol 7 no 1 pp 69ndash90 1973

Complexity 15

[12] P M Horan ldquoInformation-theoretic measures and the analysisof social structuresrdquo Sociological Methods and Research vol 3no 3 pp 321ndash340 1975

[13] O E Klapp ldquoOpening and closing in open systemsrdquo BehavioralScience vol 20 no 4 pp 251ndash257 1975

[14] J Galtung ldquoEntropy and the general theory of peacerdquo in Essaysin Peace Research J Galtung Ed vol L Humanities PressAtlantic Highlands NJ USA 1975

[15] T Parsons The System of Modern Societies Englewood CliffsNew Jersey NJ USA 1972

[16] K D Bailey ldquoSociological entropy theory Toward a statisticaland verbal congruencerdquo Quality amp Quantity vol 18 no 1 pp113ndash133 1983

[17] K D Bailey Social EntropyTheory State University of NewYorkPress Albany NY USA 1990

[18] J G Miller Living Systems McGraw-Hill New York NY USA1978

[19] K D Bailey Sociology and the New Systems Theory Toward aTheoretical Synthesis State University of New York PressAlbany NY USA 1994

[20] K D Bailey ldquoSociocybernetics and social entropy theoryrdquoKybernetes vol 35 no 3-4 pp 375ndash384 2006

[21] F Geyer ldquoThe challenge of sociocyberneticsrdquo Kybernetes vol24 no 4 pp 6ndash32 1995

[22] F Geyer ldquoThe march of self-referencerdquo Kybernetes vol 31 no7-8 pp 1021ndash1042 2002

[23] J HabermasTheTheory of Communicative Action Polity PressCambridge UK 1987

[24] H Haken and J Portugali Information Adaptation The Inter-play between Shannon Information and Semantic Information inCognition Springer New York NY USA 2015

[25] M Weber Economy and Society vol 1 Berkeley University ofCalifornia Press 1978

[26] L Wittgenstein Philosophical Investigations Blackwell Publish-ers Oxford London 3rd edition 2001

[27] L Vygotsky Mind in Society The Development of HigherPsychological Processes Harvard University Press CambridgeMA 1978

[28] M Wilson ldquoSix Views of Embodied Cognitionrdquo PsychonomicBulletin amp Review vol 9 no 4 pp 625ndash636 2002

[29] A Clark and D Chalmers ldquoThe extended mindrdquo Analysis vol58 no 1 pp 7ndash19 1998

[30] R PassiniWayfinding in Architecture Van Nostrand ReinholdNew York NY USA 1992

[31] E Rosch ldquoPrinciples of categorizationrdquo in Cognition and Cate-gorization E Rosch and B Lloyd Eds Hillsdale LawrenceErlbaum Ass 1978

[32] W KintschMemory and Cognition John Wiley and Sons NewYork NY USA 1970

[33] U Neisser ldquoMultiple Systems a New Approach to CognitiveTheoryrdquo European Journal of Cognitive Psychology vol 6 no 3pp 225ndash241 1994

[34] A Faria and R Krafta ldquoRepresenting urban cognitive structurethrough spatial differentiationrdquo in Proceedings of 4th Space Syn-tax International Symposium pp 531-518 UCL Press London2003

[35] J Portugali Complexity Cognition and the City Springer NewYork NY USA 2011

[36] D Kirsh and P Maglio ldquoOn distinguishing epistemic frompragmatic actionrdquo Cognitive Science vol 18 no 4 pp 513ndash5491994

[37] S M Kosslyn Image and Brain The Resolution of the ImageryDebate MIT Press Cambridge MA

[38] R Brooks Cambrian Intelligence The Early History of the NewAI MIT Press Cambridge MA 1999

[39] A Clark BeinGThere Putting Brain Body AndWorld TogetherAgain MIT Press Cambridge MA 1997

[40] R Pfeifer and C ScheierUnderstanding Intelligence MIT PressCambridge MA 1999

[41] A M Glenberg and D A Robertson ldquoIndexical understandingof instructionsrdquoDiscourse Processes vol 28 no 1 pp 1ndash26 1999

[42] JM Iverson and SGoldin-Meadow ldquoWhypeople gesturewhenthey speakrdquo Nature vol 396 no 6708 p 228 1998

[43] R M Krauss ldquoWhy do we gesture when we speakrdquo CurrentDirections in Psychological Science vol 7 no 2 pp 54ndash60 1998

[44] S Franklin Artificial Minds MIT Press Cambridge MA 1995[45] J Gibson The Ecological Approach to Visual Perception

Houghton-Mifflin Boston 1979[46] G Lakoff andM JohnsonPhilosophy in the FleshTheEmbodied

Mind and Its Challenge to Western Thought Basic Books NewYork NY USA 1999

[47] E HutchinsCognition in theWild MIT Press CambridgeMA1995

[48] E Hutchins ldquoThe cultural ecosystem of human cognitionrdquoPhilosophical Psychology vol 27 no 1 pp 34ndash49 2014

[49] V M Netto ldquoPractice space and the duality of meaningrdquoEnvironment and Planning D Society and Space vol 26 no 2pp 359ndash379 2008

[50] V M Netto The Social Fabric of Cities Routledge New YorkNY USA 2017

[51] T Parsons The Structure of Social Action The Free Press NewYork NY USA 1968

[52] T Allen Managing the Flow of Technology MIT Press Cam-bridge MA 1977

[53] F L Ribeiro J Meirelles F F Ferreira and R C Neto ldquoAmodel of urban scaling laws based on distance-dependent inter-actionsrdquo Royal Society Open Science vol 4 Article ID 1609262017

[54] W Alonso Location and Land Use Toward a General Theory ofLand Rent Harvard University Press Cambridge MA 1964

[55] WGHansen ldquoHow accessibility shapes land userdquo Journal of theAmerican Institute of Planners vol 25 no 2 pp 73ndash76 1959

[56] B Hillier Space Is the Machine Cambridge University PressCambridge UK 1996

[57] S Farber M OrsquoKelly H J Miller and T Neutens ldquoMeasuringsegregation using patterns of daily travel behavior a socialinteraction based model of exposurerdquo Journal of TransportGeography vol 49 pp 26ndash38 2015

[58] T C Schelling ldquoDynamic models of segregationrdquo Journal ofMathematical Sociology vol 1 no 2 pp 143ndash186 1971

[59] R Krafta V M Netto and L Lima ldquoUrban built form growscriticalrdquo CyberGeo vol 2011 article 565 2011

[60] N Lanchier and S Scarlatos ldquoFixation in the one-dimensionalAxelrod modelrdquo The Annals of Applied Probability vol 23 no6 pp 2538ndash2559 2013

[61] F L Ribeiro and K N Ribeiro ldquoA one dimensional modelof population growthrdquo Physica A Statistical Mechanics and itsApplications vol 434 pp 201ndash210 2015

[62] F L Ribeiro ldquoA non-phenomenological model of competitionand cooperation to explain population growth behaviorsrdquo Bul-letin of Mathematical Biology vol 77 no 3 pp 409ndash433 2015

16 Complexity

[63] R V dos Santos F L Ribeiro and A S Martinez ldquoModels forAllee effect based on physical principlesrdquo Journal of TheoreticalBiology vol 385 pp 143ndash152 2015

[64] M BattyThe New Science of Cities The MIT Press CambridgeMA 2013

[65] A G Wilson Entropy in Urban and Regional Modelling PionLondon 1970

[66] M Batty R Morphet P Masucci and K Stanilov ldquoEntropycomplexity and spatial informationrdquo Journal of GeographicalSystems vol 16 no 4 pp 363ndash385 2014

[67] R Axelrod ldquoThe dissemination of culture a model with localconvergence and global polarizationrdquo Journal of Conflict Reso-lution vol 41 no 2 pp 203ndash226 1997

[68] B H Hodges and RM Baron ldquoValues as constraints on afford-ances perceiving and acting properlyrdquo Journal for the Theory ofSocial Behaviour vol 22 no 3 pp 263ndash294 1992

[69] J JacobsThe Economy of Cities Random house New York NYUSA 1969

[70] E Glaeser H Kallal J Scheinkman and A Shleifer ldquoGrowthin Citiesrdquo Journal of Political Economy vol 100 no 6 pp 1126ndash1152 1992

[71] S Rosenthal and W Strange ldquoEvidence on the nature andsources of agglomeration economiesrdquo in Handbook of Urbanand Regional Economics J V Henderson and J-F Thisse Edsvol 4 pp 2119-2171 NorthHollandNewYork NYUSA 2004

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Complexity

Start of simulation

00000

00005

00010

00015

00020

o

f tot

al ac

tions

250 500 750 10000Type of action

Materially inactive spaceMaterially active space

t = 0

(a)

