Cities and Cellular Automata · Discrete Dynamicsin NatureandSociety, Vol. 2, pp. 111-125 Reprints...

Discrete Dynamics in Nature and Society, Vol. 2, pp. 111-125

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Cities and Cellular AutomataROGER WHITE

Department of Geography, Memorial University of Newfoundland, St. John’s, Newfoundland, A1B 3X9, Canada

(Received 7 December 1997)

Cellular automata provide a high-resolution representation of urban spatial dynamics.Consequently they give the most realistic predictions of urban structural evolution, and inparticular they are able to replicate the various fractal dimensionalities of actual cities.However, since these models do not readily incorporate certain phenomena like densitymeasures and long-distance (as opposed to neighbourhood) spatial interactions, theirperformance may be enhanced by integrating them with other types of urban models.Cellular automata based models promise deeper theoretical insights into the nature of citiesas self-organizing structures.

Keywords." Cellular automata, Urban spatial dynamics, Model integration, Fractal dimensions,Complexity

1 INTRODUCTION

As computing power continues to expand, micro-simulation models become increasingly interestingas ways to understand complex systems. Theapproach has much to recommend it, since it getsto the level at which objects or agents act tostructure a system, and thus allows us to explorethe phenomenon realistically while imposing aminimum of arbitrary simplifying assumptions.Nevertheless, the strength of the approach is alsoits weakness: as the scale of the map approaches1:1, it no longer provides an overview of thecountry. In other words, the detailed nature ofexplanation inherent in micro-simulation modelsmay make it difficult to understand the generalnature of emergent properties. On the other hand,macro-scale models may give extremely useful

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descriptions of emergent behaviour, even if theydo not show how these macroscopic structuresarise. Clearly it is desirable to have both levels ofdescription, since each complements the other, andthe micro-level provides a deeper understanding ofthe reasons for the macro-scale phenomena. Buteven better is to have an approach that has proper-ties of both levels, and provides a link between thetwo. Cellular automata (CA) are such an approach.

Cellular automata are well suited to urban model-ling, being inherently dynamic and intrinsicallyspatial, with good spatial resolution. Although theyare remarkably simple mechanisms they can gen-erate spatio-temporal patterns of unlimited com-

plexity (Wolfram, 1984; Langton, 1990; 1992; BakChen and Creutz, 1989), and that is one reasonwhy they are such powerful modelling tools.Nevertheless, cities are even more complex, being

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generated by a large number of processes operatingat a variety of spatial and temporal scales. Thus inorder to use CA for modelling cities with a usefuldegree of realism, it is necessary to use more

complex CA that is, CA with more states, a

larger neighbourhood, and more complex transi-tion rules. It is also necessary to constrain the CAin various ways in order to represent the effect ofprocesses operating at scales that cannot be directlyincluded in the CA transition rules by in effectmodifying the CA transition rules as the CA runs.

DEFINING CELLULAR AUTOMATAFOR URBAN MODELLING

A CA may be defined as (1) a discrete cell space,together with (2) a set ofpossible cell states and (3) aset oftransition rules that determine the state ofeachcell as afunction of the states ofall cells within (4) a

defined cell-space neighbourhood of the cell; (5) time

is discrete and all cell states are updated simulta-neously at each iteration. While the concept is re-

markably simple, and many of the most intensivelystudied CA, like Game of Life are, in fact, almosttrivially simple as mechanisms, the definition leavesquite a bit of room for variety, and the CA thathave been most useful in urban modelling tend todepart significantly from the ideal of simplicity. Itis useful, therefore, to elaborate on this definitionfrom the point of view of urban modelling.

(1) Cell Space. The most important character-istics of cell space are dimensionality, cell size andcell shape, and homogeneity or heterogeneity. CAdeveloped for urban modelling are, not sur-prisingly, typically defined on a two dimensionalcell space, though it is easy to imagine using athree dimensional cell space in order to be able torepresent building height, or, more generally,density of land use. One dimensional CA, whileless common in modelling applications, are beingused for urban traffic modelling (Nagel et al.,1997; Di Gregorio et al., 1996). Cell size in currentapplications ranges from 250m (e.g. White et al.,1997) down to tens of metres (Batty and Xie,

1994). At the latter scale, cells may representcadastral units (i.e. actual land lots), which mayhave any shape; thus while it is often convenientfor various reasons (e.g. run time) to use a regularcell space, that is not necessary. In conventionalCA the cell space on which the cell state dynamicsis defined is homogeneous that is, all cells areidentical except for their state. In modelling cities,however, where it is natural to let cell statesrepresent land use or land cover, it is useful towork on a heterogeneous cell space, where eachcell may have an intrinsic quality (or a vector ofqualities) representing such characteristics of thereal geographical space as soil quality, slope, or

land use regulations.(2) Cell State. In most urban modelling appli-

cations, cell states represent land use and landcover, but depending on the focus of the model,they may represent other features of the urbanarea; Portugali and Benenson (1995; 1996), forexample, use cell states to represent social cate-gories, e.g. immigrant vs. native populations. And,broadening the idea of CA, the conventionalBoolean cell state could acquire a cardinal mea-sure in order to represent such characteristics as

building quality or density. Finally, it is often con-venient to define some cell states as fixed, so thatwhile cells in these states can influence the statetransitions of other cells, they are not themselvessubject to changes of state (e.g. White et al., 1997).Rivers and parks are examples of land uses thatcould well be represented by fixed states.

