““Modularity” of Social Modularity” of Social NetworksNetworks

Sitabhra SinhaSitabhra SinhaThe Institute of Mathematical Sciences, The Institute of Mathematical Sciences,

ChennaiChennai

Image: Ramki (www.wildventures.com)

Image : R K Pan

Outline or Why ? What ? How ? Outline or Why ? What ? How ?

WhyWhy networks ? networks ?

WhatWhat structures underlie real life networks ? structures underlie real life networks ?

HowHow do such structures affect network do such structures affect network dynamics ?dynamics ?

In particular, In particular, HowHow does it affect ordering dynamics in spin does it affect ordering dynamics in spin models ?models ?

Significant for understanding social Significant for understanding social coordination processes such as consensus coordination processes such as consensus formation & adoption of innovationsformation & adoption of innovations

R K Pan & SS, EPL (2009)R K Pan & SS, EPL (2009)

S Dasgupta, R K Pan & SS, PRE Rapid (2009)S Dasgupta, R K Pan & SS, PRE Rapid (2009)

• In a In a networknetwork of interacting components of interacting components• EmergenceEmergence• of of qualitatively different behaviorqualitatively different behavior from from individual componentsindividual components..

E.g., component = neuron, system = brainE.g., component = neuron, system = brain

Interactions add a new layer of complexity!Interactions add a new layer of complexity!

The aim of studying networksThe aim of studying networks: : how interactions → complexity at systems-how interactions → complexity at systems-levellevel

NodesNodes

LinksLinksWhy Networks ? Why Networks ?

Networks appear at all scalesNetworks appear at all scales

Ubiquity of NetworksUbiquity of Networks

1010-3-3 m m1010-6-6 m m1010-9-9 m m 1 m1 m 101033 m m 101066 m m

MoleculesMolecules CellsCells OrganismsOrganisms PopulationsPopulations EcologiesEcologies

ProteinsProteins Intra-cellular Intra-cellular signallingsignalling

Neuronal Neuronal communicatiocommunicatio

nn

Social Social contactcontact

Food websFood webs

Empirical networks are not Empirical networks are not randomrandom – many have certain – many have certain structural patternsstructural patterns

Theoretical understanding of Theoretical understanding of networksnetworks• Regular lattice or grid (Regular lattice or grid (PhysicsPhysics))

• average path length ~ average path length ~ NN (no. of nodes) (no. of nodes)• clustering clustering highhigh • delta function distribution of degree delta function distribution of degree (links/node)(links/node)

•Random networks (Random networks (Graph theoryGraph theory))

• average path length ~ log average path length ~ log NN• clustering lowclustering low• Poisson distribution of degreePoisson distribution of degree

Regular NetworkRegular Network Random NetworkRandom Network““Small-world” NetworkSmall-world” Network

Increasing RandomnessIncreasing Randomness

p = 0p = 0 p = 1p = 10 < p < 10 < p < 1

Example: Example: small-worldsmall-world networks networks

p: fraction of random, long-range p: fraction of random, long-range connectionsconnections

Watts and Strogatz (1998): Many biological, technological Watts and Strogatz (1998): Many biological, technological and social networks have connection topologies that lie and social networks have connection topologies that lie between the two extremes of completely regular and between the two extremes of completely regular and completely random.completely random.

High cluste

ring,

High cluste

ring,

Larg

e path length

Larg

e path length

Low cl

usterin

g,

Low cl

usterin

g,

Short path le

ngth

Short path le

ngth

High cluste

ring,

High cluste

ring,

Short path le

ngth

Short path le

ngth

Small-worldSmall-world networks can be networks can be highly clusteredhighly clustered (like (like regular networks ), regular networks ), yet have yet have small characteristic path lengthssmall characteristic path lengths (as in (as in random networks ).random networks ).

Characteristic path lengthCharacteristic path length ( ( ℓ ( p )ℓ ( p ) ) and ) and clustering clustering coefficientcoefficient ( (C( p )C( p ) ) as network randomness ) as network randomness increases.increases.