End of simulation

00000

00005

00010

00015

00020

o

f tot

al ac

tions

250 500 750 10000Type of action

Materially inactive spaceMaterially active space

t = 50

(b)

Figure 7 Histogram of probability distribution of actors performing different types of action at the start (a) and at the end (b) of simulationsin scenarios where distance is not considered (blue) and distance is considered aligning orientations in action coordination (red) Resultsare averaged for 30 runs for 125 parameter combinations

activityplace

currentaction

latentorientation

225

250

275

300

325

Entr

opy

10 20 30 40 500Time

Figure 8 Evolution of entropy colors in a RGB scale refer toparameter weights in decisions on actions in a next time-step andtheir impact on entropy levels in different simulations (under thefriction of distance) In the palette triangle each color representsa specific combination of the three parameters In each vertex aparameter reaches maximum value while the others have minimumvalue Results are averaged for 30 different runs for 125 parametercombinations

It just keeps what is already happening workingas ldquoreinforcement feedbackrdquo for either direction theagent (and the system) is going Current actions aremeans to conserve tendencies in the system

(iv) Activity places play a key role in the reduction ofentropy (red lines in Figures 8 and 9)The social infor-mation in space ldquocontaminatesrdquo agents they aligntheir actions through the social contents of placesWe have seen that space matters as extension Nowwe see that space also has a very active informationalpresence

(v) Different combinations of factors have differenteffects over entropy Figure 8 shows the transitions

between the three parameters and how nuancedweights resulting from blending factors matter inthe reduction of entropy For instance blending theweight of activity places and latent orientation on thenext action displays a strong reduction of entropy(pink lines in Figure 8) That means that when agentsdo not keep information from previous selections(ie 120573 asymp 0) the action system reduces more entropythan in other parameter combinations A strongshort-term memory leads the system into conservingitself and into a poorer capability to coordinateactions Also when current actions 120573 and activityplaces 120574 share a similar weight over the next action(orange lines on top) the social system does notexperience a great reduction of entropy as agents tendto reproduce their actions

(vi) Finally the reduction of entropy implies that theprobability of certain actions and interactionsincreases In practical terms this means morealignments between agents and more connectionsbetween actions (interactions) However if all agentsin the system reached the same orientation andentropy dropped to zero the system would lose inter-nal differentiation Agents would behave in the sameway say in a world with no personal differentiationspecialization or division of labour New orientations(therefore entropy) are necessary if the social systemis to keep differentiated agencies On the otherhand a system with maximum entropy would createdifference in actions to a point where no action coor-dination or interaction would be possible Clearly thissituation cannot be the case A social system requiresbalances neither full entropy nor total predictability

These results also suggest that we are dealing with distinctldquosocial memoriesrdquo a long-term memory is active in what we

Complexity 13

20

25

30En

trop

y

04 0602-latent orientation

04 0602-current action

20

25

30

Entr

opy

20

25

30

Entr

opy

04 0602-activity place

Figure 9 Parameter exploration isolated importance of each parameter in entropy (under the friction of distance) Each vertical set of dotsrepresents one fixed value for the highlighted parameter with the two other parameters assuming every possible combination within thetested range Each dot corresponds to the entropy value when a sufficiently large time-frame is used

call latent orientations Short-term memory is active in theinfluence of a current orientation over a new action In turnthe social information latent in activity places is a result ofsocial arrangements (say as firms a local economy and soon) which find certain stability changing much slower thenactions It is an extension of the social system projected ontourban space stabilizing the system to some extent

7 Cities and Social Interaction Conclusions

What does this approach bring to the state of the art on therelations of social interaction information and space and onsocial entropy and ABM in particular We have seen thatprevious approaches overlooked the coordination of actionas a key empirical and analytical problem Furthermorethey tend to have the thin spatiality of a passive territorialbackground (eg [16 20]) In turn although the conceptof ldquocollective actionrdquo is finally getting attention in urbanstudies [64] the problem of social organization is still largelyunderestimated The problem of entropy has been dealt withsince Wilsonrsquos [65] work on spatial interaction but has notreached the mainstream of the discipline and deals mostlywith urban form (eg [24]) and spatial distributions [66]not with social information and action Regarding agentbased models we are not aware of an approach that dealswith the problem of how cities and space are part of socialorganization In a sense our model is closer to Axelrodrsquos[67] ABM model of dissemination of culture His agentsexchange ldquoculturerdquo through direct contact with neighboursin a cellular automata model whereas our agents are mobileand actively deal with distance and social information inspace recognizing types of activities in places The intensityof exchanges between agents in Axelrodrsquos model is a functionof similarities between them In our model exchanges aremediated by the selection and interaction with space In ourunderstanding our contribution in terms of ABM lies onthe fact that a new and simple model is proposed to dealwith the material and informational dimensions of socialorganization not just as an illustration but as a proof ofconcept space can have causal presence if agents generatepreferences as detailed in a domain that coevolves with them

Now does the simulation model corroborate the theoryintroduced in this paper Our proposition cannot be empir-ically validated at this stage given that it is very hard to

assess peoplersquos choices and the whole panorama of actionsin a city Entropy may well be beyond observation In thesecircumstances simulations of the behaviour of agents underdifferent spatial conditions become useful to assess entitiesevents and entropic forces at play One expected result fromthe model is that space matters for coordinating actions butsimulations surprisingly showed that space is a necessary butnot sufficient condition for the reduction of entropy Neithermaterial cognitive resources nor individual powers of agentsare enough Information in activity places and a bit of agentsrsquopersonal history also play roles in the way social systemsdeal with their own entropy That would mean a causal butnondeterministic presence of space

Substantively this approach explores a subtle but funda-mental presence of cities in social organizationWe argue thaturban space materializes gradients of difference in potentialinteractions from less to more recognizable costly or likelyPlacing simple criteria including distance as a factor in select-ing activities our ABM showed that space becomes a meansfor producing differences in the probabilities of interactionincreasing chances of certain selections and convergences incollective action

In reality people recognize the social information ofactivity places along with uses of busy streets or local central-ities We are able to retrieve these differences from the builtenvironment and relate to them One of the key points of ourargument is that these differences help us make selectionsEvery time we move through the city and select places toperform we participate in the large-scale coordination ofaction We enact space as a ldquoreferential systemrdquo [49 50]a set of semantic bits indexing social activities informingand guiding daily decisions Space becomes inherently partof the conversion of orientations into interactions that isconnections that crisscross sequences of actions recreatingthe interaction system By means of a structure that bothmaterializes and restricts the quasi-endless combinatorial pos-sibilities of interaction this system can acquire sufficientinternal guidance to make its own reproduction possibleBy distributing sufficiently recognizable differences in theprobability of interaction cities express and release localforces of reproduction of social systems The interfaces ofaction cognitive and spatial systems transform entropiccomplexity into structured complexity constantly reorderingactions in time

14 Complexity

This materialistic viewpoint can be integrated with otherviews of social organization Further work may include therelational roles of cultures language social norms and othersteering media along with explorations of possibly similareffects of communication technologies over entropy At thisstage we attempted to integrate a social dimension (agentsengaged in coordination) a spatial dimension (an extensiveenvironment) and an informational dimension (as differ-ences in action and in the environment itself) a demandinginterdisciplinary effort The approach identifies a central rolefor coevolving agents actualizing aims as they coordinate (cf[68]) and a central role for the environment exterior butresponsive to agents leading to cumulative changes alongtheir historyThemodel also identifies roles for cognition andmemory in the ways agents deal with their own orientationsand choose their actions and activity places changing theirspatial environment with consequences in the overall levelsof entropy All this suggests that the behaviour of the modelcannot be reduced to or fully predicted from the behaviour ofindividual variables which is a key aspect of complexity

So one of the main aims of this paper was to assesswhether space as information environment had any effecton the coordination and entropy of interaction However ifphysical distance and social information in activity placesmatter what about city size and the internal spatial struc-tures of cities What about rural areas or rarefied suburbsSince our model explores an abstract circular city it onlybegins to answer such questions Our findings indicate thatshorter distances between activity places tend to reducesocial entropy Density seems to matter denser spatialities(as opposed to rarefied ones) have a role to play A pathfor further development here is Jacobsrsquos [69] idea that thediversity of activities generates positive externalities Anddiversity as spatial economics shows us has to do withthe size of cities and density of population (eg [70 71])Entropy may also be influenced by diversity Batty et al [66]have shown that information increases as cities get biggerThe remaining question is whether larger denser and moreinternally structured cities could create process and reduceentropy with more intensity than smaller cities transformingbig pools of activities into differences in the probability ofinteraction Although we proposed a unidimensional modelour agentsrsquo behaviour can be explored in more realisticrepresentations of cities testing the roles of density topologyand diversity while going beyond perfectly informed agentsBy proposing the ring model we intended to simplify acity to a minimum system where the spatial dimensionof social organization becomes intelligible without losingfundamental properties