(3) Transition Rules. These are the heart of CAsince they represent the process that is beingmodelled; they are thus the key to a good applica-tion. They may be deterministic or stochastic; verysimple (e.g. a cell changes to the modal state ofits neighbourhood), or quite elaborate, like thetransition rules of the urban model describedbelow in Section 4. Typically the rules are fixed,but some work has been done with simple CAthat evolve to solve a test problem, and there is

potential for use of CA with rules that evolveendogenously to represent such phenomena asurban real estate speculation.

CITIES AND CELLULAR AUTOMATA 113

(4) Neighbourhood. Conventionally, the neigh-bourhood to which the transition rules are appliedis small. In two dimensional rectilinear CA themost commonly used neighbourhoods are theVon Neumann (4-cell) and the Moore (8-cell).These are convenient, and for many physicalprocesses, where field effects are absent and inter-action occurs only through contiguity, they are

entirely appropriate. They are occasionally usedfor urban modelling (e.g. Cecchini and Viola,1990; 1992); but in general larger neighbourhoodswould seem to be more appropriate, since landuse decisions are made by people who are awareof and able to take into account conditions overa larger area than that represented by a Mooreneighbourhood. In general, the size of the neigh-bourhood should be defined with reference to thedistance over which the processes represented bythe transition rules operate.

(5) Time. Discrete time with simultaneous cellstate transitions is standard. However, there maybe situations where it is convenient to nest timescales, with some parts of the CA running anumber of time steps for each step in the rest ofthe model. For example, in a model of a region inwhich some areas are subject to seasonal inunda-tion, the low-lying areas are modelled on a

monthly basis to capture the dynamics of theflooding, while upland areas are modelled with atime step of one year (Uljee et al., 1996). A ratherdifferent approach to time would be to updatecell states sequentially; such an approach woulddepend on having a method for choosing thesequence of cells that was inherently meaningfulin terms of the modelling problem.

CELLULAR AUTOMATA BASEDURBAN MODELS

A number of CA models of urban systems havebeen developed, together with several modellingsystems. Tobler (1979) was the first to propose acellular approach to geographical modelling, andhis idea was followed up by Couclelis (1985; 1988;

1989; 1996) and later Takeyama (1996). Couclelisuses the ideas of CA modelling to explore thenature of space and spatial relations in the contextof dynamics, but does not attempt applications tospecific cities. However, she points out that whilestandard spatial interaction based approaches tothe modelling of urban and regional systems pos-tulate a relational space, usually of low resolution,in which absolute location is virtually irrelevant,GIS (Geographical Information Systems), on theother hand, presuppose an absolute space, andone of very high resolution. Both approaches areappropriate for certain problems, but an adequaterepresentation of a city will involve both relationaland absolute spaces. As she says,

On the one hand is absolute space and the concrete

geo-referenced location or object with attributes,that knows nothing of its surroundings," on the otheris relative space and the varified complex spatialrelation that is so innocent of the specifics ofplace as

to be dubbed "the geography ofnowhere". In betweenis proximal space and the generalized CA models,that can now...partake of the earthy data richness

of GIS while probing hypotheses about the largescale effects of micro-scale interactions. That thesemodels can now.., be extended into the realm ofrelative space, where the most robust theories ofurban and regional growth are still to be found, is

a[n].., exciting prospect. (Couclelis, 1996, p. 174)

Couclelis (1996) and Takeyama (1996) thus pro-pose a generalized modelling language which wouldpermit integrated dynamic spatial modelling at allscales within a GIS framework.

Batty and Xie (1996) and Xie (1996) have devel-oped a CA for urban modelling which generatesboth land use patterns and an associated transpor-tation network. Their use of a CA to generate anetwork is interesting and novel, but not yetentirely successful; one problem may be that CAprocesses are inherently local, whereas the evolu-tion of a transportation network must reflect bothlocal and long range interactions. Their models arenot developed to model actual cities, nor do theycarry out sensitivity analysis to determine general

114 R. WHITE

characteristics of model behaviour. They concludethat

It is unlikely that current approaches can reallygenerate acceptable levels ofperformance in terms ofthe goodness-of-fit to existing patterns in the same

way in which less complex, traditional models are

able, and thus the emphasis in application should notbe on fit but on feasibility and plausibility. (Battyand Xie, 1996, p. 203)

