Networks, Modules & Networks, Modules & PersistencePersistenceDynamics of agent interactions on Dynamics of agent interactions on complex networks with mesoscopic complex networks with mesoscopic organizationorganization

Sitabhra SinhaSitabhra Sinha in collaboration within collaboration with

R K Pan and T Jesan (IMSc Chennai) R K Pan and T Jesan (IMSc Chennai) S Dasgupta (Calcutta Univ)S Dasgupta (Calcutta Univ)

Social Network Social Network Research Research

@ IMSc@ IMScEpidemics or contagion spreading

Primate Networks: Genesis of hierarchy

Market Dynamics: Bubble & Crashes

Opinion or Consensus formation

Popularity: Diffusion of products or

ideas

Games on Graphs

ThemesThemes

•Are there common structural mesoscopic Are there common structural mesoscopic features in social networks ? features in social networks ? Statics: Statics: modularitymodularity•What role do such network features play in What role do such network features play in governing network function ? governing network function ? Response & Response & dynamics: dynamics: polarization polarization •How can such structures arise in general ? How can such structures arise in general ? Evolution:Evolution: multi-constraint optimization multi-constraint optimization•Are there specific processes in social Are there specific processes in social networks promoting such motifs ? networks promoting such motifs ? Social Social games: games: Emergence of co-operationEmergence of co-operation

Contagion propagation in society through contact

Spread of SARS Spread of SARS from Taiwan, from Taiwan, 20032003

Importance of social network structureImportance of social network structure

“The small-worlds of public health” (CDC director)Chen et al, Chen et al, Lect Notes Comp SciLect Notes Comp Sci 4506 (2009) 23 4506 (2009) 23

We have shown that structurallyWe have shown that structurally

Why small-world pattern in Why small-world pattern in complex networks at all ?complex networks at all ?

All the classic “small-world” structural All the classic “small-world” structural properties of properties of Watts-Strogatz small world Watts-Strogatz small world networksnetworkse.g., high clustering, short average distance, e.g., high clustering, short average distance, etc.etc.are are also seenalso seen in in Modular networksModular networks

Watts-Watts-Strogatz Strogatz networknetwork

Modular Modular networknetwork

≡≡

R K Pan and SS, EPL 2009R K Pan and SS, EPL 2009

Modular NetworksModular Networks: dense connections : dense connections withinwithin certain sub-networks (certain sub-networks (modulesmodules) & relatively few ) & relatively few connections connections betweenbetween modules modules

Modules: A Modules: A mesoscopicmesoscopic organizational principle of organizational principle of networksnetworksGoing beyond Going beyond motifsmotifs but more detailed than but more detailed than globalglobal description (description (LL, , CC etc.) etc.)

Kim & Park, WIREs Syst Biol & Med, 2010

MicroMicro MesoMeso MacroMacro

Chesapeake Bay Chesapeake Bay foodweb (Ulanowicz foodweb (Ulanowicz et al)et al)

Metabolic network of Metabolic network of E coliE coli (Guimera & (Guimera & Amaral)Amaral)

Modular Biology Modular Biology (Hartwell et al, (Hartwell et al, Nature 1999)Nature 1999)Functional modules as a critical level of Functional modules as a critical level of biological organizationbiological organization

Ubiquity of modular networksUbiquity of modular networks

Modules in biological networks Modules in biological networks are often associated with are often associated with specific functionsspecific functions

Are social networks modular ?Are social networks modular ?Let’s look at real re-constructed human Let’s look at real re-constructed human social contact networksocial contact network: : A mobile phone A mobile phone interaction networkinteraction network

Data:Data:• A mobile phone operator in an European country, 20% coverageA mobile phone operator in an European country, 20% coverage• Aggregated from a period of 18 weeksAggregated from a period of 18 weeks• > 7 million private mobile phone subscriptions> 7 million private mobile phone subscriptions• Voice calls within the operatorVoice calls within the operator

• Require reciprocity of calls for a linkRequire reciprocity of calls for a link• Quantify tie strength (link weight) Quantify tie strength (link weight)

15 min (3 calls)5 min

7 min

3 min

Aggregate call duration

Total number of calls

Onnela et al. PNAS,104,7332 (2007)Onnela et al. PNAS,104,7332 (2007)