Shouldwe conclude fromall this that entropy is a problemfor social systems Our approach suggests that it is notEntropy is a necessary force It means that novel orientationsare entering the interaction system and must be dealt withEntropy only becomes a problem if not converted into orga-nization only if numerous potential actions are not convertedinto smaller sets of actual interactions The effects of newactions over entropy would be positive once actualizedhaving to dowith the diversity of agencyConservative systemsare likely to face less entropy but also likely to produce

less novelty having homogeneity as a problematic horizonFluctuations of entropy seem vital to complex societies

Finally our intention was to explore entropy as a meansto think about the conditions of social organization ldquofromanother anglerdquo so to speak from the viewpoint of challengesinvolved in social reproduction Hence the approach focusedon the idea of uncertainty surrounding agents dealing withaims and decisions and the different probabilities of inter-action that follow an attempt to bring social cognitiveinformational and spatial theories under a single roof Ourapproach sees cities as connective systems produced to createthe delicate fabric of interaction that keeps large numbersof agents coherently living together It also suggests that thevarying states of entropy reveal deep connections betweensocial informational and spatial systems

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

Theauthors would like to thank the Ecole Polytechnique Fed-erale de Lausanne (EPFL) CAPES FAPEMIG FAPESP andCNPq for financial support Maıra Pinheiro Henrique Loreaand colleagues at ABMUS2017 for earlier discussions andMike Batty for exchanges and inspiration Finally the authorsthank Romulo Krafta maverick of the complexity studiesof cities in Brazil for his input and support throughout theyears This work is dedicated to him

References

[1] C W Mills The Sociological Imagination Oxford UniversityPress New York NY USA 1959

[2] M Castells ldquoThe information age economy society and cul-turerdquo inThe Rise of the Network Blackwell London 1996

[3] C Hidalgo Why Information Grows The Evolution of Orderfrom Atoms to Economies Basic Books New York NY USA2015

[4] D Helbing The Automation of Society Is Next ETH ZurichSwitzerland 2016

[5] N Luhmann Social Systems University Press Stanford 1995[6] I Prigogine and I Stengers Order out of Chaos Manrsquos New

Dialogue with Nature Bantam Books 1984[7] C E Shannon ldquoAmathematical theory of communicationrdquo Bell

System Technical Journal vol 27 no 4 pp 623ndash656 1948[8] W Buckley Sociology and Modern Systems Theory Prentice-

Hall Englewood Cliffs NJ USA 1967[9] D D McFarland ldquoMeasuring the Permeability of Occupational

Structures An Information-Theoretic Approachrdquo AmericanJournal of Sociology vol 75 no 1 pp 41ndash61 1969

[10] F Charvat ldquoOn philosophical aspects of the system conceptionin contemporary sociological knowledgerdquo Quality and Quan-tity vol 6 no 1 pp 3ndash16 1972

[11] F Charvat J Kucera and M Soukup ldquoToward the systemtheory of dependence further general theoretical remarksrdquoQuality and Quantity vol 7 no 1 pp 69ndash90 1973

Complexity 15

[12] P M Horan ldquoInformation-theoretic measures and the analysisof social structuresrdquo Sociological Methods and Research vol 3no 3 pp 321ndash340 1975

[13] O E Klapp ldquoOpening and closing in open systemsrdquo BehavioralScience vol 20 no 4 pp 251ndash257 1975

[14] J Galtung ldquoEntropy and the general theory of peacerdquo in Essaysin Peace Research J Galtung Ed vol L Humanities PressAtlantic Highlands NJ USA 1975

[15] T Parsons The System of Modern Societies Englewood CliffsNew Jersey NJ USA 1972

[16] K D Bailey ldquoSociological entropy theory Toward a statisticaland verbal congruencerdquo Quality amp Quantity vol 18 no 1 pp113ndash133 1983

[17] K D Bailey Social EntropyTheory State University of NewYorkPress Albany NY USA 1990

[18] J G Miller Living Systems McGraw-Hill New York NY USA1978

[19] K D Bailey Sociology and the New Systems Theory Toward aTheoretical Synthesis State University of New York PressAlbany NY USA 1994

[20] K D Bailey ldquoSociocybernetics and social entropy theoryrdquoKybernetes vol 35 no 3-4 pp 375ndash384 2006

[21] F Geyer ldquoThe challenge of sociocyberneticsrdquo Kybernetes vol24 no 4 pp 6ndash32 1995

[22] F Geyer ldquoThe march of self-referencerdquo Kybernetes vol 31 no7-8 pp 1021ndash1042 2002

[23] J HabermasTheTheory of Communicative Action Polity PressCambridge UK 1987

[24] H Haken and J Portugali Information Adaptation The Inter-play between Shannon Information and Semantic Information inCognition Springer New York NY USA 2015

[25] M Weber Economy and Society vol 1 Berkeley University ofCalifornia Press 1978

[26] L Wittgenstein Philosophical Investigations Blackwell Publish-ers Oxford London 3rd edition 2001

[27] L Vygotsky Mind in Society The Development of HigherPsychological Processes Harvard University Press CambridgeMA 1978

[28] M Wilson ldquoSix Views of Embodied Cognitionrdquo PsychonomicBulletin amp Review vol 9 no 4 pp 625ndash636 2002

[29] A Clark and D Chalmers ldquoThe extended mindrdquo Analysis vol58 no 1 pp 7ndash19 1998

[30] R PassiniWayfinding in Architecture Van Nostrand ReinholdNew York NY USA 1992

[31] E Rosch ldquoPrinciples of categorizationrdquo in Cognition and Cate-gorization E Rosch and B Lloyd Eds Hillsdale LawrenceErlbaum Ass 1978

[32] W KintschMemory and Cognition John Wiley and Sons NewYork NY USA 1970

[33] U Neisser ldquoMultiple Systems a New Approach to CognitiveTheoryrdquo European Journal of Cognitive Psychology vol 6 no 3pp 225ndash241 1994

[34] A Faria and R Krafta ldquoRepresenting urban cognitive structurethrough spatial differentiationrdquo in Proceedings of 4th Space Syn-tax International Symposium pp 531-518 UCL Press London2003

[35] J Portugali Complexity Cognition and the City Springer NewYork NY USA 2011

[36] D Kirsh and P Maglio ldquoOn distinguishing epistemic frompragmatic actionrdquo Cognitive Science vol 18 no 4 pp 513ndash5491994

[37] S M Kosslyn Image and Brain The Resolution of the ImageryDebate MIT Press Cambridge MA

[38] R Brooks Cambrian Intelligence The Early History of the NewAI MIT Press Cambridge MA 1999

[39] A Clark BeinGThere Putting Brain Body AndWorld TogetherAgain MIT Press Cambridge MA 1997

[40] R Pfeifer and C ScheierUnderstanding Intelligence MIT PressCambridge MA 1999

[41] A M Glenberg and D A Robertson ldquoIndexical understandingof instructionsrdquoDiscourse Processes vol 28 no 1 pp 1ndash26 1999

[42] JM Iverson and SGoldin-Meadow ldquoWhypeople gesturewhenthey speakrdquo Nature vol 396 no 6708 p 228 1998

[43] R M Krauss ldquoWhy do we gesture when we speakrdquo CurrentDirections in Psychological Science vol 7 no 2 pp 54ndash60 1998

[44] S Franklin Artificial Minds MIT Press Cambridge MA 1995[45] J Gibson The Ecological Approach to Visual Perception

Houghton-Mifflin Boston 1979[46] G Lakoff andM JohnsonPhilosophy in the FleshTheEmbodied

Mind and Its Challenge to Western Thought Basic Books NewYork NY USA 1999

[47] E HutchinsCognition in theWild MIT Press CambridgeMA1995

[48] E Hutchins ldquoThe cultural ecosystem of human cognitionrdquoPhilosophical Psychology vol 27 no 1 pp 34ndash49 2014

[49] V M Netto ldquoPractice space and the duality of meaningrdquoEnvironment and Planning D Society and Space vol 26 no 2pp 359ndash379 2008

[50] V M Netto The Social Fabric of Cities Routledge New YorkNY USA 2017

[51] T Parsons The Structure of Social Action The Free Press NewYork NY USA 1968

[52] T Allen Managing the Flow of Technology MIT Press Cam-bridge MA 1977

[53] F L Ribeiro J Meirelles F F Ferreira and R C Neto ldquoAmodel of urban scaling laws based on distance-dependent inter-actionsrdquo Royal Society Open Science vol 4 Article ID 1609262017

[54] W Alonso Location and Land Use Toward a General Theory ofLand Rent Harvard University Press Cambridge MA 1964

[55] WGHansen ldquoHow accessibility shapes land userdquo Journal of theAmerican Institute of Planners vol 25 no 2 pp 73ndash76 1959