It is significant that, like Couclelis, they discusstheir CA models in the context of "traditional" (i.e."Lowry-type") spatial interaction based models,thus recognizing a complementarity. The issue of"goodness-of-fit" is two pronged. On the one hand,CA, by virtue of their spatial detail, permit ex-tremely realistic results; the traditional approachescapture only highly aggregated measures of urbanstructure, though they may be "realistic" in thesense that an appropriately specified and well-calibrated model may yield a very good value fora measure of goodness-of-fit. However, this merelybrings into stark relief the fact that measures ofgoodness-of-fit are highly generalized measures,whereas a city is an extremely complex entity.A model capable of producing truly realisticresults will thus produce a complex output, andat present there are no appropriate measures ofgoodness-of-fit. As modelling techniques advance,and with them the quality of the model output,this problem is becoming critical; we will return toit below.

Portugali and Benenson (1995; 1997) andPortugali et al. (1994), while investigating generalprinciples of urban self-organization by means ofCA models, have gone farther in the search forempirical realism. They are concerned primarilywith urban social structure, and the attitudes whichengender and are in turn modified by that structure.They work with models which they characterize asmodels ofhuman agents in cell space. The cell spacerepresents individual residential lots, and cell statesare actually vectors representing such qualities asstatus and value of the dwelling as well as the char-acteristics of the family inhabiting it. Transition

rules relate changes in dwelling characteristics tothe characteristics of families in the cell neigh-bourhood, and also relate changes in familycharacteristics (e.g. ethnicity or attitude towardneighbours) to the characteristics of families anddwellings in the neighbourhood. These modelsclearly demonstrate the close link between basictheoretical ideas and empirical realism that is oneof the useful characteristics of cellular automatamodelling.

Other applications of CA to urban structure

(Cecchini, 1996; Cecchini and Viola 1990; 1992)have emphasized simple automata, for exampletwo state (urban, rural) or small neighbourhood(Von Neumann or Moore) formulations. Makseet al. (1995) have adapted a physical model, amodel of correlated percolation in the presence of adensity gradient, to the problem of urban form.They use the two state model to simulate thegrowth of an urbanized area, and show that theresult has a fractal structure. The fractal dimen-sions they generate are almost identical to thosecharacterizing the output of other cellular models(see Section 4 below) (White and Engelen, 1993b;White et al., 1997). Since the latter models do not

impose a density gradient, it would seem that thatfeature of the correlated percolation model is not

necessary: a global density gradient can be gener-ated by the local dynamics of the CA itself, a notuncommon phenomenon.

CONSTRAINED CA-BASEDURBAN LAND USE MODELS

White and Engelen (1993a,b; 1994; 1997b), Whiteet al. (1997), and Engelen et al. (1997a) have alsodeveloped CA based urban models thatjoin theoret-ical concerns with empirical realism. These are theonly CA based urban models that have been sub-jected to extensive sensitivity analysis, and also to a

degree of empirical testing. It is thus useful to focuson these models both in order to get an indicationof the quality of the results possible with CA basedurban models, and also to examine more closely


some of the major issues that arise in urbanmodelling with CA.

Unlike the models of Portugali and Benenson,which focus on socio-economic phenomena, orthose of Batty and Xie which use a cellular mechan-ism to generate a transport network, these are

essentially land use models. The focus is specificallyon the dynamics of land use and land cover, whilethe transport network is taken as given (though itmay be altered or extended exogenously during a

simulation). Land use and land cover types aredivided into two categories: fixed features, likewater bodies and parks, and active land use

functions like housing, commerce, and industry.In the neighbourhood dynamics defined for theCA, fixed features appear as arguments in thetransition rule functions, and so affect the changeof state from one active land use function toanother, but they are not themselves subject tostate changes (other than changes determinedexogenously to the CA mechanism itself). Likethe fixed features, the transportation network mayaffect the transition probabilities of active cells, butis not itself changed by the cellular dynamics.The transition rules applied to active cells

incorporate five types of factors:

intrinsic suitabilities representing inhomogene-ities in the geographic space being modelled(e.g. soil quality or legal restrictions);the neighbourhood effect, representing the attrac-tive or repulsive effects of the various land uses

(cell states) in the neighbourhood consisting ofthe 113 cells lying within a radius of 6 cells of thetarget cell;a local accessibility effect, representing ease ofaccess to the transport network;a stochastic perturbation capturing the effect ofimperfect knowledge and varying needs andtastes among the implicit actors whose decisionsare represented by cell state transitions;global constraints on the number of cells in eachstate, reflecting the fact that the requirements forland for various activities are largely determinedexogenously to the internal land use dynamics of

the city being modelled, depending rather onsuch factors as the size of the city and the needfor a balance among various activities. Thus ateach iteration, a target number of cells for eachstate is specified, and the CA determines not thenumber of cells in each state, but only theirlocation. The target numbers for each cell statemay simply be read from a file, or generated by alinked model as described below.