A prominent example of small-world A prominent example of small-world

networksnetworks: Social networks: Social networks

Nodes : Nodes : individuals (individuals (NN = 3 – 10 = 3 – 1099))

Links : Links : social interactions (< social interactions (< 150/node)150/node)

• Represent the “underlying” Represent the “underlying” socialsocial network (e.g. a knows b) network (e.g. a knows b)

• Approximated by real Approximated by real interactions interactions between a & b (e.g. through between a & b (e.g. through phone calls, emails, trades, …)phone calls, emails, trades, …)

• Weight represents interactionWeight represents interaction strengthstrength

Example of social interaction Example of social interaction

networknetwork Collaboration Collaboration NetworksNetworksIMSc Physics co-authorship network (2000-08)IMSc Physics co-authorship network (2000-08)

1 2 3 4 5

DCBA

AuthorsAuthors

PapersPapers

1

2

3

4 5

Co-authorship networkCo-authorship network

Bipartite Paper-Author networkBipartite Paper-Author network

UAS-GKVK Campus, BangaloreUAS-GKVK Campus, BangaloreData: Anindya Sinha (NIAS,B’lore)Data: Anindya Sinha (NIAS,B’lore)

The Bonnet Macaque (The Bonnet Macaque (Macaca radiataMacaca radiata) seen widely in southern India ) seen widely in southern India

Re-constructing social Re-constructing social networksnetworks

Example: Macaque troupe social networkExample: Macaque troupe social network

Usually live in large (~ Usually live in large (~ 40) multi-male, multi-40) multi-male, multi-female troops where the female troops where the adult individuals (~ 10) adult individuals (~ 10) develop strong affiliative develop strong affiliative relationshipsrelationships

Image: Ramki (www.wildventures.com)

Image: Arunkumar

Macaque Social Networks can be defined in Macaque Social Networks can be defined in terms ofterms ofgrooming frequencygrooming frequencytotal grooming time total grooming time approach frequencyapproach frequency

approach frequencyapproach frequency

grooming timegrooming time

grooming frequencygrooming frequencyNumbers refer to rank Numbers refer to rank among the adult among the adult females from 11 (most females from 11 (most dominant) to 1 (least dominant) to 1 (least dominant)dominant)

Data: 1993-1997Data: 1993-1997

Female bonnet macaques Female bonnet macaques •usually remain in the group throughout their lifeusually remain in the group throughout their life•as adults, form strong linear matrilineal dominance as adults, form strong linear matrilineal dominance hierarchies that are hierarchies that are stablestable over time over time

Male bonnet macaques Male bonnet macaques •as adults, form as adults, form unstableunstable dominance hierarchies dominance hierarchies•occupy low ranks when young, high when mature and at occupy low ranks when young, high when mature and at peak of health peak of health

Network analysis can predict group Network analysis can predict group dynamics !dynamics !

Community detection generates consistent partitions for females, Community detection generates consistent partitions for females, notnot for males for males

Predictive power: Predictive power: Observation in 1998 showed the group had split Observation in 1998 showed the group had split into two (11,10,9,8,7,2) and (6,5,4,3) [1 had died]into two (11,10,9,8,7,2) and (6,5,4,3) [1 had died]

Contagion propagation in society through contact

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

Importance of SW social network structureImportance of SW 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

Worldwide spread of SARS through international airline Worldwide spread of SARS through international airline networknetwork

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

How about How about socialsocial small-world small-world networks ?networks ?Let’s look at real re-constructed Let’s look at real re-constructed social contact patterns: social contact patterns: 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 epidemic information or epidemic spreadingspreading consensus or opinion formationconsensus or opinion formation adoption of innovationsadoption of innovations

A simple model of modular A simple model of modular networksnetworks

Model parameter Model parameter r r : Ratio of inter- to intra-modular connectivity: Ratio of inter- to intra-modular connectivity