[56] B Hillier Space Is the Machine Cambridge University PressCambridge UK 1996

[57] S Farber M OrsquoKelly H J Miller and T Neutens ldquoMeasuringsegregation using patterns of daily travel behavior a socialinteraction based model of exposurerdquo Journal of TransportGeography vol 49 pp 26ndash38 2015

[58] T C Schelling ldquoDynamic models of segregationrdquo Journal ofMathematical Sociology vol 1 no 2 pp 143ndash186 1971

[59] R Krafta V M Netto and L Lima ldquoUrban built form growscriticalrdquo CyberGeo vol 2011 article 565 2011

[60] N Lanchier and S Scarlatos ldquoFixation in the one-dimensionalAxelrod modelrdquo The Annals of Applied Probability vol 23 no6 pp 2538ndash2559 2013

[61] F L Ribeiro and K N Ribeiro ldquoA one dimensional modelof population growthrdquo Physica A Statistical Mechanics and itsApplications vol 434 pp 201ndash210 2015

[62] F L Ribeiro ldquoA non-phenomenological model of competitionand cooperation to explain population growth behaviorsrdquo Bul-letin of Mathematical Biology vol 77 no 3 pp 409ndash433 2015

16 Complexity

[63] R V dos Santos F L Ribeiro and A S Martinez ldquoModels forAllee effect based on physical principlesrdquo Journal of TheoreticalBiology vol 385 pp 143ndash152 2015

[64] M BattyThe New Science of Cities The MIT Press CambridgeMA 2013

[65] A G Wilson Entropy in Urban and Regional Modelling PionLondon 1970

[66] M Batty R Morphet P Masucci and K Stanilov ldquoEntropycomplexity and spatial informationrdquo Journal of GeographicalSystems vol 16 no 4 pp 363ndash385 2014

[67] R Axelrod ldquoThe dissemination of culture a model with localconvergence and global polarizationrdquo Journal of Conflict Reso-lution vol 41 no 2 pp 203ndash226 1997

[68] B H Hodges and RM Baron ldquoValues as constraints on afford-ances perceiving and acting properlyrdquo Journal for the Theory ofSocial Behaviour vol 22 no 3 pp 263ndash294 1992

[69] J JacobsThe Economy of Cities Random house New York NYUSA 1969

[70] E Glaeser H Kallal J Scheinkman and A Shleifer ldquoGrowthin Citiesrdquo Journal of Political Economy vol 100 no 6 pp 1126ndash1152 1992

[71] S Rosenthal and W Strange ldquoEvidence on the nature andsources of agglomeration economiesrdquo in Handbook of Urbanand Regional Economics J V Henderson and J-F Thisse Edsvol 4 pp 2119-2171 NorthHollandNewYork NYUSA 2004

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Complexity 13

20

25

30En

trop

y

04 0602-latent orientation

04 0602-current action

20

25

30

Entr

opy

20

25

30

Entr

opy

04 0602-activity place

Figure 9 Parameter exploration isolated importance of each parameter in entropy (under the friction of distance) Each vertical set of dotsrepresents one fixed value for the highlighted parameter with the two other parameters assuming every possible combination within thetested range Each dot corresponds to the entropy value when a sufficiently large time-frame is used

call latent orientations Short-term memory is active in theinfluence of a current orientation over a new action In turnthe social information latent in activity places is a result ofsocial arrangements (say as firms a local economy and soon) which find certain stability changing much slower thenactions It is an extension of the social system projected ontourban space stabilizing the system to some extent

7 Cities and Social Interaction Conclusions

What does this approach bring to the state of the art on therelations of social interaction information and space and onsocial entropy and ABM in particular We have seen thatprevious approaches overlooked the coordination of actionas a key empirical and analytical problem Furthermorethey tend to have the thin spatiality of a passive territorialbackground (eg [16 20]) In turn although the conceptof ldquocollective actionrdquo is finally getting attention in urbanstudies [64] the problem of social organization is still largelyunderestimated The problem of entropy has been dealt withsince Wilsonrsquos [65] work on spatial interaction but has notreached the mainstream of the discipline and deals mostlywith urban form (eg [24]) and spatial distributions [66]not with social information and action Regarding agentbased models we are not aware of an approach that dealswith the problem of how cities and space are part of socialorganization In a sense our model is closer to Axelrodrsquos[67] ABM model of dissemination of culture His agentsexchange ldquoculturerdquo through direct contact with neighboursin a cellular automata model whereas our agents are mobileand actively deal with distance and social information inspace recognizing types of activities in places The intensityof exchanges between agents in Axelrodrsquos model is a functionof similarities between them In our model exchanges aremediated by the selection and interaction with space In ourunderstanding our contribution in terms of ABM lies onthe fact that a new and simple model is proposed to dealwith the material and informational dimensions of socialorganization not just as an illustration but as a proof ofconcept space can have causal presence if agents generatepreferences as detailed in a domain that coevolves with them

Now does the simulation model corroborate the theoryintroduced in this paper Our proposition cannot be empir-ically validated at this stage given that it is very hard to

assess peoplersquos choices and the whole panorama of actionsin a city Entropy may well be beyond observation In thesecircumstances simulations of the behaviour of agents underdifferent spatial conditions become useful to assess entitiesevents and entropic forces at play One expected result fromthe model is that space matters for coordinating actions butsimulations surprisingly showed that space is a necessary butnot sufficient condition for the reduction of entropy Neithermaterial cognitive resources nor individual powers of agentsare enough Information in activity places and a bit of agentsrsquopersonal history also play roles in the way social systemsdeal with their own entropy That would mean a causal butnondeterministic presence of space

Substantively this approach explores a subtle but funda-mental presence of cities in social organizationWe argue thaturban space materializes gradients of difference in potentialinteractions from less to more recognizable costly or likelyPlacing simple criteria including distance as a factor in select-ing activities our ABM showed that space becomes a meansfor producing differences in the probabilities of interactionincreasing chances of certain selections and convergences incollective action

In reality people recognize the social information ofactivity places along with uses of busy streets or local central-ities We are able to retrieve these differences from the builtenvironment and relate to them One of the key points of ourargument is that these differences help us make selectionsEvery time we move through the city and select places toperform we participate in the large-scale coordination ofaction We enact space as a ldquoreferential systemrdquo [49 50]a set of semantic bits indexing social activities informingand guiding daily decisions Space becomes inherently partof the conversion of orientations into interactions that isconnections that crisscross sequences of actions recreatingthe interaction system By means of a structure that bothmaterializes and restricts the quasi-endless combinatorial pos-sibilities of interaction this system can acquire sufficientinternal guidance to make its own reproduction possibleBy distributing sufficiently recognizable differences in theprobability of interaction cities express and release localforces of reproduction of social systems The interfaces ofaction cognitive and spatial systems transform entropiccomplexity into structured complexity constantly reorderingactions in time

14 Complexity

This materialistic viewpoint can be integrated with otherviews of social organization Further work may include therelational roles of cultures language social norms and othersteering media along with explorations of possibly similareffects of communication technologies over entropy At thisstage we attempted to integrate a social dimension (agentsengaged in coordination) a spatial dimension (an extensiveenvironment) and an informational dimension (as differ-ences in action and in the environment itself) a demandinginterdisciplinary effort The approach identifies a central rolefor coevolving agents actualizing aims as they coordinate (cf[68]) and a central role for the environment exterior butresponsive to agents leading to cumulative changes alongtheir historyThemodel also identifies roles for cognition andmemory in the ways agents deal with their own orientationsand choose their actions and activity places changing theirspatial environment with consequences in the overall levelsof entropy All this suggests that the behaviour of the modelcannot be reduced to or fully predicted from the behaviour ofindividual variables which is a key aspect of complexity

So one of the main aims of this paper was to assesswhether space as information environment had any effecton the coordination and entropy of interaction However ifphysical distance and social information in activity placesmatter what about city size and the internal spatial struc-tures of cities What about rural areas or rarefied suburbsSince our model explores an abstract circular city it onlybegins to answer such questions Our findings indicate thatshorter distances between activity places tend to reducesocial entropy Density seems to matter denser spatialities(as opposed to rarefied ones) have a role to play A pathfor further development here is Jacobsrsquos [69] idea that thediversity of activities generates positive externalities Anddiversity as spatial economics shows us has to do withthe size of cities and density of population (eg [70 71])Entropy may also be influenced by diversity Batty et al [66]have shown that information increases as cities get biggerThe remaining question is whether larger denser and moreinternally structured cities could create process and reduceentropy with more intensity than smaller cities transformingbig pools of activities into differences in the probability ofinteraction Although we proposed a unidimensional modelour agentsrsquo behaviour can be explored in more realisticrepresentations of cities testing the roles of density topologyand diversity while going beyond perfectly informed agentsBy proposing the ring model we intended to simplify acity to a minimum system where the spatial dimensionof social organization becomes intelligible without losingfundamental properties

