1 / 36

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.

ghalib
Télécharger la présentation

Dynamic Models of On-Line Social Networks

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related