Adjacency matrixAdjacency matrixModule ≡ random Module ≡ random networknetwork

The modularity of the network is changed keeping avg degree constantThe modularity of the network is changed keeping avg degree constant

Comparison with Watts-Strogatz Comparison with Watts-Strogatz modelmodel

E = [avg path length, ℓ ]E = [avg path length, ℓ ]-1-1 = 2 /N(N-1) = 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 WS and Modular networks behave networks behave similarly as function of similarly as function of pp or or rr(Also for between-ness (Also for between-ness centrality, edge clustering, centrality, edge clustering, etc)etc)In fact, for same N and In fact, for same N and <k>, we can find <k>, we can find pp and and rr such that the WS and such that the WS and Modular networks have the Modular networks have the same same “modularity”“modularity” Q Q

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)

Then how can you tell them Then how can you tell them apart ?apart ?Dynamics on Watts-Strogatz network Dynamics on Watts-Strogatz network different from that on Modular networksdifferent from that on Modular networksNetwork 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

Consider synchronization on modular networksConsider synchronization on modular networkse.g., Kuramoto oscillators: de.g., Kuramoto oscillators: di i /dt = w + (1/k/dt = w + (1/kii))KKijij sin ( sin (jj - - ii))

UniversalityUniversalityAlmost identical two time scale behavior is Almost identical two time scale behavior is seen for oscillator synchronization (viz., seen for oscillator synchronization (viz., nonlinear relaxation oscillators)nonlinear relaxation oscillators) Network topologyNetwork topology

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

globalglobal

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

Localization of Localization of eigenmodes of transition eigenmodes of transition

matrixmatrix

Within modulesWithin modulesdiffusiondiffusion

Between modulesBetween modulesdiffusiondiffusion

Diffusion process on modular Diffusion process on modular networksnetworksAlso shows the existence of 2 distinct time scales:Also shows the existence of 2 distinct time scales:• • fast intra-modular diffusion fast intra-modular diffusion • • slower inter-modular diffusionslower inter-modular diffusion

ERER

WSWS

ModularModular

Distrn of first passage times for rnd Distrn of first passage times for rnd walks to reach a target node walks to reach a target node starting from a source nodestarting from a source node

Random walker moving from one node to randomly chosen nghbring nodeRandom walker moving from one node to randomly chosen nghbring node

Relevant for diffusion of Relevant for diffusion of innovation or epidemicsinnovation or epidemics

How about How about realreal small-world small-world networks ?networks ?

The networks of The networks of cortical connections cortical connections in mammalian brain in mammalian brain have been shown to have been shown to have small-world have small-world structural structural propertiesproperties

Our analysis Our analysis reveals their reveals their dynamical dynamical properties to be properties to be consistent with consistent with modular “small-modular “small-world” networks world” networks

gagapp

gagapp

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

We again turn attention back to We again turn attention back to consensus formation dynamicsconsensus formation dynamics

•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 +/-)

The emergence of novel phase The emergence of novel phase of collective behaviorof collective behavior

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

Types of possible order in Types of possible order in modular network of Ising spinsmodular network of Ising spins

Modular Modular orderorder

Global orderGlobal order

Avg magnetic moment / Avg magnetic moment / module module Total or global magnetic Total or global magnetic

momentmoment

NN spins, spins, nnmm modules modules

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

But how do the different ordered phases But how do the different ordered phases occur as a function of modularity parameter r occur as a function of modularity parameter r and temperature T ?and temperature T ?

Long transient to global order can be mistaken as Long transient to global order can be mistaken as modular ordermodular order

r

T

Tgc

0 1 r0 1 r0 1

T T

Can modular order be seen as a phase at Can modular order be seen as a phase at all ?all ?

