1. On-line Social Networks - Anthony Bonato 1 Dynamic Models of On-Line Social Networks Anthony Bonato Ryerson University WAW’2009 February 13, 2009 nt

2. On-line Social Networks - Anthony Bonato 2 Complex Networks web graph, social networks, biological networks, internet networks, …

3. On-line Social Networks - Anthony Bonato 3 Social Networks nodes: people edges: social interaction

4. On-line Social Networks - Anthony Bonato 4 On-line networks: Flickr graph

5. On-line Social Networks - Anthony Bonato 5 Facebook graph

6. On-line Social Networks - Anthony Bonato 6 On-line gaming networks

7. On-line Social Networks - Anthony Bonato 7 Properties of Complex Networks observed properties: –massive, power law, small world, decentralized –many bipartite subgraphs, high clustering, sparse cuts, strong conductance, eigenvalue power law, … (Broder et al, 01)

8. On-line Social Networks - Anthony Bonato 8 Small World Property small world networks introduced by social scientists Watts & Strogatz in 1998 –low diameter/average distance (“6 degrees of separation”) –globally sparse, locally dense (high clustering coefficient)

9. On-line Social Networks - Anthony Bonato 9 Social network analysis Milgram (67): average distance between two Americans is 6 Watts and Strogatz (98): introduced small world property Adamic et al. (03): early study of on-line social networks Liben-Nowell et al. (05): small world property in LiveJournal Kumar et al. (06): Flickr, Yahoo!360; average distances decrease with time Golder et al. (06): studied 4 million users of Facebook Ahn et al. (07): studied Cyworld in South Korea, along with MySpace and Orkut Mislove et al. (07): studied Flickr, YouTube, LiveJournal, Orkut On-line

10. On-line Social Networks - Anthony Bonato 10 Key parameters power law degree distributions: average distance: clustering coefficient: Wiener index, W(G)

11. On-line Social Networks - Anthony Bonato 11 Facebook Golder et al (06): current number of users (nodes): > 120 million

12. On-line Social Networks - Anthony Bonato 12 Flickr and Yahoo!360 Kumar et al (06): shrinking diameters

13. On-line Social Networks - Anthony Bonato 13 Sample data: Flickr, YouTube, LiveJournal, Orkut Mislove et al (07): short average distances and high clustering coefficients

14. On-line Social Networks - Anthony Bonato 14 Leskovec, Kleinberg, Faloutsos (05): –many complex networks obey two laws: 1.Densification Power Law –networks are becoming more dense over time; i.e. average degree is increasing e(t) ≈ n(t) a where 1 < a ≤ 2: densification exponent –a=1: linear growth – constant average degree, such as in web graph models –a=2: quadratic growth – cliques

15. On-line Social Networks - Anthony Bonato 15 Densification – Physics Citations n(t) e(t) 1.69

16. On-line Social Networks - Anthony Bonato 16 Densification – Autonomous Systems n(t) e(t) 1.18

17. On-line Social Networks - Anthony Bonato 17 2.Decreasing distances distances (diameter and/or average distances) decrease with time –noted by Kumar et al in Flickr and Yahoo!360 in contrast with Preferential attachment model –a.a.s. diameter O(log t)

18. On-line Social Networks - Anthony Bonato 18 Diameter – ArXiv citation graph time [years] diameter

19. On-line Social Networks - Anthony Bonato 19 Diameter – Autonomous Systems number of nodes diameter

20. On-line Social Networks - Anthony Bonato 20 Models for the laws Leskovec, Kleinberg, Faloutsos (05, 07): –Forest Fire model stochastic densification power law, decreasing diameter, power law degree distribution Leskovec, Chakrabarti, Kleinberg,Faloutsos (05, 07): –Kronecker Multiplication deterministic densification power law, decreasing diameter, power law degree distribution

21. On-line Social Networks - Anthony Bonato 21 Models of On-line Social Networks many models exist for general complex networks (preferential attachment, random power law, copying, duplication, geometric, rank-based, Forest fire, …) few models for on-line social networks goal: design and analyze a model which simulates many of the observed properties of on-line social networks –model should be simple and evolve in a natural way