J-P OnnelaJ-P Onnela

Reconstructed networkReconstructed network

Onnela et al. PNAS,104,7332 (2007)Onnela et al. PNAS,104,7332 (2007)

Modularity of social networksModularity of social networks

Modularity: Cohesive Modularity: Cohesive groups groups

communities with dense communities with dense internal & sparse internal & sparse external connectionsexternal connections

Other examples of Other examples of modular social modular social networksnetworks– Scientific collaborators Scientific collaborators – e-mail communicatione-mail communication– PGP encryption ”web-PGP encryption ”web-

of-trust”of-trust”– non-human animalsnon-human animals Onnela et al. PNAS,104,7332 (2007)Onnela et al. PNAS,104,7332 (2007)

4.6 4.6 101066 nodes nodes 7.0 7.0 101066 links links

Next…Next…HowHow do such do such structuresstructures affect affect dynamicsdynamics ? ?

Q. Q. WhatWhat structure ? structure ?Ans.Ans. Modular Modular

Over social networks, such dynamics Over social networks, such dynamics can be of can be of

information or contagion information or contagion spreadingspreading consensus or opinion formationconsensus or opinion formation adoption of innovationsadoption of innovations

Social Network of a Karnataka Village Data: microfinance institution Bharatha Swamukti

SamstheNodes: IndividualsLinks: Social relationsDescribed in Banerjee et al, Science (2013)

Node colors represent the community to which they belong

Village “52”Population:1525 individualsLargest connected component 1497 individuals

43 modulesLargest module contains 15.5% of nodes32 modules contain less than 1% of nodes

Measuring modularityMeasuring modularity

(Newman, EPJB, 2004) (Newman, EPJB, 2004)

A: Adjacency matrixL : Total number of links ki : degree of i-th nodeci : label of module to which i-th node belongs

One suggested measure:One suggested measure:

A

BC

D

Modules determined through a Modules determined through a generalization of the spectral method generalization of the spectral method (Leicht & Newman, 2008)(Leicht & Newman, 2008)

For directed & weighted networks:

W: Weight matrixsi : strength of i-th node

( ) ( )

How to quantify the degree of modularity ?How to quantify the degree of modularity ?

SSIIRRSS dynamics dynamics

To model the transmission of contagion on the To model the transmission of contagion on the network we usenetwork we use

Basic reproduction number Basic reproduction number RR00

Mean number of new infections caused by a single Mean number of new infections caused by a single infectious individual in a infectious individual in a wholly susceptiblewholly susceptible population population (as in the beginning of an epidemic): (as in the beginning of an epidemic): If each infected If each infected person on average infects more than one other person on average infects more than one other individual, Rindividual, R00 > 1 > 1 EpidemicEpidemic

The “infectiousness” of a The “infectiousness” of a contagioncontagion

For a contagion spreading on a contact networkFor a contagion spreading on a contact network• if the if the probability of infection probability of infection is is , and,, and,• if k is the if k is the average degreeaverage degree of the contact network of the contact network Then RThen R00 kkgg where where gg : : mean generation intervalmean generation interval

To measure the force of infection of a contagion (how To measure the force of infection of a contagion (how quickly it can spread) we use the empirically measurable quickly it can spread) we use the empirically measurable quantityquantity

PersistencePersistencei.e., existence (circulation) of contagion in i.e., existence (circulation) of contagion in the population for an indefinite period, i.e., the population for an indefinite period, i.e., I(t) > 0 as tI(t) > 0 as t

A contagion that starts out A contagion that starts out as an epidemic, eventually as an epidemic, eventually either either dies outdies out [I(t)=0] or [I(t)=0] or becomes becomes endemicendemic

We investigate the long-term transmission of contagia in modular contact networksIn particular,

Hollingsworth, J Pub Health Policy, 2009Hollingsworth, J Pub Health Policy, 2009

Contagia in Contagia in empirical network empirical network comprising comprising communities are communities are surprisingly surprisingly persistent persistent in in contrast to those contrast to those in degree-in degree-preserved preserved randomized randomized networks which do networks which do not have modular not have modular organizationorganization

Difference even Difference even more more pronounced if pronounced if modularity is modularity is enhanced by enhanced by decreasing the decreasing the inter-modular inter-modular connectivityconnectivity

A simple model of modular A simple model of modular networksnetworksModel parameter Model parameter r r : Ratio of inter- to intra-modular connection density: Ratio of inter- to intra-modular connection density

Can modular organization of contact network have Can modular organization of contact network have significant impact on the long-term dynamics of significant impact on the long-term dynamics of contagia spreading ?contagia spreading ?