The first four factors are incorporated into thecalculation of a vector of transition potentials foreach active cell that is, a potential is calculatedfor each possible (active) cell state. Then thetransition rule is applied: each cell is changed tothe state for which it has the highest potential,subject to the global constraints.The potentials are calculated as follows:

Phj vajsj(1 + ZZZmkdlia + Hj (1)k d

where:

Phi is the transition potential from state h tostate j;

rnkd-- a weighting parameter applied to cells withstate k in distance zone d (cells that are closerusually are weighted more heavily; but weightsmay be positive, representing an attractive effect,or negative, if two states are incompatible);Iid-- if the state of cell i-k, otherwise Iid= 0,where is the index of cells within the distancezone d (the role of Iid is simply to ensure that onlythe relevant weight rnka for the cell at locationi,d- i.e. the weight corresponding to cell’sactual state is included in the calculation);/-/j.- an inertia parameter, with /-/j. > 0 if j- h;otherwise, /-/j.-O (/-/j. increases the likelihoodthat a cell will remain in its current state);sj-the suitability of the cell state for j, whereO_<sj_< 1;aj.-the accessibility parameter,

aj (1 + O/(j) -1 (2)

where D is the Euclidean distance from the cell to

116 R. WHITE

the nearest link of the network andis a coefficient expressing the importance ofaccessibility on the desirability of the cell forland use activity j. Accessibility is thus definedstrictly locally, in terms of distance to thenetwork, rather than globally in terms ofdistances through the network to distant points;v--the stochastic disturbance term,

v + (-ln(r)) (3)

where (0 < r < 1) is a uniform random variateand c is a parameter that allows the size of theperturbation to be adjusted.

The random term permits the cellular city to

explore its state space, since cells that are appar-ently unsuited to a particular activity will never-theless occasionally be converted to that state. Sincethe inertia term will normally prevent a cell fromreverting immediately to its previous state, these

cells may function as nuclei of new developmentsby increasing the transition potentials of other cellsin their neighbourhood.

This generic cellular model has been used to

generate test cities (Fig. 1) on both a homogeneouscell space (i.e. space with no suitabilities or trans-

portation network) and a cell space homogeneousexcept for the presence ofa transportation network,for purposes of sensitivity analysis. It has also beenused as the basis for simulations of actual citiesand regions (Figs. 2, 4). The results are surprisinglyrealistic for such a simple model, especially forone lacking any representation of long-distanceinteractions.The most striking feature is that all cellular cities

generated with versions of this generic modelexhibit a fractal form:

The perimeter of the urbanized area (as definedby cells in any state representing an urban land

FIGURE Simulation of a generic city generated on a homogeneous cell space, with a transport system represented by "road"cells. Land uses: dark, commerce; medium, industry; light, housing.


use like housing or commerce) is a fractal(White and Engelen, 1993b).The radial dimensions, calculated as the deriva-tive of log(area) with respect to log(radius), ofboth the urbanized area and of individual urbanland uses are bi-fractals. For example, the urban-ized area is characterized by a radial dimensionof approximately 1.92 out to a critical radius;beyond that radius the dimension falls to a valuein the neighbourhood of 1.1. Inside the criticalradius the cellular pattern is relatively stable,representing an urbanization process that isessentially complete; outside it, the cell stateconfigurations change rapidly, as the CAgenerates the future urban form (White andEngelen, 1993b; White et al., 1997).The radial dimensions of the individual land.uses are significantly different from each other,reflecting a relative concentration of the variousland uses in successive concentric zones, with,for example, commerce relatively concentratedtoward the centre and housing toward theperiphery (White and Engelen, 1993b; Whiteet al., 1997).The size-frequency spectra of land use clustersare log-linear, another measure of fractal struc-ture (White and Engelen 1993b; White et al.,1997).

In all cases, actual cities are found to exhibit thesame fractal dimensionality (White and Engelen,1.993b; White et al., 1997; Frankhauser, 1991;1994). Frankhauser (1994), for example, foundthat for 17 of 19 world cities, the radial dimensionof the urbanized area within the critical radius("rayon de s{grgation") ranged in value between1.94 and 1.99. 1960 Land use data for Cincinnati,USA, yielded a log-linear size-frequency spectrumwith a slope of 1.29 for clusters of commercial land;a cellular simulation of the city to the same yearalso gave a log-linear spectrum, with a slope of 1.44(White et al., 1997). And Batty and Longley (1994)determined the fractal dimension of the perimeterof the Cardiff, UK, urbanized area to lie in therange of 1.2-1.3. Thus in terms of their variousfractal dimensions, cellular cities would appear tobe highly similar to actual cities.