22. On-line Social Networks - Anthony Bonato 22 “All models are wrong, but some are more useful.” – G.P.E. Box

23. On-line Social Networks - Anthony Bonato 23 Iterated Local Transitivity (ILT) model (Bonato, Hadi, Horn, Prałat, Wang, 08) key paradigm is transitivity: friends of friends are more likely friends; eg (Frank,80), (White, Harrison, Breiger, 76), (Scott, 00) –iterative cloning of closed neighbour sets deterministic: amenable to analysis local: nodes often only have local influence evolves over time, but retains memory of initial graph

24. On-line Social Networks - Anthony Bonato 24 ILT model parameter: finite simple undirected graph G = G 0 to form the graph G t+1 for each vertex x from time t, add a vertex x’, the clone of x, so that xx’ is an edge, and x’ is joined to each neighbour of x

25. On-line Social Networks - Anthony Bonato 25 G 0 = C 4

26. On-line Social Networks - Anthony Bonato 26 Properties of ILT model average degree increasing to ∞ with time average distance bounded by constant and converging, and in many cases decreasing with time; diameter does not change clustering higher than in a random generated graph with same average degree cop and domination numbers do not change with time bad expansion: small gaps between 1 st and 2 nd eigenvalues in adjacency and normalized Laplacian matrices of G t

27. On-line Social Networks - Anthony Bonato 27 Densification n t = order of G t, e t = size of G t Lemma: For t > 0, n t = 2 t n 0, e t = 3 t (e 0 +n 0 ) - 2 t n 0. → densification power law: e t ≈ n t a, where a = log(3)/log(2).

28. On-line Social Networks - Anthony Bonato 28 Average distance Theorem 2: If t > 0, then average distance bounded by a constant, and converges; for many initial graphs (large cycles) it decreases diameter does not change from time 0

29. On-line Social Networks - Anthony Bonato 29 Clustering Coefficient Theorem 3: If t > 0, then c(G t ) = n t log(7/8)+o(1). higher clustering than in a random graph G(n t,p) with same order and average degree as G t, which satisfies c(G(n t,p)) = n t log(3/4)+o(1)

30. On-line Social Networks - Anthony Bonato 30 Sketch of proof each node x at time t has a binary sequence corresponding to descendants from time 0, with a clone indicated by 1 let e(x,t) be the number of edges in N(x) at time t we show that e(x,t+1) = 3e(x,t) + 2deg t (x) e(x’,t+1) = e(x,t) + deg t (x) if there are k many 1’s in the binary sequence of x, then e(x,t) ≥ 3 k-2 e(x,2) = Ω(3 k )

31. On-line Social Networks - Anthony Bonato 31 Sketch of proof, continued there are many nodes with k many 0’s in their binary sequence hence,

32. On-line Social Networks - Anthony Bonato 32 example: (Zachary, 72) observed social ties and rivalries in a university karate club (34 nodes,78 edges) during his observation, conflicts intensified and group split see also (Girvan, Newman, 02) Community structure in social networks

33. On-line Social Networks - Anthony Bonato 33 Spectral results the spectral gap λ of G is defined by min{λ 1, 2 - λ n-1 }, where 0 = λ 0 ≤ λ 1 ≤ … ≤ λ n-1 ≤ 2 are the eigenvalues of the normalized Laplacian of G: I-D -1/2 AD 1/2 (Chung, 97) for random graphs, λ tends to 1 as order grows in the ILT model, λ < ½ similar results for adjacency matrix A bad expansion/small spectral gaps in the ILT model found in social networks but not in the web graph or biological networks (Estrada, 06) –in social networks, there are a higher number of intra- rather than inter-community links

34. On-line Social Networks - Anthony Bonato 34 Random model randomize the ILT model –add random edges independently to new nodes, with probability a function of t –makes densification tuneable densification exponent becomes log(3 + ε) / log(2), where ε is any fixed real number in (0,1) –gives any exponent in (log(3)/log(2), 2) similar (or better) distance, clustering and spectral results as in deterministic case

35. On-line Social Networks - Anthony Bonato 35 Missing ingredient: Power laws –generate power law graphs from ILT? deterministic ILT model gives a binomial-type distribution

36. On-line Social Networks - Anthony Bonato 36 preprints, reprints, contact: Google: “Anthony Bonato”