1. On-line Social Networks - Anthony Bonato 1 Dynamic Models of On-line Social Networks Anthony Bonato Ryerson University ICMCM’09 December, 2009

2. On-line Social Networks - Anthony Bonato 2 Toronto in December…

3. On-line Social Networks - Anthony Bonato 3 Complex Networks web graph, social networks, biological networks, internet networks, …

4. On-line Social Networks - Anthony Bonato 4 The web graph nodes: web pages edges: links over 1 trillion nodes, with billions of nodes added each day

5. On-line Social Networks - Anthony Bonato 5 Social Networks nodes: people edges: social interaction (eg friendship)

6. On-line Social Networks - Anthony Bonato 6 On-line Social Networks (OSNs) Facebook, Twitter, Orkut, LinkedIn, GupShup…

7. A new paradigm half of all users of internet on some OSN –250 million users on Facebook, 45 million on Twitter unprecedented, massive record of social interaction unprecedented access to information/news/gossip On-line Social Networks - Anthony Bonato 7

8. 8 Properties of Complex Networks observed properties: –massive, power law, small world, decentralized (Broder et al, 01)

9. On-line Social Networks - Anthony Bonato 9 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)

10. Paths in Twitter On-line Social Networks - Anthony Bonato 10 Dalai Lama Ashton Kutcher Christianne Amanpour Queen Rania of Jordan Arnold Schwarzenegger

11. On-line Social Networks - Anthony Bonato 11 Why model complex networks? uncover the generative mechanisms underlying complex networks models are a predictive tool nice mathematical challenges models can uncover the hidden reality of networks –in OSNs: community detection advertising security

12. Many different models On-line Social Networks - Anthony Bonato 12

13. On-line Social Networks - Anthony Bonato 13 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 Java et al. (07): studied Twitter: power laws, small world On-line

14. On-line Social Networks - Anthony Bonato 14 Key parameters power law degree distributions: average distance: clustering coefficient: Wiener index, W(G)

15. Power laws in OSNs On-line Social Networks - Anthony Bonato 15

16. On-line Social Networks - Anthony Bonato 16 Flickr and Yahoo!360 (Kumar et al,06): shrinking diameters

17. On-line Social Networks - Anthony Bonato 17 Sample data: Flickr, YouTube, LiveJournal, Orkut (Mislove et al,07): short average distances and high clustering coefficients

18. On-line Social Networks - Anthony Bonato 18 (Leskovec, Kleinberg, Faloutsos,05): –many complex networks (including on-line social networks) obey two additional 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

19. On-line Social Networks - Anthony Bonato 19 Densification – Physics Citations n(t) e(t) 1.69

20. On-line Social Networks - Anthony Bonato 20 Densification – Autonomous Systems n(t) e(t) 1.18

21. On-line Social Networks - Anthony Bonato 21 2.Decreasing distances distances (diameter and/or average distances) decrease with time –noted by Kumar et al. in Flickr and Yahoo!360 Preferential attachment model (Barabási, Albert, 99), (Bollobás et al, 01) –diameter O(log t) Random power law graph model (Chung, Lu, 02) –average distance O(log log t)

22. On-line Social Networks - Anthony Bonato 22 Diameter – ArXiv citation graph time [years] diameter

23. On-line Social Networks - Anthony Bonato 23 Diameter – Autonomous Systems number of nodes diameter

24. On-line Social Networks - Anthony Bonato 24 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

25. On-line Social Networks - Anthony Bonato 25 Models of OSNs many models exist for general complex networks few models for on-line social networks goal: find a model which simulates many of the observed properties of OSNs –must be simple and evolve in a natural way –must be different than previous complex network models: densification and constant diameter!

26. On-line Social Networks - Anthony Bonato 26 “All models are wrong, but some are more useful.” – G.P.E. Box

27. On-line Social Networks - Anthony Bonato 27 Iterated Local Transitivity (ILT) model (Bonato, Hadi, Horn, Prałat, Wang, 08) key paradigm is transitivity: friends of friends are more likely friends; eg (Girvan and Newman, 03) –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

28. On-line Social Networks - Anthony Bonato 28 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 order of G t is 2 t n 0

29. On-line Social Networks - Anthony Bonato 29 G 0 = C 4

30. On-line Social Networks - Anthony Bonato 30 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 bad expansion: small gaps between 1 st and 2 nd eigenvalues in adjacency and normalized Laplacian matrices of G t

31. On-line Social Networks - Anthony Bonato 31 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).

32. On-line Social Networks - Anthony Bonato 32 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

33. On-line Social Networks - Anthony Bonato 33 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)

34. On-line Social Networks - Anthony Bonato 34 Sketch of proof of lower bound 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 may 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 0’s in the binary sequence of x, then e(x,t) ≥ 3 k-2 e(x,2) = Ω(3 k )