Module ≡ random Module ≡ random networknetwork

Comparison with Watts-Strogatz Comparison with Watts-Strogatz modelmodel

E = 1 /avg path length, ℓ = 2 /N(N-1) E = 1 /avg path length, ℓ = 2 /N(N-1) i>ji>jddijij

CommunicatiCommunication efficiencyon efficiency

Clustering Clustering coefficientcoefficient

Structural measures used:Structural measures used:

C = fraction of observed to potential C = fraction of observed to potential triads triads = (1 /N) = (1 /N) ii2n2ni i / k/ kii (k (kii - 1) - 1)

WS and Modular networks behave similarly as function of p or r(Also for between-ness centrality, edge clustering, etc)

Consider Consider orderingordering or alignment of orientation on such or alignment of orientation on such networksnetworkse.g., Ising spin model: dynamics minimizes H= - e.g., Ising spin model: dynamics minimizes H= - J Jijij S Sii (t) (t) SSjj (t) (t)

How can you tell them apart ?How can you tell them apart ?Dynamics on Modular networks different Dynamics on Modular networks different from that on other small-world (Watts-from that on other small-world (Watts-Strogatz) networksStrogatz) networks

Network topologyNetwork topology

2 distinct time scales in Modular networks: t 2 distinct time scales in Modular networks: t modularmodular & t & t

globalglobal Global orderGlobal order

Modular orderModular order

Time required Time required for global for global ordering ordering diverges as diverges as rr →0 →0

Eigenvalue spectra of the Eigenvalue spectra of the LaplacianLaplacianShows the existence of spectral gap Shows the existence of spectral gap distinct time distinct time scalesscalesModular network Laplacian spectraModular network Laplacian spectra

Existence of distinct time-scales in Modular Existence of distinct time-scales in Modular networksnetworksNo such distinction in Watts-Strogatz small-world networksNo such distinction in Watts-Strogatz small-world networks

gapgap

No gapNo gap

WS network Laplacian spectraWS network Laplacian spectra

Spectral gap in Spectral gap in modular networks modular networks diverges with diverges with decreasing rdecreasing r

Shows the existence of Shows the existence of two two distinctdistinct time scales: time scales:• • fast intra-modular fast intra-modular diffusion diffusion • • slower inter-modular slower inter-modular diffusiondiffusionwhile random networks while random networks show a continuous range show a continuous range of time scalesof time scales

Distrn of first passage Distrn of first passage times for random walkstimes for random walks

Diffusion process on modular Diffusion process on modular networksnetworksE.g., Random walker moving from one node to randomly chosen neighboring nodeE.g., Random walker moving from one node to randomly chosen neighboring node

Signal propagation on contact networks with Signal propagation on contact networks with community organization can take extremely long community organization can take extremely long timestimes

In modular networks, signal spreads slowly from module In modular networks, signal spreads slowly from module to module ! to module !

Pan & Sinha, EPL 2009

SIRS Dynamics: SIRS Dynamics: Existence of Existence of communities can make highly infectious communities can make highly infectious contagia persistentcontagia persistent

For isolated For isolated modules (r=0) modules (r=0) and and homogeneous homogeneous networks (r=1) networks (r=1) epidemic with epidemic with high Rhigh R00 dies out dies out quicklyquickly

m=64 modules of size m=64 modules of size n=16 avg degree <k> n=16 avg degree <k> = 12= 12Avgd over 100 rlznsAvgd over 100 rlzns

However, forHowever, fora critical range a critical range of modularity of modularity for contact for contact network network (r ~ 10(r ~ 10-3-3), ), highly highly infectious infectious contagia are contagia are persistent. persistent.

r = 0.0002: rapid extinction r = 0.002: persistence r = 0.02: rapid extinction

Persistence in a critical range of Persistence in a critical range of modularitymodularity