Shouldwe conclude fromall this that entropy is a problemfor social systems Our approach suggests that it is notEntropy is a necessary force It means that novel orientationsare entering the interaction system and must be dealt withEntropy only becomes a problem if not converted into orga-nization only if numerous potential actions are not convertedinto smaller sets of actual interactions The effects of newactions over entropy would be positive once actualizedhaving to dowith the diversity of agencyConservative systemsare likely to face less entropy but also likely to produce

less novelty having homogeneity as a problematic horizonFluctuations of entropy seem vital to complex societies

Finally our intention was to explore entropy as a meansto think about the conditions of social organization ldquofromanother anglerdquo so to speak from the viewpoint of challengesinvolved in social reproduction Hence the approach focusedon the idea of uncertainty surrounding agents dealing withaims and decisions and the different probabilities of inter-action that follow an attempt to bring social cognitiveinformational and spatial theories under a single roof Ourapproach sees cities as connective systems produced to createthe delicate fabric of interaction that keeps large numbersof agents coherently living together It also suggests that thevarying states of entropy reveal deep connections betweensocial informational and spatial systems

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

Theauthors would like to thank the Ecole Polytechnique Fed-erale de Lausanne (EPFL) CAPES FAPEMIG FAPESP andCNPq for financial support Maıra Pinheiro Henrique Loreaand colleagues at ABMUS2017 for earlier discussions andMike Batty for exchanges and inspiration Finally the authorsthank Romulo Krafta maverick of the complexity studiesof cities in Brazil for his input and support throughout theyears This work is dedicated to him

References

[1] C W Mills The Sociological Imagination Oxford UniversityPress New York NY USA 1959

[2] M Castells ldquoThe information age economy society and cul-turerdquo inThe Rise of the Network Blackwell London 1996

[3] C Hidalgo Why Information Grows The Evolution of Orderfrom Atoms to Economies Basic Books New York NY USA2015

[4] D Helbing The Automation of Society Is Next ETH ZurichSwitzerland 2016

[5] N Luhmann Social Systems University Press Stanford 1995[6] I Prigogine and I Stengers Order out of Chaos Manrsquos New

Dialogue with Nature Bantam Books 1984[7] C E Shannon ldquoAmathematical theory of communicationrdquo Bell

System Technical Journal vol 27 no 4 pp 623ndash656 1948[8] W Buckley Sociology and Modern Systems Theory Prentice-

Hall Englewood Cliffs NJ USA 1967[9] D D McFarland ldquoMeasuring the Permeability of Occupational

Structures An Information-Theoretic Approachrdquo AmericanJournal of Sociology vol 75 no 1 pp 41ndash61 1969

[10] F Charvat ldquoOn philosophical aspects of the system conceptionin contemporary sociological knowledgerdquo Quality and Quan-tity vol 6 no 1 pp 3ndash16 1972

[11] F Charvat J Kucera and M Soukup ldquoToward the systemtheory of dependence further general theoretical remarksrdquoQuality and Quantity vol 7 no 1 pp 69ndash90 1973

Complexity 15

[12] P M Horan ldquoInformation-theoretic measures and the analysisof social structuresrdquo Sociological Methods and Research vol 3no 3 pp 321ndash340 1975

[13] O E Klapp ldquoOpening and closing in open systemsrdquo BehavioralScience vol 20 no 4 pp 251ndash257 1975

[14] J Galtung ldquoEntropy and the general theory of peacerdquo in Essaysin Peace Research J Galtung Ed vol L Humanities PressAtlantic Highlands NJ USA 1975

[15] T Parsons The System of Modern Societies Englewood CliffsNew Jersey NJ USA 1972

[16] K D Bailey ldquoSociological entropy theory Toward a statisticaland verbal congruencerdquo Quality amp Quantity vol 18 no 1 pp113ndash133 1983

[17] K D Bailey Social EntropyTheory State University of NewYorkPress Albany NY USA 1990

[18] J G Miller Living Systems McGraw-Hill New York NY USA1978

[19] K D Bailey Sociology and the New Systems Theory Toward aTheoretical Synthesis State University of New York PressAlbany NY USA 1994

[20] K D Bailey ldquoSociocybernetics and social entropy theoryrdquoKybernetes vol 35 no 3-4 pp 375ndash384 2006

[21] F Geyer ldquoThe challenge of sociocyberneticsrdquo Kybernetes vol24 no 4 pp 6ndash32 1995

[22] F Geyer ldquoThe march of self-referencerdquo Kybernetes vol 31 no7-8 pp 1021ndash1042 2002

[23] J HabermasTheTheory of Communicative Action Polity PressCambridge UK 1987

[24] H Haken and J Portugali Information Adaptation The Inter-play between Shannon Information and Semantic Information inCognition Springer New York NY USA 2015

[25] M Weber Economy and Society vol 1 Berkeley University ofCalifornia Press 1978

[26] L Wittgenstein Philosophical Investigations Blackwell Publish-ers Oxford London 3rd edition 2001

[27] L Vygotsky Mind in Society The Development of HigherPsychological Processes Harvard University Press CambridgeMA 1978

[28] M Wilson ldquoSix Views of Embodied Cognitionrdquo PsychonomicBulletin amp Review vol 9 no 4 pp 625ndash636 2002

[29] A Clark and D Chalmers ldquoThe extended mindrdquo Analysis vol58 no 1 pp 7ndash19 1998

[30] R PassiniWayfinding in Architecture Van Nostrand ReinholdNew York NY USA 1992

[31] E Rosch ldquoPrinciples of categorizationrdquo in Cognition and Cate-gorization E Rosch and B Lloyd Eds Hillsdale LawrenceErlbaum Ass 1978

[32] W KintschMemory and Cognition John Wiley and Sons NewYork NY USA 1970

[33] U Neisser ldquoMultiple Systems a New Approach to CognitiveTheoryrdquo European Journal of Cognitive Psychology vol 6 no 3pp 225ndash241 1994

[34] A Faria and R Krafta ldquoRepresenting urban cognitive structurethrough spatial differentiationrdquo in Proceedings of 4th Space Syn-tax International Symposium pp 531-518 UCL Press London2003

[35] J Portugali Complexity Cognition and the City Springer NewYork NY USA 2011

[36] D Kirsh and P Maglio ldquoOn distinguishing epistemic frompragmatic actionrdquo Cognitive Science vol 18 no 4 pp 513ndash5491994

[37] S M Kosslyn Image and Brain The Resolution of the ImageryDebate MIT Press Cambridge MA

[38] R Brooks Cambrian Intelligence The Early History of the NewAI MIT Press Cambridge MA 1999

[39] A Clark BeinGThere Putting Brain Body AndWorld TogetherAgain MIT Press Cambridge MA 1997

[40] R Pfeifer and C ScheierUnderstanding Intelligence MIT PressCambridge MA 1999

[41] A M Glenberg and D A Robertson ldquoIndexical understandingof instructionsrdquoDiscourse Processes vol 28 no 1 pp 1ndash26 1999

[42] JM Iverson and SGoldin-Meadow ldquoWhypeople gesturewhenthey speakrdquo Nature vol 396 no 6708 p 228 1998

[43] R M Krauss ldquoWhy do we gesture when we speakrdquo CurrentDirections in Psychological Science vol 7 no 2 pp 54ndash60 1998

[44] S Franklin Artificial Minds MIT Press Cambridge MA 1995[45] J Gibson The Ecological Approach to Visual Perception

Houghton-Mifflin Boston 1979[46] G Lakoff andM JohnsonPhilosophy in the FleshTheEmbodied

Mind and Its Challenge to Western Thought Basic Books NewYork NY USA 1999

[47] E HutchinsCognition in theWild MIT Press CambridgeMA1995

[48] E Hutchins ldquoThe cultural ecosystem of human cognitionrdquoPhilosophical Psychology vol 27 no 1 pp 34ndash49 2014

[49] V M Netto ldquoPractice space and the duality of meaningrdquoEnvironment and Planning D Society and Space vol 26 no 2pp 359ndash379 2008

[50] V M Netto The Social Fabric of Cities Routledge New YorkNY USA 2017

[51] T Parsons The Structure of Social Action The Free Press NewYork NY USA 1968

[52] T Allen Managing the Flow of Technology MIT Press Cam-bridge MA 1977

[53] F L Ribeiro J Meirelles F F Ferreira and R C Neto ldquoAmodel of urban scaling laws based on distance-dependent inter-actionsrdquo Royal Society Open Science vol 4 Article ID 1609262017

[54] W Alonso Location and Land Use Toward a General Theory ofLand Rent Harvard University Press Cambridge MA 1964