Fractal dimensions are, however, highly generalmeasures. One of the potential strengths of cellularmodelling lies in the spatial detail, and hence therealism, that it can provide; and while fractility is ameasure of the complexity of the urban form, it is a

very abstract one, and thus does not capture theparticular form of specific cities or their cellularsimulations. Cellular simulations of actual cities,for example Cincinnati (Fig. 2), using the actualtransportation network of major roads and

FIGURE 2 Simulation of Cincinnati (left) and actual land use (simplified), 1960 (right); the road system used in the simulationis not shown. See Colour Plate I.

118 R. WHITE

suitability maps representing the slope of the land,give results that are visually quite similar in form tothe land use map of the actual city.

In the case of Cincinnati, the correspondence ofthe simulated map to the actual map of the city isconfirmed by Kappa index values (see Monserudand Leemans, 1992) ranging from 0.51 (fair) forcommerce to 0.63-0.69 (good) for the other landuses. However, Kappa is based on a cell-by-cellcoincidence matrix, and so frequently does not

capture the more general similarity of patterns.This problem is evident in the comparison of thesimulated and actual locations of commercialactivity (Fig. 3), where it is clear that the two loca-tional patterns are quite similar, even though veryfew of the cells coincide. A recently developed mapcomparison technique making use of fuzzy set

theory (Power, 1998) is based on a comparison oflocal configurations as opposed to individual cellcoincidences. An application to the Cincinnati casegives similarity values about 15% higher than theKappa index. Thus while the various quantitativemeasures of the modelling results support the

FIGURE 3 Comparison of simulated and actual locationsof commerce, for simulation shown in Fig. 2. See ColourPlate II.

conclusion that the cellular approach is worth-while, they serve primarily to reveal the inadequacyof the current kit of techniques for assessing spatialpatterns a problem noted by Mandelbrot (1983)among others. In other words, our ability to modelreality is outrunning our ability to evaluate theresults.

LIMITATIONS OF CA DUE TOSHORT RANGE INTERACTIONS

While it is gratifying that the results of modellingwith cellular automata are as realistic as they appearto be, it is also, in one respect, somewhat puzzling.The fractal nature of the patterns generated bycalibrated CA models implies spatial structure atall scales i.e. spatial autocorrelation up to thedimension of the grid (e.g. 80 x 80 cells) even

though the processes that generate the structureoperate only over short distances (e.g. 6 cell units).There is no question about how this happens: theiterative CA process can, given sufficient time,transmit a signal from any cell on the grid to anyother. Rather, the mystery is why the results are so

realistic, since in real cities the spatial interactionsthat generate spatial structure occur not only at a

local scale corresponding to the CA neighbour-hood, but also at regional or long distance scaleswhich are not represented at all in the CA models.Must we to conclude, counter intuitively, that theenormous volume of long distance interactionswithin cities shopping trips, for example, or thedaily trips to work are irrelevant to the spatialstructuring of the city?Most existing models of the dynamics of urban

spatial structure (e.g. Allen, 1983; 1997; White,1977; 1978; Wilson, 1974; 1976) are based on repre-sentations of these long distance flows; and as wehave seen, Couclelis (1996), Takeyama (1996), andBatty and Xie (1996) have reaffirmed the need formodels of this sort, as have White and Engelen(1994). Fortunately for tradition and intuition, thereare indications that the cellular models do nothandle all aspects of urban spatial structure as well


as the spatial interaction based models. Forexample, cellular models of the sort characterizedby Eqs. (1)-(3) represent land use qualitatively, interms of land use type, and do so with excellentspatial resolution; but they do not representquantities such as density. In other words, lackinga measure of intensity, the cellular model, in spiteof its spatial resolution, cannot give a good quanti-tative representation of the location of activity.Thus, if only for such mundane reasons, it willfrequently be desirable to supplement a CA modelwith a traditional spatial interaction based repre-sentation of urban structural dynamics. But inaddition, there is some evidence that intuition is

correct, and that long range interactions are impor-tant in shaping the urban spatial structure; this isseen most clearly in the commercial sector.Major retail centres compete for business at the

scale of the entire urban area. The process ofcompetition for customers, which determines therevenues of the centres and thus ultimately theirsuccess or failure, must therefore be modelled in a

way that captures the long distance spatial inter-action involved, for example by means of a Lowry-type model, as mentioned by Batty and Xie. Themodels of urban retail centre dynamics due toWilson (1974; 1976) and White (1977; 1978) areboth based on long distance spatial interactionmodels. Extensive simulation experiments with asimilar model (White and Engelen, 1997b) indicatethat the system of major retail centres is character-ized by two qualitatively different types of stablestates, depending on the values of relevant param-eters: in one of these a single centre is dominant; inthe other (convex) state, a number of centres ofsimilar size dominate the hierarchy. If the system iscaused to pass through this bifurcation, it mayspend a significant time in a meta-stable phasetransition state in which the distribution of centresizes is log linear, representing a fractal structure(Nicolis, 1989). Between the late 1950s and the early1980s, most US cities passed through this bifurca-tion, spending a decade or two in the log linear,fractal transition, and then ending up clearly in theconvex state (White and Engelen, 1997b). By

contrast, in all cases where the CA model basedon Eqs. (1)-(3) was realistically calibrated, com-

merce was fractally distributed indeed, with thismodel it is difficult to generate distributions thatmatch either of the stable configurations. The goodagreement between the simulated and actual sizedistributions of commerce clusters for Cincinnatiwould thus seem to be a fluke, due to the fact thatthe simulation was tested against 1960 data, whenthe city was coincidentally in the transition betweensteady states.

