“Modularity” of Social Networks

1. “Modularity” of Social Networks Image: Ramki (www.wildventures.com) Sitabhra SinhaThe Institute of Mathematical Sciences, Chennai Image : R K Pan

2. Outline or Why ? What ? How ? • Why networks ? • What structures underlie real life networks ? • How do such structures affect network dynamics ? In particular, • How does it affect ordering dynamics in spin models ? Significant for understanding social coordination processes such as consensus formation & adoption of innovations R K Pan & SS, EPL (2009) S Dasgupta, R K Pan & SS, PRE Rapid (2009)

3. Nodes Why Networks ? Links • In a network of interacting components • Emergence • of qualitatively different behavior from individual components. E.g., component = neuron, system = brain Interactions add a new layer of complexity! The aim of studying networks: how interactions → complexity at systems-level

4. Ubiquity of Networks Networks appear at all scales Proteins Social contact Food webs Neuronal communication Intra-cellular signalling 10-9 m 10-6 m 10-3 m 1 m 103 m 106 m Molecules Cells Organisms Populations Ecologies

5. Theoretical understanding of networks • Regular lattice or grid (Physics) • average path length ~ N (no. of nodes) • clustering high • delta function distribution of degree (links/node) • Random networks (Graph theory) • average path length ~ log N • clustering low • Poisson distribution of degree Empirical networks are not random – many have certain structural patterns

6. Example: small-world networks Low clustering, Short path length High clustering, Short path length High clustering, Large path length Regular Network “Small-world” Network Random Network p = 1 0 < p < 1 p = 0 Increasing Randomness p: fraction of random, long-range connections Watts and Strogatz (1998): Many biological, technological and social networks have connection topologies that lie between the two extremes of completely regular and completely random.

7. Small-world networks can be highly clustered (like regular networks ), yet have small characteristic path lengths (as in random networks ). Characteristic path length ( ℓ ( p ) ) and clustering coefficient (C( p ) ) as network randomness increases.

8. A prominent example of small-world networks: Social networks Nodes : individuals (N = 3 – 109) Links : social interactions (< 150/node) • Represent the “underlying” social network (e.g. a knows b) • Approximated by real interactions between a & b (e.g. through phone calls, emails, trades, …) • Weight represents interaction strength

9. Example of social interaction networkCollaboration Networks Authors IMSc Physics co-authorship network (2000-08) 1 2 3 4 5 B C D A Papers Bipartite Paper-Author network Co-authorship network 1 2 4 5 3

10. Re-constructing social networksExample: Macaque troupe social network UAS-GKVK Campus, Bangalore Data: Anindya Sinha (NIAS,B’lore) The Bonnet Macaque (Macaca radiata) seen widely in southern India Usually live in large (~ 40) multi-male, multi-female troops where the adult individuals (~ 10) develop strong affiliative relationships Image: Ramki (www.wildventures.com) Image: Arunkumar

11. Macaque Social Networks can be defined in terms ofgrooming frequencytotal grooming time approach frequency Data: 1993-1997 grooming frequency Numbers refer to rank among the adult females from 11 (most dominant) to 1 (least dominant) grooming time approach frequency

12. Network analysis can predict group dynamics ! • Female bonnet macaques • usually remain in the group throughout their life • as adults, form strong linear matrilineal dominance hierarchies that are stable over time • Male bonnet macaques • as adults, form unstable dominance hierarchies • occupy low ranks when young, high when mature and at peak of health Community detection generates consistent partitions for females, not for males Predictive power: Observation in 1998 showed the group had split into two (11,10,9,8,7,2) and (6,5,4,3) [1 had died]

13. Importance of SW social network structure Contagion propagation in society through contact Spread of SARS from Taiwan, 2003 “The small-worlds of public health” (CDC director) Chen et al, Lect Notes Comp Sci 4506 (2009) 23

14. Worldwide spread of SARS through international airline network

15. Why small-world pattern in complex networks at all ? R K Pan and SS, EPL 2009 We have shown that structurally Watts-Strogatz network ≡ Modular network All the classic “small-world” structural properties of Watts-Strogatz small world networks e.g., high clustering, short average distance, etc. are also seen in Modular networks

16. Modular Networks: dense connections within certain sub-networks (modules) & relatively few connections between modules Modules: A mesoscopic organizational principle of networks Going beyond motifs but more detailed than globaldescription (L, C etc.) Micro Meso Macro Kim & Park, WIREs Syst Biol & Med, 2010

17. Ubiquity of modular networks Modular Biology (Hartwell et al, Nature 1999) Functional modules as a critical level of biological organization Modules in biological networks are often associated with specific functions Metabolic network of E coli (Guimera & Amaral) Chesapeake Bay foodweb (Ulanowicz et al)