[55] WGHansen ldquoHow accessibility shapes land userdquo Journal of theAmerican Institute of Planners vol 25 no 2 pp 73ndash76 1959

[56] B Hillier Space Is the Machine Cambridge University PressCambridge UK 1996

[57] S Farber M OrsquoKelly H J Miller and T Neutens ldquoMeasuringsegregation using patterns of daily travel behavior a socialinteraction based model of exposurerdquo Journal of TransportGeography vol 49 pp 26ndash38 2015

[58] T C Schelling ldquoDynamic models of segregationrdquo Journal ofMathematical Sociology vol 1 no 2 pp 143ndash186 1971

[59] R Krafta V M Netto and L Lima ldquoUrban built form growscriticalrdquo CyberGeo vol 2011 article 565 2011

[60] N Lanchier and S Scarlatos ldquoFixation in the one-dimensionalAxelrod modelrdquo The Annals of Applied Probability vol 23 no6 pp 2538ndash2559 2013

[61] F L Ribeiro and K N Ribeiro ldquoA one dimensional modelof population growthrdquo Physica A Statistical Mechanics and itsApplications vol 434 pp 201ndash210 2015

[62] F L Ribeiro ldquoA non-phenomenological model of competitionand cooperation to explain population growth behaviorsrdquo Bul-letin of Mathematical Biology vol 77 no 3 pp 409ndash433 2015

16 Complexity

[63] R V dos Santos F L Ribeiro and A S Martinez ldquoModels forAllee effect based on physical principlesrdquo Journal of TheoreticalBiology vol 385 pp 143ndash152 2015

[64] M BattyThe New Science of Cities The MIT Press CambridgeMA 2013

[65] A G Wilson Entropy in Urban and Regional Modelling PionLondon 1970

[66] M Batty R Morphet P Masucci and K Stanilov ldquoEntropycomplexity and spatial informationrdquo Journal of GeographicalSystems vol 16 no 4 pp 363ndash385 2014

[67] R Axelrod ldquoThe dissemination of culture a model with localconvergence and global polarizationrdquo Journal of Conflict Reso-lution vol 41 no 2 pp 203ndash226 1997

[68] B H Hodges and RM Baron ldquoValues as constraints on afford-ances perceiving and acting properlyrdquo Journal for the Theory ofSocial Behaviour vol 22 no 3 pp 263ndash294 1992

[69] J JacobsThe Economy of Cities Random house New York NYUSA 1969

[70] E Glaeser H Kallal J Scheinkman and A Shleifer ldquoGrowthin Citiesrdquo Journal of Political Economy vol 100 no 6 pp 1126ndash1152 1992

[71] S Rosenthal and W Strange ldquoEvidence on the nature andsources of agglomeration economiesrdquo in Handbook of Urbanand Regional Economics J V Henderson and J-F Thisse Edsvol 4 pp 2119-2171 NorthHollandNewYork NYUSA 2004

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

14 Complexity

This materialistic viewpoint can be integrated with otherviews of social organization Further work may include therelational roles of cultures language social norms and othersteering media along with explorations of possibly similareffects of communication technologies over entropy At thisstage we attempted to integrate a social dimension (agentsengaged in coordination) a spatial dimension (an extensiveenvironment) and an informational dimension (as differ-ences in action and in the environment itself) a demandinginterdisciplinary effort The approach identifies a central rolefor coevolving agents actualizing aims as they coordinate (cf[68]) and a central role for the environment exterior butresponsive to agents leading to cumulative changes alongtheir historyThemodel also identifies roles for cognition andmemory in the ways agents deal with their own orientationsand choose their actions and activity places changing theirspatial environment with consequences in the overall levelsof entropy All this suggests that the behaviour of the modelcannot be reduced to or fully predicted from the behaviour ofindividual variables which is a key aspect of complexity

So one of the main aims of this paper was to assesswhether space as information environment had any effecton the coordination and entropy of interaction However ifphysical distance and social information in activity placesmatter what about city size and the internal spatial struc-tures of cities What about rural areas or rarefied suburbsSince our model explores an abstract circular city it onlybegins to answer such questions Our findings indicate thatshorter distances between activity places tend to reducesocial entropy Density seems to matter denser spatialities(as opposed to rarefied ones) have a role to play A pathfor further development here is Jacobsrsquos [69] idea that thediversity of activities generates positive externalities Anddiversity as spatial economics shows us has to do withthe size of cities and density of population (eg [70 71])Entropy may also be influenced by diversity Batty et al [66]have shown that information increases as cities get biggerThe remaining question is whether larger denser and moreinternally structured cities could create process and reduceentropy with more intensity than smaller cities transformingbig pools of activities into differences in the probability ofinteraction Although we proposed a unidimensional modelour agentsrsquo behaviour can be explored in more realisticrepresentations of cities testing the roles of density topologyand diversity while going beyond perfectly informed agentsBy proposing the ring model we intended to simplify acity to a minimum system where the spatial dimensionof social organization becomes intelligible without losingfundamental properties

Shouldwe conclude fromall this that entropy is a problemfor social systems Our approach suggests that it is notEntropy is a necessary force It means that novel orientationsare entering the interaction system and must be dealt withEntropy only becomes a problem if not converted into orga-nization only if numerous potential actions are not convertedinto smaller sets of actual interactions The effects of newactions over entropy would be positive once actualizedhaving to dowith the diversity of agencyConservative systemsare likely to face less entropy but also likely to produce

less novelty having homogeneity as a problematic horizonFluctuations of entropy seem vital to complex societies

Finally our intention was to explore entropy as a meansto think about the conditions of social organization ldquofromanother anglerdquo so to speak from the viewpoint of challengesinvolved in social reproduction Hence the approach focusedon the idea of uncertainty surrounding agents dealing withaims and decisions and the different probabilities of inter-action that follow an attempt to bring social cognitiveinformational and spatial theories under a single roof Ourapproach sees cities as connective systems produced to createthe delicate fabric of interaction that keeps large numbersof agents coherently living together It also suggests that thevarying states of entropy reveal deep connections betweensocial informational and spatial systems

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

Theauthors would like to thank the Ecole Polytechnique Fed-erale de Lausanne (EPFL) CAPES FAPEMIG FAPESP andCNPq for financial support Maıra Pinheiro Henrique Loreaand colleagues at ABMUS2017 for earlier discussions andMike Batty for exchanges and inspiration Finally the authorsthank Romulo Krafta maverick of the complexity studiesof cities in Brazil for his input and support throughout theyears This work is dedicated to him

References

[1] C W Mills The Sociological Imagination Oxford UniversityPress New York NY USA 1959

[2] M Castells ldquoThe information age economy society and cul-turerdquo inThe Rise of the Network Blackwell London 1996

[3] C Hidalgo Why Information Grows The Evolution of Orderfrom Atoms to Economies Basic Books New York NY USA2015

[4] D Helbing The Automation of Society Is Next ETH ZurichSwitzerland 2016

[5] N Luhmann Social Systems University Press Stanford 1995[6] I Prigogine and I Stengers Order out of Chaos Manrsquos New

Dialogue with Nature Bantam Books 1984[7] C E Shannon ldquoAmathematical theory of communicationrdquo Bell

System Technical Journal vol 27 no 4 pp 623ndash656 1948[8] W Buckley Sociology and Modern Systems Theory Prentice-

Hall Englewood Cliffs NJ USA 1967[9] D D McFarland ldquoMeasuring the Permeability of Occupational

Structures An Information-Theoretic Approachrdquo AmericanJournal of Sociology vol 75 no 1 pp 41ndash61 1969

[10] F Charvat ldquoOn philosophical aspects of the system conceptionin contemporary sociological knowledgerdquo Quality and Quan-tity vol 6 no 1 pp 3ndash16 1972

[11] F Charvat J Kucera and M Soukup ldquoToward the systemtheory of dependence further general theoretical remarksrdquoQuality and Quantity vol 7 no 1 pp 69ndash90 1973

Complexity 15

[12] P M Horan ldquoInformation-theoretic measures and the analysisof social structuresrdquo Sociological Methods and Research vol 3no 3 pp 321ndash340 1975

[13] O E Klapp ldquoOpening and closing in open systemsrdquo BehavioralScience vol 20 no 4 pp 251ndash257 1975

[14] J Galtung ldquoEntropy and the general theory of peacerdquo in Essaysin Peace Research J Galtung Ed vol L Humanities PressAtlantic Highlands NJ USA 1975

[15] T Parsons The System of Modern Societies Englewood CliffsNew Jersey NJ USA 1972

[16] K D Bailey ldquoSociological entropy theory Toward a statisticaland verbal congruencerdquo Quality amp Quantity vol 18 no 1 pp113ndash133 1983

[17] K D Bailey Social EntropyTheory State University of NewYorkPress Albany NY USA 1990