INTEGRATING CELLULAR AUTOMATAWITH OTHER MODELS

In view of these problems, and in recognition of thefact that activity patterns in cities are dominated byinteractions beyond the scale of a cellular neigh-bourhood, it would seem obvious to link CAmodels of urban structure with other, standarddynamical models (Uljee et al., 1996; White andEngelen, 1997b). In this approach the dynamicalmodel operates at a macro-scale, in contrast withthe micro-scale of the cellular model. It may beeither non-spatial, consisting, for example, of a

model of economic and demographic change in theurban area as a whole, or spatial, as in the case ofthe standard spatial interaction based modelsoperating on statistical units like census tracts.Only in the latter case, of course, can it possiblymake up for the deficiencies of the cellular modeldescribed in the previous section. In either case,however, the macro-scale model constrains thecellular dynamics by providing target numbersfor cells of each state, but in the case of the spatialmodel the targets may be regionally specific.For example, a macro-scale model may generate

a demand for a certain number of housing cells,with the actual locations to be determined by thecellular model; but if the macro-scale model isregionalized and so contains its own representationof spatial dynamics, then the model will allocatethe total predicted demand among the regions, so

that the cellular model then works with regionalrather than global targets.

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The discussion in this section is based on

experience with an integrated model consisting ofthe cellular model described above (Section 4)linked to a dynamic, regionalized spatial interac-tion based model of urban structure (Fig. 4). Theinteresting problems arise, of course, in linking thetwo levels. While sets of cells can be associatedunambiguously with regions, the same is notnecessarily true of the activities associated withthe spatial units. State variables and parametersassociated with cells can generally be aggregated insome appropriate way over all cells correspondingto a region and then assigned to that region in themacro-scale model. But if regional data is to bepassed to the cellular model, it is frequently not

possible to assign it exclusively to the set of cellscorresponding to the region, as we will see below.Thus in general the relationship between an activ-ity variable in the regional model and the re-

gional expression of that variable in cell spaceis one-to-many. Furthermore, while in simple

models there may be a one-to-one correspondencebetween the activity variables as defined in theregional model and the cell states defined in the CAmodel (e.g. population housing), in general thisis not the case: the population variable in theregional model may correspond to several CA cellstates (e.g. population [(1) single family housing,(2) low rise apartments, (3) high-rise apartments]),and vice versa.A typical macro-scale model, specifically a re-

gionalized, spatial interaction based model ofurban structure dynamics, will re-allocate activity

(X0.) e.g. population, industry, offices, jobs, etc.

from one region (i) to another (j) on the basis ofthe quantity of activity in each of the two regions(Xi, .), the distance between them (d0-), and thedistance weighted attraction of all competingregions (Vk):

x0. f(x;, d0,i,j, k 1,..., R number of regions. (4)

FIGURE 4 Simulation of St. John’s (note the cellular structure of the land use map), with inset showing the 40 regions onwhich the macro-scale regional model operates. See Colour Plate llI.


But as Couclelis pointed out, one of the majorshortcomings of such a model is that its space ispurely relational; the model makes no use ofdata with absolute spatial coordinates i.e.the sort of data that is stored in a GIS and isused to generate suitabilities for the CAmodel described above. Neither does the modelincorporate possible effects of the micro-scaleconfigurations generated by the cellular dynamics,nor does it take into account possible effects ofcrowding or land shortages. Thus when such amodel is linked to a CA model there is animmediate potential for reformulating it so thatit includes additional variables that reflect thesecellular level phenomena:

Xij g(Xi, Xj, do., VO., wji,Pji, sji) (5)

where wji is the ratio of the density of activity inthe destination region j to that in the origin regioni, Pji is the ratio of the means of the cellularpotentials for the activity (as in Eq. (1)) in the tworegions, and sji is the ratio of the mean suitabilitiesof cells in the two regions for the activity.What is gained by introducing these new vari-

ables that summarize information from the cellularmodel? Typical urban dynamics models as repre-sented by Eq. (4) embody only positive feedbackeffects: in most formulations the movement ofactivity into a region is positively related to theexisting level of activity, with no negative orinhibitory terms included. In particular, with theexception of the models due to Allen (1997), theytypically do not include a density term. Yet densityis probably one of the most important controlson urban dynamics, since as density increases, theassociated costs, particularly the rising cost of landand the indirect costs of congestion, will eventuallyinhibit the continued growth of the most attractiveregions.Of course such effects can be included in the