Distribution of Distribution of persistence time persistence time shows shows bimodalbimodal character for character for large r – with the large r – with the upper branch upper branch diverging for a diverging for a critical range of critical range of modularity…modularity…

……while for lower r while for lower r the distribution is the distribution is unimodalunimodal with with avg avg decreasing decreasing rapidly as the rapidly as the modules are modules are effectively effectively isolated isolated (R(R00 = 6) = 6)

Effect of increasing Effect of increasing the number of the number of modulesmodules

The mechanism of enhanced The mechanism of enhanced persistencepersistenceTime-scale for Time-scale for global diffusion global diffusion (inverse of (inverse of smallest finite smallest finite Laplacian Laplacian eigenvalue) eigenvalue) decreases with decreases with r…r………. So does the . So does the time-scale time-scale separation separation between inter- between inter- and intra-modular and intra-modular events (the events (the Laplacian spectral Laplacian spectral gap) gap)

However – the However – the ratio of the two ratio of the two show non-show non-monotonic monotonic dependence on dependence on modularitymodularity

contagion spreads slowly from module to contagion spreads slowly from module to module, allowing parts of the network to module, allowing parts of the network to recover before return of infection! recover before return of infection!

Q.Q. How does individual behavior at How does individual behavior at micro-level relate to social phenomena micro-level relate to social phenomena at macro level ?at macro level ?

Order-disorder transitions in Social Order-disorder transitions in Social Coordination Coordination How does existence of different time-scales affect How does existence of different time-scales affect consensus formation dynamics on networks with consensus formation dynamics on networks with communities communities

E.g., can social polarization occur as a result E.g., can social polarization occur as a result of modular network structure even when all of modular network structure even when all relations are “friendly” ?relations are “friendly” ?

•Spin orientationSpin orientation: mutually : mutually exclusive choicesexclusive choices

Ising model with FM interactions: Ising model with FM interactions: each each agent can only be in one of 2 states agent can only be in one of 2 states (Yes/No or +/-)(Yes/No or +/-)

Spin models of statistical physics: simple Spin models of statistical physics: simple models of coordination or consensus models of coordination or consensus

formation formation

•Choice dynamicsChoice dynamics: decision based : decision based on information about choice of on information about choice of majority in local neighborhoodmajority in local neighborhood

Simplest case: 2 possible choicesSimplest case: 2 possible choices

•TemperatureTemperature: noise or degree of : noise or degree of uncertainty among agentsuncertainty among agents

possible order in modular possible order in modular network of “spins”network of “spins”

Modular Modular orderorder

Global orderGlobal order

Avg modular order Avg modular order parameterparameterSystem or global order System or global order

parameterparameter

NN spins, spins, nnmm modules modules

FM interactions:FM interactions: J J > 0> 0

ConsensusConsensusNo consensusNo consensus

(“Friendly” interactions)(“Friendly” interactions)

r = 0.002

There will be a phase corresponding to There will be a phase corresponding to modular modular butbut no global order ( no global order (coexistence of coexistence of contrary opinionscontrary opinions) ) even when all mutual even when all mutual interactions are FM (interactions are FM (favor consensusfavor consensus)) ! !

Phase diagram: two Phase diagram: two transitionstransitions

T

Even when global order is possible, i.e.,Even when global order is possible, i.e.,strongly modular network takes very long to strongly modular network takes very long to show global order show global order Time required to achieve consensus Time required to achieve consensus increases rapidly for a strongly modular increases rapidly for a strongly modular social organizationsocial organizationBut then …But then …How do certain innovations get adopted How do certain innovations get adopted rapidly ?rapidly ?Possible modifications to the dynamics:Possible modifications to the dynamics:Positive feedbackPositive feedbackDifferent strengths for inter/intra Different strengths for inter/intra couplingscouplings

Divergence of relaxation time with Divergence of relaxation time with modularitymodularity

I. The effect of Positive I. The effect of Positive FeedbackFeedback

Brian ArthurBrian Arthur

The case of “counter-clockwise” clocksThe case of “counter-clockwise” clocks

Microsoft & the 7 DwarvesMicrosoft & the 7 Dwarves

Effectively increases inter-modular interactions

Drives system away from critical line by increasing Tg

c

reduces

Introduce a field H = hM (proportional to order parameter)