[18] J G Miller Living Systems McGraw-Hill New York NY USA1978

[19] K D Bailey Sociology and the New Systems Theory Toward aTheoretical Synthesis State University of New York PressAlbany NY USA 1994

[20] K D Bailey ldquoSociocybernetics and social entropy theoryrdquoKybernetes vol 35 no 3-4 pp 375ndash384 2006

[21] F Geyer ldquoThe challenge of sociocyberneticsrdquo Kybernetes vol24 no 4 pp 6ndash32 1995

[22] F Geyer ldquoThe march of self-referencerdquo Kybernetes vol 31 no7-8 pp 1021ndash1042 2002

[23] J HabermasTheTheory of Communicative Action Polity PressCambridge UK 1987

[24] H Haken and J Portugali Information Adaptation The Inter-play between Shannon Information and Semantic Information inCognition Springer New York NY USA 2015

[25] M Weber Economy and Society vol 1 Berkeley University ofCalifornia Press 1978

[26] L Wittgenstein Philosophical Investigations Blackwell Publish-ers Oxford London 3rd edition 2001

[27] L Vygotsky Mind in Society The Development of HigherPsychological Processes Harvard University Press CambridgeMA 1978

[28] M Wilson ldquoSix Views of Embodied Cognitionrdquo PsychonomicBulletin amp Review vol 9 no 4 pp 625ndash636 2002

[29] A Clark and D Chalmers ldquoThe extended mindrdquo Analysis vol58 no 1 pp 7ndash19 1998

[30] R PassiniWayfinding in Architecture Van Nostrand ReinholdNew York NY USA 1992

[31] E Rosch ldquoPrinciples of categorizationrdquo in Cognition and Cate-gorization E Rosch and B Lloyd Eds Hillsdale LawrenceErlbaum Ass 1978

[32] W KintschMemory and Cognition John Wiley and Sons NewYork NY USA 1970

[33] U Neisser ldquoMultiple Systems a New Approach to CognitiveTheoryrdquo European Journal of Cognitive Psychology vol 6 no 3pp 225ndash241 1994

[34] A Faria and R Krafta ldquoRepresenting urban cognitive structurethrough spatial differentiationrdquo in Proceedings of 4th Space Syn-tax International Symposium pp 531-518 UCL Press London2003

[35] J Portugali Complexity Cognition and the City Springer NewYork NY USA 2011

[36] D Kirsh and P Maglio ldquoOn distinguishing epistemic frompragmatic actionrdquo Cognitive Science vol 18 no 4 pp 513ndash5491994

[37] S M Kosslyn Image and Brain The Resolution of the ImageryDebate MIT Press Cambridge MA

[38] R Brooks Cambrian Intelligence The Early History of the NewAI MIT Press Cambridge MA 1999

[39] A Clark BeinGThere Putting Brain Body AndWorld TogetherAgain MIT Press Cambridge MA 1997

[40] R Pfeifer and C ScheierUnderstanding Intelligence MIT PressCambridge MA 1999

[41] A M Glenberg and D A Robertson ldquoIndexical understandingof instructionsrdquoDiscourse Processes vol 28 no 1 pp 1ndash26 1999

[42] JM Iverson and SGoldin-Meadow ldquoWhypeople gesturewhenthey speakrdquo Nature vol 396 no 6708 p 228 1998

[43] R M Krauss ldquoWhy do we gesture when we speakrdquo CurrentDirections in Psychological Science vol 7 no 2 pp 54ndash60 1998

[44] S Franklin Artificial Minds MIT Press Cambridge MA 1995[45] J Gibson The Ecological Approach to Visual Perception

Houghton-Mifflin Boston 1979[46] G Lakoff andM JohnsonPhilosophy in the FleshTheEmbodied

Mind and Its Challenge to Western Thought Basic Books NewYork NY USA 1999

[47] E HutchinsCognition in theWild MIT Press CambridgeMA1995

[48] E Hutchins ldquoThe cultural ecosystem of human cognitionrdquoPhilosophical Psychology vol 27 no 1 pp 34ndash49 2014

[49] V M Netto ldquoPractice space and the duality of meaningrdquoEnvironment and Planning D Society and Space vol 26 no 2pp 359ndash379 2008

[50] V M Netto The Social Fabric of Cities Routledge New YorkNY USA 2017

[51] T Parsons The Structure of Social Action The Free Press NewYork NY USA 1968

[52] T Allen Managing the Flow of Technology MIT Press Cam-bridge MA 1977

[53] F L Ribeiro J Meirelles F F Ferreira and R C Neto ldquoAmodel of urban scaling laws based on distance-dependent inter-actionsrdquo Royal Society Open Science vol 4 Article ID 1609262017

[54] W Alonso Location and Land Use Toward a General Theory ofLand Rent Harvard University Press Cambridge MA 1964

[55] WGHansen ldquoHow accessibility shapes land userdquo Journal of theAmerican Institute of Planners vol 25 no 2 pp 73ndash76 1959

[56] B Hillier Space Is the Machine Cambridge University PressCambridge UK 1996

[57] S Farber M OrsquoKelly H J Miller and T Neutens ldquoMeasuringsegregation using patterns of daily travel behavior a socialinteraction based model of exposurerdquo Journal of TransportGeography vol 49 pp 26ndash38 2015

[58] T C Schelling ldquoDynamic models of segregationrdquo Journal ofMathematical Sociology vol 1 no 2 pp 143ndash186 1971

[59] R Krafta V M Netto and L Lima ldquoUrban built form growscriticalrdquo CyberGeo vol 2011 article 565 2011

[60] N Lanchier and S Scarlatos ldquoFixation in the one-dimensionalAxelrod modelrdquo The Annals of Applied Probability vol 23 no6 pp 2538ndash2559 2013

[61] F L Ribeiro and K N Ribeiro ldquoA one dimensional modelof population growthrdquo Physica A Statistical Mechanics and itsApplications vol 434 pp 201ndash210 2015

[62] F L Ribeiro ldquoA non-phenomenological model of competitionand cooperation to explain population growth behaviorsrdquo Bul-letin of Mathematical Biology vol 77 no 3 pp 409ndash433 2015

16 Complexity

[63] R V dos Santos F L Ribeiro and A S Martinez ldquoModels forAllee effect based on physical principlesrdquo Journal of TheoreticalBiology vol 385 pp 143ndash152 2015

[64] M BattyThe New Science of Cities The MIT Press CambridgeMA 2013

[65] A G Wilson Entropy in Urban and Regional Modelling PionLondon 1970

[66] M Batty R Morphet P Masucci and K Stanilov ldquoEntropycomplexity and spatial informationrdquo Journal of GeographicalSystems vol 16 no 4 pp 363ndash385 2014

[67] R Axelrod ldquoThe dissemination of culture a model with localconvergence and global polarizationrdquo Journal of Conflict Reso-lution vol 41 no 2 pp 203ndash226 1997

[68] B H Hodges and RM Baron ldquoValues as constraints on afford-ances perceiving and acting properlyrdquo Journal for the Theory ofSocial Behaviour vol 22 no 3 pp 263ndash294 1992

[69] J JacobsThe Economy of Cities Random house New York NYUSA 1969

[70] E Glaeser H Kallal J Scheinkman and A Shleifer ldquoGrowthin Citiesrdquo Journal of Political Economy vol 100 no 6 pp 1126ndash1152 1992

[71] S Rosenthal and W Strange ldquoEvidence on the nature andsources of agglomeration economiesrdquo in Handbook of Urbanand Regional Economics J V Henderson and J-F Thisse Edsvol 4 pp 2119-2171 NorthHollandNewYork NYUSA 2004

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Complexity 15

[12] P M Horan ldquoInformation-theoretic measures and the analysisof social structuresrdquo Sociological Methods and Research vol 3no 3 pp 321ndash340 1975

[13] O E Klapp ldquoOpening and closing in open systemsrdquo BehavioralScience vol 20 no 4 pp 251ndash257 1975

[14] J Galtung ldquoEntropy and the general theory of peacerdquo in Essaysin Peace Research J Galtung Ed vol L Humanities PressAtlantic Highlands NJ USA 1975

[15] T Parsons The System of Modern Societies Englewood CliffsNew Jersey NJ USA 1972

[16] K D Bailey ldquoSociological entropy theory Toward a statisticaland verbal congruencerdquo Quality amp Quantity vol 18 no 1 pp113ndash133 1983

[17] K D Bailey Social EntropyTheory State University of NewYorkPress Albany NY USA 1990

[18] J G Miller Living Systems McGraw-Hill New York NY USA1978

[19] K D Bailey Sociology and the New Systems Theory Toward aTheoretical Synthesis State University of New York PressAlbany NY USA 1994

[20] K D Bailey ldquoSociocybernetics and social entropy theoryrdquoKybernetes vol 35 no 3-4 pp 375ndash384 2006