Tgc Tg

c

Possible Phase DiagramsPossible Phase Diagrams

Magnetic moment of a single moduleMagnetic moment of a single module

fraction of “up” spins in modulefraction of “up” spins in module

At eqlbm, for strong modularity (r << 1)At eqlbm, for strong modularity (r << 1)

Total magnetic momentTotal magnetic moment

Minimizing free energy w.r.t. Minimizing free energy w.r.t. ff++

““Modular” critical tempModular” critical temp

fraction of modules with +fraction of modules with +

# links within module# links within module

Continuous transition to “modular order” phase below Continuous transition to “modular order” phase below

As T is lowered, As T is lowered, Another continuous transition to “global order” phase below Another continuous transition to “global order” phase below

““Global” critical tempGlobal” critical tempconn prob betn modulesconn prob betn modules

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 …Even when global order is possible …

Divergence of relaxation time with Divergence of relaxation time with modularitymodularityTo switch from + to - , module crosses free energy barrier: To switch from + to - , module crosses free energy barrier:

At low T:At low T:

++--

Relaxation time to global Relaxation time to global order:order:

Thus, even whenThus, even whenstrongly 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

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 magnetization)

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)

Ongoing and Future workOngoing and Future work

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

Effect of modular contact structure in formal games -Effect of modular contact structure in formal games -Is it easier for cooperation to emerge for Prisoners Is it easier for cooperation to emerge for Prisoners Dilemma on modular networks ? Dilemma on modular networks ?

Role of multiple (>2) choices : q-state Potts Role of multiple (>2) choices : q-state Potts modelmodel

Having different types of interactions : spin glass Having different types of interactions : spin glass behavior on modular networksbehavior on modular networks

Question:Question: Why Modular Networks ? Why Modular 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 ?

Similar mechanisms explain the Similar mechanisms explain the emergence of hierarchical emergence of hierarchical

structures in societystructures in societyMany complex networks that we see in Many complex networks that we see in society (and also nature) have hierarchical society (and also nature) have hierarchical structures. Why ?structures. Why ?

Consider social networks as solutions to multi-Consider social networks as solutions to multi-constraint problems:constraint problems:• Increase communication efficiency Increase communication efficiency Decrease Decrease average path length average path length ll in a network in a network• Minimize information load at each node Minimize information load at each node Decrease max degree kDecrease max degree kmaxmax

• Minimize overall communication needs Minimize overall communication needs Decrease Decrease total number of links total number of links LL

Emergence of hierarchyEmergence of hierarchy

Emergence of hierarchyEmergence of hierarchyMinimize the energy Minimize the energy function, function, E = a E = a ll + g + g LL + (1-a- + (1-a-g) g) kkmaxmax

• For a = 0, g =1 For a = 0, g =1 optimal network is a optimal network is a chainchain• For a = 1, g = 0, For a = 1, g = 0, optimal network is a optimal network is a cliqueclique• For a = g =0.5, For a = g =0.5, optimal network is a optimal network is a starstar• Over large range of Over large range of values of a and g, values of a and g, hierarchical networks hierarchical networks emerge – hierarchy emerge – hierarchy measured by HQmeasured by HQ

Cycles

Hierarchical network

Modular networks are small-world networks: Modular networks are small-world networks: indistinguishable from WS model generated indistinguishable from WS model generated networks by using structural measures networks by using structural measures

Dynamics on modular networks (but not in WS Dynamics on modular networks (but not in WS networks) show distinct, separate time-scales; networks) show distinct, separate time-scales; manifested as Laplacian spectral gap seen in real manifested as Laplacian spectral gap seen in real networks (cortico-cortical brain networks)networks (cortico-cortical brain networks)

Collective behavior such as ordering in spin Collective behavior such as ordering in spin models show a novel phase corresponding to models show a novel phase corresponding to modular order, in addition to global order and no modular order, in addition to global order and no order – even when mutual interactions favor order – even when mutual interactions favor consensusconsensus

Conclusions: Conclusions:

Thanks: Thanks:

Raj Kumar Pan

Sumithra Surendralal

Abhishek Dasgupta

Subinay Dasgupta

Anindya Sinha