The effect of Positive The effect of Positive FeedbackFeedback

II. Varying II. Varying Strength of Strength of Coupling Coupling Between & Within Between & Within ModulesModules

Marc GranovetterMarc Granovetter

JJ00 : strength of inter-modular : strength of inter-modular connectionsconnections JJii : strength of intra-modular : strength of intra-modular connectionsconnections

increasingincreasing J J00 / / JJii increasing increasing rr Non-monotonic behavior Non-monotonic behavior of of

relaxation time vs ratio of relaxation time vs ratio of strengths of short- & long-strengths of short- & long-range couplingsrange couplings

Not seen in Watts-Strogatz SW Not seen in Watts-Strogatz SW networksnetworks(Jeong et al, PRE 2005)(Jeong et al, PRE 2005)

Why Modular Why Modular Networks ? Networks ?

As random As random networks are networks are divided into more divided into more modules (m) they modules (m) they become more become more unstableunstable… …

… … however, we see modular networks all around us. however, we see modular networks all around us. Why ?Why ?

Suggestion:Suggestion: modularity imparts robustness modularity imparts robustnessE.g.,Variano et al, PRL 2004E.g.,Variano et al, PRL 2004

N=256N=256

Not quite !Not quite !Consider stability of a Consider stability of a random network with random network with a modular structure.a modular structure.

Most real networks have non-trivial degree Most real networks have non-trivial degree distributiondistribution Degree: total number of connections for a nodeDegree: total number of connections for a node

HubsHubs: nodes having high degree relative to other : nodes having high degree relative to other nodesnodes

Clue: Clue: Many of these modular networks also possess multiple Many of these modular networks also possess multiple hubs hubs !!

Question:Question: Why Modular Networks ? Why Modular Networks ?

Why Modular Networks ? Why Modular Networks ? R K Pan & SS, R K Pan & SS, PRE PRE 76 045103(R)76 045103(R) (2007)(2007)

Hypothesis:Hypothesis: Real networks optimize Real networks optimize between several constraints,between several constraints,

•Minimizing link cost, i.e., total # links LMinimizing link cost, i.e., total # links L•Minimizing average path length ℓ Minimizing average path length ℓ •Minimizing instability Minimizing instability maxmax

Minimizing link cost and avg path length yields …Minimizing link cost and avg path length yields …… … a a starstar-shaped network with single hub -shaped network with single hub

But But unstableunstable ! !Instability measured by Instability measured by maxmax ~ ~ max max degree (i.e., degree of the hub)degree (i.e., degree of the hub)In fact, for star network, In fact, for star network, maxmax ~ ~ NN

Increasing stability, average path lengthIncreasing stability, average path length

N = 64, L = N-1

= 0.4 = 0.78

= 1

As star-shaped networks As star-shaped networks are divided into more are divided into more modules they become modules they become more more stablestable … …

……as stability increases by as stability increases by decreasing the degree of decreasing the degree of hub nodes hub nodes maxmax ~ ~ [N/m][N/m]

How to satisfy all three constraints ?How to satisfy all three constraints ?Answer:Answer: go modular ! go modular !

Shown explicitly byShown explicitly byNetwork OptimizationNetwork Optimization

Fix link cost to min (L=N-1) Fix link cost to min (L=N-1) and minimize the energy fnand minimize the energy fn

E (E () = ) = ℓ + (1- ℓ + (1- ) ) maxmax

[0,1] :[0,1] : relative importance relative importance of path length constraint over of path length constraint over stability constraintstability constraint Transition to star configurationTransition to star configuration

Tanizawa et al, PRE 2005

We have considered dynamical instability criterion We have considered dynamical instability criterion for network robustness – how about stability against for network robustness – how about stability against structural perturbations ?structural perturbations ?E.g., w.r.t. random or targeted removal of nodesE.g., w.r.t. random or targeted removal of nodes

The robustness of modular structures The robustness of modular structures

Surprisingly YES ! Surprisingly YES !