[21] F Geyer ldquoThe challenge of sociocyberneticsrdquo Kybernetes vol24 no 4 pp 6ndash32 1995

[22] F Geyer ldquoThe march of self-referencerdquo Kybernetes vol 31 no7-8 pp 1021ndash1042 2002

[23] J HabermasTheTheory of Communicative Action Polity PressCambridge UK 1987

[24] H Haken and J Portugali Information Adaptation The Inter-play between Shannon Information and Semantic Information inCognition Springer New York NY USA 2015

[25] M Weber Economy and Society vol 1 Berkeley University ofCalifornia Press 1978

[26] L Wittgenstein Philosophical Investigations Blackwell Publish-ers Oxford London 3rd edition 2001

[27] L Vygotsky Mind in Society The Development of HigherPsychological Processes Harvard University Press CambridgeMA 1978

[28] M Wilson ldquoSix Views of Embodied Cognitionrdquo PsychonomicBulletin amp Review vol 9 no 4 pp 625ndash636 2002

[29] A Clark and D Chalmers ldquoThe extended mindrdquo Analysis vol58 no 1 pp 7ndash19 1998

[30] R PassiniWayfinding in Architecture Van Nostrand ReinholdNew York NY USA 1992

[31] E Rosch ldquoPrinciples of categorizationrdquo in Cognition and Cate-gorization E Rosch and B Lloyd Eds Hillsdale LawrenceErlbaum Ass 1978

[32] W KintschMemory and Cognition John Wiley and Sons NewYork NY USA 1970

[33] U Neisser ldquoMultiple Systems a New Approach to CognitiveTheoryrdquo European Journal of Cognitive Psychology vol 6 no 3pp 225ndash241 1994

[34] A Faria and R Krafta ldquoRepresenting urban cognitive structurethrough spatial differentiationrdquo in Proceedings of 4th Space Syn-tax International Symposium pp 531-518 UCL Press London2003

[35] J Portugali Complexity Cognition and the City Springer NewYork NY USA 2011

[36] D Kirsh and P Maglio ldquoOn distinguishing epistemic frompragmatic actionrdquo Cognitive Science vol 18 no 4 pp 513ndash5491994

[37] S M Kosslyn Image and Brain The Resolution of the ImageryDebate MIT Press Cambridge MA

[38] R Brooks Cambrian Intelligence The Early History of the NewAI MIT Press Cambridge MA 1999

[39] A Clark BeinGThere Putting Brain Body AndWorld TogetherAgain MIT Press Cambridge MA 1997

[40] R Pfeifer and C ScheierUnderstanding Intelligence MIT PressCambridge MA 1999

[41] A M Glenberg and D A Robertson ldquoIndexical understandingof instructionsrdquoDiscourse Processes vol 28 no 1 pp 1ndash26 1999

[42] JM Iverson and SGoldin-Meadow ldquoWhypeople gesturewhenthey speakrdquo Nature vol 396 no 6708 p 228 1998

[43] R M Krauss ldquoWhy do we gesture when we speakrdquo CurrentDirections in Psychological Science vol 7 no 2 pp 54ndash60 1998

[44] S Franklin Artificial Minds MIT Press Cambridge MA 1995[45] J Gibson The Ecological Approach to Visual Perception

Houghton-Mifflin Boston 1979[46] G Lakoff andM JohnsonPhilosophy in the FleshTheEmbodied

Mind and Its Challenge to Western Thought Basic Books NewYork NY USA 1999

[47] E HutchinsCognition in theWild MIT Press CambridgeMA1995

[48] E Hutchins ldquoThe cultural ecosystem of human cognitionrdquoPhilosophical Psychology vol 27 no 1 pp 34ndash49 2014

[49] V M Netto ldquoPractice space and the duality of meaningrdquoEnvironment and Planning D Society and Space vol 26 no 2pp 359ndash379 2008

[50] V M Netto The Social Fabric of Cities Routledge New YorkNY USA 2017

[51] T Parsons The Structure of Social Action The Free Press NewYork NY USA 1968

[52] T Allen Managing the Flow of Technology MIT Press Cam-bridge MA 1977

[53] F L Ribeiro J Meirelles F F Ferreira and R C Neto ldquoAmodel of urban scaling laws based on distance-dependent inter-actionsrdquo Royal Society Open Science vol 4 Article ID 1609262017

[54] W Alonso Location and Land Use Toward a General Theory ofLand Rent Harvard University Press Cambridge MA 1964

[55] WGHansen ldquoHow accessibility shapes land userdquo Journal of theAmerican Institute of Planners vol 25 no 2 pp 73ndash76 1959

[56] B Hillier Space Is the Machine Cambridge University PressCambridge UK 1996

[57] S Farber M OrsquoKelly H J Miller and T Neutens ldquoMeasuringsegregation using patterns of daily travel behavior a socialinteraction based model of exposurerdquo Journal of TransportGeography vol 49 pp 26ndash38 2015

[58] T C Schelling ldquoDynamic models of segregationrdquo Journal ofMathematical Sociology vol 1 no 2 pp 143ndash186 1971

[59] R Krafta V M Netto and L Lima ldquoUrban built form growscriticalrdquo CyberGeo vol 2011 article 565 2011

[60] N Lanchier and S Scarlatos ldquoFixation in the one-dimensionalAxelrod modelrdquo The Annals of Applied Probability vol 23 no6 pp 2538ndash2559 2013

[61] F L Ribeiro and K N Ribeiro ldquoA one dimensional modelof population growthrdquo Physica A Statistical Mechanics and itsApplications vol 434 pp 201ndash210 2015

[62] F L Ribeiro ldquoA non-phenomenological model of competitionand cooperation to explain population growth behaviorsrdquo Bul-letin of Mathematical Biology vol 77 no 3 pp 409ndash433 2015

16 Complexity

[63] R V dos Santos F L Ribeiro and A S Martinez ldquoModels forAllee effect based on physical principlesrdquo Journal of TheoreticalBiology vol 385 pp 143ndash152 2015

[64] M BattyThe New Science of Cities The MIT Press CambridgeMA 2013

[65] A G Wilson Entropy in Urban and Regional Modelling PionLondon 1970

[66] M Batty R Morphet P Masucci and K Stanilov ldquoEntropycomplexity and spatial informationrdquo Journal of GeographicalSystems vol 16 no 4 pp 363ndash385 2014

[67] R Axelrod ldquoThe dissemination of culture a model with localconvergence and global polarizationrdquo Journal of Conflict Reso-lution vol 41 no 2 pp 203ndash226 1997

[68] B H Hodges and RM Baron ldquoValues as constraints on afford-ances perceiving and acting properlyrdquo Journal for the Theory ofSocial Behaviour vol 22 no 3 pp 263ndash294 1992

[69] J JacobsThe Economy of Cities Random house New York NYUSA 1969

[70] E Glaeser H Kallal J Scheinkman and A Shleifer ldquoGrowthin Citiesrdquo Journal of Political Economy vol 100 no 6 pp 1126ndash1152 1992

[71] S Rosenthal and W Strange ldquoEvidence on the nature andsources of agglomeration economiesrdquo in Handbook of Urbanand Regional Economics J V Henderson and J-F Thisse Edsvol 4 pp 2119-2171 NorthHollandNewYork NYUSA 2004

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

16 Complexity

[63] R V dos Santos F L Ribeiro and A S Martinez ldquoModels forAllee effect based on physical principlesrdquo Journal of TheoreticalBiology vol 385 pp 143ndash152 2015

[64] M BattyThe New Science of Cities The MIT Press CambridgeMA 2013

[65] A G Wilson Entropy in Urban and Regional Modelling PionLondon 1970

[66] M Batty R Morphet P Masucci and K Stanilov ldquoEntropycomplexity and spatial informationrdquo Journal of GeographicalSystems vol 16 no 4 pp 363ndash385 2014

[67] R Axelrod ldquoThe dissemination of culture a model with localconvergence and global polarizationrdquo Journal of Conflict Reso-lution vol 41 no 2 pp 203ndash226 1997

[68] B H Hodges and RM Baron ldquoValues as constraints on afford-ances perceiving and acting properlyrdquo Journal for the Theory ofSocial Behaviour vol 22 no 3 pp 263ndash294 1992

[69] J JacobsThe Economy of Cities Random house New York NYUSA 1969

[70] E Glaeser H Kallal J Scheinkman and A Shleifer ldquoGrowthin Citiesrdquo Journal of Political Economy vol 100 no 6 pp 1126ndash1152 1992

[71] S Rosenthal and W Strange ldquoEvidence on the nature andsources of agglomeration economiesrdquo in Handbook of Urbanand Regional Economics J V Henderson and J-F Thisse Edsvol 4 pp 2119-2171 NorthHollandNewYork NYUSA 2004

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of