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Dynamic Models of On-Line Social Networks

WAW’2009 February 13, 2009. Dynamic Models of On-Line Social Networks. nt. Anthony Bonato Ryerson University. Complex Networks. web graph, social networks, biological networks, internet networks , …. Social Networks. nodes : people edges : social interaction.

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Dynamic Models of On-Line Social Networks

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

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

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

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

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

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

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

  9. Social network analysis On-line • 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): studiedCyworld in South Korea, along with MySpace and Orkut • Mislove et al. (07): studiedFlickr, YouTube, LiveJournal, Orkut On-line Social Networks - Anthony Bonato

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

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

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

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

  14. Leskovec, Kleinberg, Faloutsos (05): • many complex networks obey two laws: • 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 On-line Social Networks - Anthony Bonato

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

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

  17. 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) On-line Social Networks - Anthony Bonato

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

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

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

  24. ILT model • parameter: finite simple undirected graph G = G0 • to form the graph Gt+1 for each vertex x from time t, add a vertex x’, the clone ofx, so that xx’ is an edge, and x’ is joined to each neighbour of x On-line Social Networks - Anthony Bonato

  25. G0 = C4 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 1st and 2nd eigenvalues in adjacency and normalized Laplacian matrices of Gt On-line Social Networks - Anthony Bonato

  27. Densification • nt = order of Gt, et = size of Gt Lemma: For t > 0, nt = 2tn0, et = 3t(e0+n0) - 2tn0. → densification power law: et ≈ nta, where a = log(3)/log(2). 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 On-line Social Networks - Anthony Bonato

  29. Clustering Coefficient Theorem 3: If t > 0, then c(Gt) = ntlog(7/8)+o(1). • higher clustering than in a random graph G(nt,p) with same order and average degree as Gt, which satisfies c(G(nt,p)) = ntlog(3/4)+o(1) 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) + 2degt(x) e(x’,t+1) = e(x,t) + degt(x) • if there are k many 1’s in the binary sequence of x, then e(x,t) ≥ 3k-2e(x,2) = Ω(3k) 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, On-line Social Networks - Anthony Bonato

  32. Community structure in social networks • 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) On-line Social Networks - Anthony Bonato

  33. Spectral results • the spectral gapλ of G is defined by min{λ1, 2 - λn-1}, where 0 = λ0 ≤ λ1 ≤ … ≤ λn-1 ≤ 2 are the eigenvalues of the normalized Laplacian of G: I-D-1/2AD1/2(Chung, 97) • for random graphs, λtends 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 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 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 On-line Social Networks - Anthony Bonato

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

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