35. On-line Social Networks - Anthony Bonato 35 Sketch of proof, continued there are many nodes with k many 0’s in their binary sequence hence,

36. On-line Social Networks - Anthony Bonato 36 Wayne Zachary’s Ph.D. thesis (1970-72): observed social ties and rivalries in a university karate club (34 nodes,78 edges) during his observation, conflicts intensified and group split Example of community structure

37. On-line Social Networks - Anthony Bonato 37 Adjacency matrix, A eigenvalue spectrum: (-2) 4 1 5 3 1

38. On-line Social Networks - Anthony Bonato 38 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 bad expansion/small spectral gaps in the ILT model found in social networks but not in the web graph (Estrada, 06) –in social networks, there are a higher number of intra- rather than inter-community links

39. On-line Social Networks - Anthony Bonato 39 Random ILT model randomize the ILT model –add random edges independently to new nodes, with probability a function of t –makes densification tunable 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

40. On-line Social Networks - Anthony Bonato 40 Degree distribution –generate power law graphs from ILT? deterministic ILT model gives a binomial-type distribution

41. Geometric model for social networks OSNs live in social space: proximity of nodes depends on common attributes (such as geography, gender, age, etc.) IDEA: embed OSN in m- dimensional Euclidean space On-line Social Networks - Anthony Bonato 41

42. Dimension of an OSN dimension of OSN: minimum number of attributes needed to classify nodes like game of “20 Questions”: each question narrows range of possibilities what is a credible mathematical formula for the dimension of an OSN? On-line Social Networks - Anthony Bonato 42

43. On-line Social Networks - Anthony Bonato 43 Random geometric graphs nodes are randomly distributed in Euclidean space according to a given distribution nodes are joined by an edge if and only if their distance is less than a threshold value (Penrose, 03)

44. Spatial model for OSNs we consider a spatial model of OSNs, where –nodes are embedded in m-dimensional Euclidean space –number of nodes is static –threshold value variable: a function of ranking of nodes On-line Social Networks - Anthony Bonato 44

45. Prestige-Based Spatial (PBS) Model (Bonato, Janssen, Prałat, 09) parameters: α, β in (0,1), α+β < 1; positive integer m nodes live in hypercube of dimension m, measure 1 each node is ranked 1,2, …, n by some function r –1 is best, n is worst –we use random initial ranking at each time-step, one new node v is born, one node chosen u.a.r. dies (and ranking is updated) each existing node u has a region of influence with volume add edge uv if v is in the region of influence of u On-line Social Networks - Anthony Bonato 45

46. Notes on PBS model models uses both geometry and ranking dynamical system: gives rise to ergodic (therefore, convergent) Markov chain –users join and leave OSNs number of nodes is static: fixed at n –order of OSNs has ceiling top ranked nodes have larger regions of influence On-line Social Networks - Anthony Bonato 46

47. Properties of the PBS model (Bonato, Janssen, Prałat, 09) with high probability, the PBS model generates graphs with the following properties: –power law degree distribution with exponent b = 1+1/α –average degree d = (1+o(1))n (1-α-β) /2 1-α dense graph tends to infinity with n –diameter D = (1+o(1))n β/(1-α)m depends on dimension m m = clog n, then diameter is a constant On-line Social Networks - Anthony Bonato 47

48. Dimension of an OSN, continued given the order of the network n, power law exponent b, average degree d, and diameter D, we can calculate m gives formula for dimension of OSN: On-line Social Networks - Anthony Bonato 48

49. Uncovering the hidden reality reverse engineering approach –given network data (n, b, d, D), dimension of an OSN gives smallest number of attributes needed to identify users that is, given the graph structure, we can (theoretically) recover the social space On-line Social Networks - Anthony Bonato 49

50. Examples OSNDimension Facebook6 MySpace8 Twitter4 Flickr4 On-line Social Networks - Anthony Bonato 50

51. Future directions what is a community in an OSN? –(Porter, Onnela, Mucha,09): a set of graph partitions obtained by some “reasonable” iterative hierarchical partitioning algorithm –motifs –Pott’s method from statistical mechanics –betweeness centrality lack of a formal definition, and few theorems On-line Social Networks - Anthony Bonato 51

52. Spatial ranking models rigorously analyze spatial model with ranking by –age –degree simulate PBS model –fit model to data –is theoretical estimate of the dimension of an OSN accurate? On-line Social Networks - Anthony Bonato 52

53. Who is popular? how to find popular users? PageRank in OSNs domination number –constant in ILT model –in OSN data, domination number is large (end- vertices) –which is the correct graph parameter to consider? On-line Social Networks - Anthony Bonato 53

54. On-line Social Networks - Anthony Bonato 54 preprints, reprints, contact: Google: “Anthony Bonato”

55. WOSN’2010 On-line Social Networks - Anthony Bonato 55

56. On-line Social Networks - Anthony Bonato 56