model, as Allen (1997) has demonstrated; but to doso in more than the most arbitrary way requires thesort of detailed spatial data which resides at thecellular level. For example, if density is obtained inthe obvious way by dividing the level of activity by

the area of the region, the measure will not becomparable from one region to another: in one

region the entire area may be available to theactivity, while in another only a small part mayeffectively be available, the rest being too steep, towet, or occupied by an activity that cannot easily bedislodged. Thus meaningful densities must bedefined at the cellular level; for example, the densi-ties Used to define wi in Eq. (5) could be calculatedas the level of activity in the region divided by thenumber of cells occupied by the activity, giving ameasure of actual operating density that would beclosely correlated with density associated costs.

Similarly, the terms Pji and sji represent the com-parative attractiveness of two regions in terms oflocal or cell scale characteristics within the regions.The relative mean potentials (Pi) of the cells in thetwo regions provide a summary measure of therelative attractiveness of the local cell neighbour-hoods in the two regions, while the relative meansuitabilities (si) of the cells generalize the site-specific qualities of Couclelis’ absolute space to theregional level. Of course, all three of these variablessummarizing cellular level properties can be speci-fied in a variety ofways, and others could be definedas well.The link from the macro-scale model to the cel-

lular level serves primarily to provide the numbersof cells required for each activity. But in the caseof the regionalized models, this is also the meansby which the effects of long range interactions onthe spatial restructuring of the city are transmittedto the cellular model, so that regional scaleinhomogeneities are introduced into the cellulardynamics. There are a variety of ways in whichthe link can be specified, but all of them involvetreating the regional activity levels generated by themacro-level model as demands (t+ 1XD,. ratherthan as actual new levels of activity, and thenpassing these demands to the cellular model forallocation in cell space and hence implicitly amongregions.A relatively loose connection between the two

levels can be established by using the changes indemand in each region to re-scale the cellular

122 R. WHITE

potentials calculated in Eq. (1). Thus the potentialvalues for cells in a region with a growing demandfor an activity are rescaled upward relative to po-tentials for cells in regions with lower incrementaldemand, but the CA then runs as an unregionalizedwhole, constrained only by the total number ofcells required for each sector. Thus in general theamount of activity implicitly allocated to eachregion by the cellular model will not be the sameas the amount called for by the regional model,although regions with higher incremental demands(/t+ 1XDi) will be relatively more attractive at thecellular level and thus will typically receive rela-tively more cells, and consequently more activity.The level of activity in a region is determinedimplicitly by the number of cells together with thedensity of activity per cell, so a final regional activ-ity level (+ 1Xi) as established in the CA can becalculated and returned to the regional model.A tighter connection can be made by treating the

regional demands for activities as regional targetsat the cellular level. Regional activity demands(t-t-1XDi) are converted to regional demands forcells, and a regionalized CA attempts to allocatethe cells within the specified regions. While in thisapproach the cellular model is regionalized withrespect to target cell numbers, the calculation ofcellular potentials (Eq. (1)) is made without regardto regional boundaries: the neighbourhood of a cellmay include cells in another region. Thus it ispossible, for example, for clusters of activity todevelop which cross regional boundaries (Engelenet al., 1997a).The regional targets approach is not so straight-

forward as it may seem, however, since it is notalways possible to satisfy all of the demands forcells within a region. Thus there must be mechan-isms for re-allocating some of the cell demand toother regions. These may take many forms, buttypically they will be based on such factors as spaceavailable, existing regional demand for the activity(t+IxDj, ji), density, and suitabilities. Theintensity of the competition for cells in a regionas measured by the ratio of cells demanded to cellsavailable can be interpreted as a surrogate for land

price. Since higher land prices lead to higher den-sities, densities are adjusted at each iteration of thecellular model to reflect the changing demand forland. Both the new densities and the derived finalregional activity levels (t+ 1Xi) are returned to themacro-scale model.We do not yet have sufficient experience with

these linked models to know to what degree theperformance of the component models is improvedby the integration. Cellular models generally per-form very well on their own, so it is to be expectedthat linking them with regionalized macro-scalemodels would not result in dramatic changes in theirperformance. The most noticeable improvementsseem to be in sectors like commerce which are highlydependent on long range interactions and whichhave proved most difficult to model well with stand-alone CA. On the other hand, the spatial interactionbased regional models may be greatly enhanced byintegration with cellular models, primarily becausethe cellular models, taking into account a vastamount of detailed, high resolution spatial dataconcerning factors important to location, act toconstrain the regional dynamics. The traditionalspatial interaction based models, popular amongplanners in the 1950s and 1960s, have fallen out offavour, in part, perhaps because their results are notlocationally specific; i.e. they have extremely poorresolution, and thus cannot easily be linked to