At the limit of extremely small At the limit of extremely small LL, , optimal modular networks ≡ optimal modular networks ≡ networks with networks with bimodal degree bimodal degree distributiondistribution……

……has been shown to be robust has been shown to be robust w.r.t.w.r.t.targeted as well as random targeted as well as random removal of nodesremoval of nodes

Scale-free network: robust w.r.t. random removalScale-free network: robust w.r.t. random removalRandom network: robust w.r.t. targeted removalRandom network: robust w.r.t. targeted removal

Is the modular network still optimal ?Is the modular network still optimal ?

Can there be specific processes at work in Can there be specific processes at work in social networks that promote the emergence social networks that promote the emergence of community structure ? of community structure ? Social games: Social games: Emergence of co-operationEmergence of co-operation

So far, we looked at rather general processes So far, we looked at rather general processes for evolving modularityfor evolving modularity

Modularity via games Modularity via games between selfish rational between selfish rational

agentsagents

Our agents are relatively “naïve”, whose rules for Our agents are relatively “naïve”, whose rules for making choices do not explicitly take into account how making choices do not explicitly take into account how the other agents choosethe other agents choose

What happens if our agents are capable of “strategic” What happens if our agents are capable of “strategic” decision making (i.e., taking into account how other decision making (i.e., taking into account how other agents will behave) ?agents will behave) ?

Prisoners DilemmaPrisoners Dilemma

CooperatCooperatee

DefectDefect

CooperatCooperatee

R,RR,R S,TS,T

DefectDefect T,ST,S P,PP,P

T: Temptation to defectT: Temptation to defectR: Reward for cooperationR: Reward for cooperationP: Punishment for mutual P: Punishment for mutual defectiondefectionS: Sucker’s payoffS: Sucker’s payoffIn general, T > R > P > SIn general, T > R > P > SUsually, R=1, P=0, S=0 and Usually, R=1, P=0, S=0 and 1<T<2 1<T<2

originally framed by Merrill Flood and originally framed by Merrill Flood and Melvin Dresher at RAND (1950)Melvin Dresher at RAND (1950)

Payoff Matrix

In the iterative setting, an In the iterative setting, an ideal model for analyzing ideal model for analyzing the conditions for the the conditions for the emergence of emergence of cooperationcooperation

Prisoners Dilemma in a networkPrisoners Dilemma in a networkNowak & May, Nature 1992Nowak & May, Nature 1992

Spatial Spatial Prisoners Prisoners Dilemma:Dilemma: Agents play with Agents play with neighbors on a neighbors on a lattice, adopting lattice, adopting strategy of strategy of neighbor with neighbor with max payoffmax payoff

1.75< T <1.8 1.8< T <2

Network Prisoners Dilemma: Network Prisoners Dilemma: agents play with nbrs in the agents play with nbrs in the network, compare payoff with randomly chosen nbr and adopt network, compare payoff with randomly chosen nbr and adopt strategy of neighbor having higher payoff with a probabilitystrategy of neighbor having higher payoff with a probabilityStability w.r.t. defection Stability w.r.t. defection Robustness w.r.t. removal Robustness w.r.t. removal of nodesof nodesRole of modules: Role of modules: Fragmenting the network into several sparsely connected Fragmenting the network into several sparsely connected communities, each with star configuration, allows communities, each with star configuration, allows cooperation to be robust – modularity beneficial if degree cooperation to be robust – modularity beneficial if degree distribution is distribution is bimodalbimodal

The evolution of cooperation The evolution of cooperation driving the emergence of modular driving the emergence of modular structures structures

Could modularity in social networks Could modularity in social networks have arisen as a result of modules have arisen as a result of modules promoting cooperation – necessary for promoting cooperation – necessary for building social organization ?building social organization ?

C

C

C C

C

C

D

D

D

D

DC

DD

C

CC Link

adaptation

dynamicsC

To SummarizeTo Summarize

• Are there common structural mesoscopic Are there common structural mesoscopic features in social networks ? features in social networks ? Statics: Statics: modularitymodularity

• What role do such network features play in What role do such network features play in governing network function ? governing network function ? Response & Response & dynamics: dynamics: polarization polarization

• How can such structures arise in general ? How can such structures arise in general ? Evolution:Evolution: multi-constraint optimization multi-constraint optimization

• Are there specific processes in social Are there specific processes in social networks promoting such motifs ? networks promoting such motifs ? Social Social games: games: Emergence of co-operationEmergence of co-operation

Thanks Thanks