18. How about social small-world networks ? Let’s look at real re-constructed social contact patterns: A mobile phone interaction network Onnela et al. PNAS,104,7332 (2007) Data: • A mobile phone operator in an European country, 20% coverage • Aggregated from a period of 18 weeks • > 7 million private mobile phone subscriptions • Voice calls within the operator • Require reciprocity of calls for a link • Quantify tie strength (link weight) Aggregate call duration Total number of calls 7 min 15 min (3 calls) 5 min 3 min J-P Onnela

19. Reconstructed network Onnela et al. PNAS,104,7332 (2007)

20. Modularity of social networks Modularity: Cohesive groups communities with dense internal & sparse external connections Other examples of modular social networks Scientific collaborators e-mail communication PGP encryption ”web-of-trust” non-human animals 4.6 106 nodes 7.0 106 links Onnela et al. PNAS,104,7332 (2007)

21. Q. What structure ? Ans. Modular Next…How do such structures affect dynamics ? • Over social networks, such dynamics can be of • information or epidemic spreading • consensus or opinion formation • adoption of innovations

22. A simple model of modular networks Module ≡ random network Adjacency matrix Model parameter r : Ratio of inter- to intra-modular connectivity The modularity of the network is changed keeping avg degree constant

23. Comparison with Watts-Strogatz model Structural measures used: Communication efficiency E = [avg path length, ℓ ]-1 = 2 /N(N-1) i>jdij Clustering coefficient C = fraction of observed to potential triads = (1 /N) i2ni / ki (ki - 1) WS and Modular networks behave similarly as function of p or r (Also for between-ness centrality, edge clustering, etc) In fact, for same N and <k>, we can find p and r such that the WS and Modular networks have the same “modularity” Q

24. Then how can you tell them apart ? Dynamics on Watts-Strogatz network different from that on Modular networks Network topology Consider ordering or alignment of orientation on such networks e.g., Ising spin model: dynamics minimizes H= -  Jij Si (t) Sj (t) 2 distinct time scales in Modular networks: t modular & t global Global order Time required for global ordering diverges as r →0 Modular order

25. Universality Almost identical two time scale behavior is seen for oscillator synchronization (viz., nonlinear relaxation oscillators) Network topology Consider synchronization on modular networks e.g., Kuramoto oscillators: di /dt = w + (1/ki)Kij sin (j -  i) 2 distinct time scales in Modular networks: t modular & t global

26. Eigenvalue spectra of the Laplacian Shows the existence of spectral gap  distinct time scales Modular network Laplacian spectra Spectral gap in modular networks diverges with decreasing r gap No gap WS network Laplacian spectra Existence of distinct time-scales in Modular networks No such distinction in Watts-Strogatz small-world networks

27. Diffusion process on modular networks Random walker moving from one node to randomly chosen nghbring node • Also shows the existence of 2 distinct time scales: • fast intra-modular diffusion • slower inter-modular diffusion Relevant for diffusion of innovation or epidemics Distrn of first passage times for rnd walks to reach a target node starting from a source node Localization of eigenmodes of transition matrix ER WS Modular Between modules diffusion Within modules diffusion

28. How about real small-world networks ? The networks of cortical connections in mammalian brain have been shown to have small-world structural properties Our analysis reveals their dynamical properties to be consistent with modular “small-world” networks gap gap

29. We again turn attention back to consensus formation dynamics Q. How does individual behavior at micro-level relate to social phenomena at macro level ? Order-disorder transitions in Social Coordination

30. The emergence of novel phase of collective behavior Spin models of statistical physics: simple models of coordination or consensus formation • Spin orientation: mutually exclusive choices • Choice dynamics: decision based on information about choice of majority in local neighborhood Simplest case: 2 possible choices Ising model with FM interactions: each agent can only be in one of 2 states (Yes/No or +/-)

31. Types of possible order in modular network of Ising spins Modular order Global order N spins, nm modules FM interactions: J > 0 Avg magnetic moment / module Total or global magnetic moment

32. But how do the different ordered phases occur as a function of modularity parameter r and temperature T ? Long transient to global order can be mistaken as modular order Possible Phase Diagrams Tgc Tgc Tgc T T T 0 r 1 0 r 1 0 r 1 Can modular order be seen as a phase at all ?

33. fraction of “up” spins in module Magnetic moment of a single module fraction of modules with + At eqlbm, for strong modularity (r << 1) Total magnetic moment Minimizing free energy w.r.t. f+ Continuous transition to “modular order” phase below “Modular” critical temp # links within module As T is lowered, Another continuous transition to “global order” phase below “Global” critical temp conn prob betn modules

34. Phase diagram: two transitions r = 0.002 T There will be a phase corresponding to modular but no global order (coexistence of contrary opinions) even when all mutual interactions are FM (favor consensus) !