specific urban phenomena. Potentially one of themajor effects of linking these models to cellularmodels will be to revive their popularity.More generally, there is no reason to limit

integration to urban models. Cities are inherentlycomplex entities, with socio-economic phenomenaunfolding in an environment that is both con-structed and natural. In the CA discussed so far,the built environment is modelled only in a generalway (as land use), and the natural environment istreated as a set of boundary conditions, repre-sented in the land cover features and the suitabil-ities. But the natural environment has its own

dynamics, and in an urban area these dynamics are

intimately linked to the socio-economic and landuse dynamics of the constructed city. For example,


a regional hydrological model may represent thedynamics of infiltration, runoff, stream discharge,and flooding as functions of precipitation and landcover. But the balance of infiltration and runoff islargely determined by land cover or land use, so as

urbanization proceeds and land use changes, infil-tration and runoff will be altered; in consequencegroundwater supplies will be affected and so willthe frequency and extent of flooding. But the lattereffects may in turn feed back on the evolution ofthe urban area, for example by discouraging devel-opment of the areas that are newly at risk offlooding, or by instigating a planning interventionestablishing a natural reserve dedicated to rechargeofthe local aquifer. Thus it is natural to link cellularurban models to models of natural processes (Uljeeet al., 1996; White and Engelen, 1997a).

In spite of the specific difficulties that are alwaysencountered, one of the great strengths of CAbased models is the relative ease with which theycan be linked to other models. Because processoriented models ofmany phenomena are inherentlyspatial and operate at local scales, as do CA, thereis frequently a basic commensurability whichfacilitates integration. In this situation the outputof one model may be passed to the other where itmodifies a boundary condition; for example, in acase where forested land was being converted tocommercial use, the output of an urban land usemodel would modify the local infiltration rateparameters of a hydrological model, which wouldthus express the effect of the land use change.Even when the models are not commensurable,

as is the case with regional models, or withnon-spatial economic models (e.g. input-outputmodels; see Engelen et al., 1997b; White andEngelen, 1997a), it is still possible, as we have seen,to establish strong links between the models. Thereason lies in the rule-based nature of CA. Cellularmodels differ from each other primarily in terms oftheir transition rules, which can take on virtuallyany form, including hierarchical rule sets, in whichone rule may over-ride another. This flexibilitypermits models to be linked through their rule sets,where the output of one model modifies the

transition rules of the other. As the need forintegrated modelling of human-natural systemsbecomes more obvious, it seems likely that theflexibility of CA will be seen as one of their mostimportant characteristics.

7 CONCLUSIONS

Cellular automata constitute a powerful tool forunderstanding cities at more than one level. At themost mundane level, their high spatial resolutionand micro-scale process modelling permit a highdegree of verisimilitude. Such highly specificmodels, linked to the GIS that currently serve asthe primary support for urban spatial planning,would radically extend the capabilities of the GISby adding a temporal, predictive dimension to their

high-resolution spatial database, thus enablingplanners to experiment with possible urban futures.

But there is a deeper significance to the realismand practicality of CA based models. The historyof urban modelling is plagued by the legacy ofoversimplification. Starting with the standard landuse models, which assumed a monocentric citylying on an isotropic plain, and continuing with themacro-scale urban dynamics models discussedabove, which are defined on highly aggregatedregions and thus lack spatial resolution, the resultshave been too general or too unrealistic to be ofpractical use.More importantly, these traditional approaches

have furnished relatively few fundamental insightsinto the nature ofurban systems, primarily, it wouldseem, because they are too divorced from the natureof cities. As self-organizing structures, cities are

inherently complex objects. Even the "simple" pro-perties of cities cannot be understood if the under-lying complexity is ignored. For example, themonocentric structure exhibited by many citiescannot be explained by a model that assumes

monocentricity; nor can the fractal nature of citiesbe discerned in the absence of a representation ofthe underlying structural complexity. Thus a deepertheoretical understanding can only be achieved

124 R. WHITE

through methods that bring together the generaland the specific methods that capture and workwith the detail and complexity that characterizereal cities.

Extensive theoretical work on the principlesof self-organization by Langton (1990; 1992),Kauffman (1989; 1990; 1993), and others (see, e.g.Forrest, 1991), using CA and related techniques,has begun to provide deep insights into the behav-iour of whole classes of complex systems. Thereare, for example, strong indications that evolvedcomplex systems will typically have fractal char-acteristics. In this light, then, it is not surprisingthat cities have a fractal structure, or that cellularmodels of cities generate such a structure (andinstructive that traditional urban models do not).The point is that cellular techniques dramaticallyshrink the distance between highly specific modelsof actual cities and models formulated to investi-gate fundamental theoretical issues. They may beexpected to lead to richer and more useful appliedmodels co-evolving with a deeper understanding ofurban systems.

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