35. Even when global order is possible … Divergence of relaxation time with modularity To switch from + to - , module crosses free energy barrier: Relaxation time to global order: - + At low T:

36. Thus, even when strongly modular network takes very long to show global order Time required to achieve consensus increases rapidly for a strongly modular social organization But then … How do certain innovations get adopted rapidly ? • Possible modifications to the dynamics: • Positive feedback • Different strengths for inter/intra couplings

37. I. The effect of Positive Feedback Brian Arthur The case of “counter-clockwise” clocks Microsoft & the 7 Dwarves

38. The effect of Positive Feedback Introduce a field H = hM (proportional to magnetization) • Effectively increases inter-modular interactions • Drives system away from critical line by increasing Tgc • reduces

39. II. Varying Strength of Coupling Between & Within Modules Marc Granovetter J0 : strength of inter-modular connections Ji : strength of intra-modular connections increasing J0 / Ji increasing r Non-monotonic behavior of relaxation time vs ratio of strengths of short- & long-range couplings Not seen in Watts-Strogatz SW networks (Jeong et al, PRE 2005)

40. Ongoing and Future work Effect of modular contact structure in formal games - Is it easier for cooperation to emerge for Prisoners Dilemma on modular networks ? Could modularity in social networks have arisen as a result of modules promoting cooperation – necessary for building social organization ? Role of multiple (>2) choices : q-state Potts model Having different types of interactions : spin glass behavior on modular networks

41. Question: Why Modular Networks ? Suggestion: modularity imparts robustness E.g.,Variano et al, PRL 2004 Not quite ! Consider stability of a random network with a modular structure. N=256 As random networks are divided into more modules (m) they become more unstable… … however, we see modular networks all around us. Why ?

42. Question: Why Modular Networks ? Clue: Many of these modular networks also possess multiple hubs ! Most real networks have non-trivial degree distribution Degree: total number of connections for a node Hubs: nodes having high degree relative to other nodes

43. Why Modular Networks ? R K Pan & SS, PRE 76 045103(R)(2007) Hypothesis:Real networks optimize between several constraints, • Minimizing link cost, i.e., total # links L • Minimizing average path length ℓ • Minimizing instability max Minimizing link cost and avg path length yields … … a star-shaped network with single hub But unstable ! Instability measured by max ~ max degree (i.e., degree of the hub) In fact, for star network, max ~ N

44. How to satisfy all three constraints ? Answer: go modular ! As star-shaped networks are divided into more modules they become more stable … …as stability increases by decreasing the degree of hub nodes max~ [N/m] Increasing stability, average path length  = 0.78  = 0.4  = 1 N = 64, L = N-1 Shown explicitly by Network Optimization Fix link cost to min (L=N-1) and minimize the energy fn E () = ℓ + (1- ) max [0,1] :relative importance of path length constraint over stability constraint Transition to star configuration

45. The robustness of modular structures We have considered dynamical instability criterion for network robustness – how about stability against structural perturbations ? E.g., w.r.t. random or targeted removal of nodes Scale-free network: robust w.r.t. random removal Random network: robust w.r.t. targeted removal Is the modular network still optimal ? Surprisingly YES ! At the limit of extremely small L, optimal modular networks ≡ networks with bimodal degree distribution… …has been shown to be robust w.r.t. targeted as well as random removal of nodes Tanizawa et al, PRE 2005

46. Similar mechanisms explain the emergence of hierarchical structures in society Many complex networks that we see in society (and also nature) have hierarchical structures. Why ?

47. Emergence of hierarchy Consider social networks as solutions to multi-constraint problems: • Increase communication efficiency  Decrease average path length l in a network • Minimize information load at each node  Decrease max degree kmax • Minimize overall communication needs  Decrease total number of links L

48. Emergence of hierarchy Minimize the energy function, E = a l+ g L + (1-a-g) kmax • For a = 0, g =1 optimal network is a chain • For a = 1, g = 0, optimal network is a clique • For a = g =0.5, optimal network is a star • Over large range of values of a and g, hierarchical networks emerge – hierarchy measured by HQ Cycles

49. Hierarchical network

50. Conclusions: • Modular networks are small-world networks: indistinguishable from WS model generated networks by using structural measures • Dynamics on modular networks (but not in WS networks) show distinct, separate time-scales; manifested as Laplacian spectral gap seen in real networks (cortico-cortical brain networks) • Collective behavior such as ordering in spin models show a novel phase corresponding to modular order, in addition to global order and no order – even when mutual interactions favor consensus