Vertex-pursuit in heirarchical social networks

1. TAMC’12 Vertex-pursuit in heirarchical social networks Anthony Bonato Ryerson University Complex Networks

2. 很高兴来北京 Complex Networks

3. Friendship networks • network of friends (some real, some virtual) form a large web of interconnected links Complex Networks

4. 6 degrees of separation • (Stanley Milgram, 67): famous chain letter experiment Complex Networks

5. 6 Degrees in Facebook? • 900 million users, > 70 billion friendship links • (Backstrom et al., 2012) • 4 degrees of separation in Facebook • when considering another person in the world, a friend of your friend knows a friend of their friend, on average • similar results for Twitter and other OSNs Complex Networks

6. Complex Networks • web graph, social networks, biological networks, internet networks, … Complex Networks

7. nodes: web pages edges: links over 1 trillion nodes, with billions of nodes added each day The web graph Complex Networks

8. nodes: proteins edges: biochemical interactions Biological networks: proteomics Yeast: 2401 nodes 11000 edges Complex Networks

9. On-line Social Networks (OSNs)Facebook, RenRen, Twitter, LinkedIn,… Complex Networks

10. Key parameters • degree distribution: • average distance: • clustering coefficient: Complex Networks

11. Properties of Complex Networks • power law degree distribution (Broder et al, 01) Complex Networks

12. Power laws in OSNs (Mislove et al,07): Complex Networks

13. Small World Property • small world networks (Watts & Strogatz,98) • low distances • diam(G) = O(log n) • L(G) = O(loglog n) • higher clustering coefficient than random graph with same expected degree Complex Networks

14. Sample data: Flickr, YouTube, LiveJournal, Orkut • (Mislove et al,07): short average distances and high clustering coefficients Complex Networks

15. Community structure • (Zachary, 72) • (Mason et al, 09) • (Fortunato, 10) • (Li, Peng, 11): • small community • property Complex Networks

16. (Leskovec, Kleinberg, Faloutsos,05): • densification power law: average degree is increasing with time • decreasing distances • (Kumar et al, 06): observed in Flickr, Yahoo! 360 Complex Networks

17. Models of complex networks • We focus on • Geometric models of OSNs. • Logarithmic dimension hypothesis • Vertex pursuit in hierarchical social networks • spread of gossip and news; disrupting terrorist networks Complex Networks

18. Why model complex networks? • uncover and explain the generative mechanisms underlying complex networks • predict the future • nice mathematical challenges • models can uncover the hidden reality of networks Complex Networks

19. Many different models Complex Networks

20. “All models are wrong, but some are more useful.” – G.P.E. Box Complex Networks

21. Geometry of OSNs? • OSNs live in social space: proximity of nodes depends on common attributes (such as geography, gender, age, etc.) • IDEA: embed OSN in 2-, 3- or higher dimensional space Complex Networks

22. 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? Complex Networks

23. Random geometric graphs, G(n,r) • nodes are randomly placed in space • each node has a constant sphere of influence • nodes are joined if their sphere of influence overlap Complex Networks

24. Simulation with 5000 nodes Complex Networks

25. Spatially Preferred Attachment (SPA) model(Aiello, Bonato, Cooper, Janssen, Prałat, 08), (Cooper, Frieze, Prałat,12) • volume of sphere of influence proportional to in-degree • nodes are added and spheres of influence shrink over time • asymptotically almost surely (a.a.s.) leads to power laws graphs, low directed diameter, and small separators Complex Networks

26. Protean graphs(Fortunato, Flammini, Menczer,06),(Łuczak, Prałat, 06), (Janssen, Prałat,09) • parameter: α in (0,1) • each node is ranked 1,2, …, n by some function r • 1 is best, n is worst • at each time-step, one new node is born, one randomly node chosen dies (and ranking is updated) • link probability proportional to r-α • many ranking schemes a.a.s. lead to power law graphs: random initial ranking, degree, age, etc. Complex Networks

27. Geometric model for OSNs • we consider a geometric model of OSNs, where • nodes are in m-dimensional Euclidean space • threshold value variable: a function of ranking of nodes Complex Networks

28. Geometric Protean (GEO-P) Model(Bonato, Janssen, Prałat, 12) • parameters: α, β in (0,1), α+β < 1; positive integer m • nodes live in m-dimensional hypercube (torus metric) • 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 randomly node chosen 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 Complex Networks

29. Notes on GEO-P model • models uses both geometry and ranking • number of nodes is static: fixed at n • order of OSNs at most number of people (roughly…) • top ranked nodes have larger regions of influence Complex Networks

30. Simulation with 5000 nodes Complex Networks

31. Simulation with 5000 nodes random geometric GEO-P Complex Networks

32. Properties of the GEO-P model (Bonato, Janssen, Prałat, 2012) • a.a.s.the GEO-P 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)/21-α • densification • diameter D = O(nβ/(1-α)m log2α/(1-α)m n) • small world: constant order if m= Clog n • clustering coefficient larger than in comparable random graph Complex Networks

33. Diameter • eminent node: • old: at least n/2 nodes are younger • highly ranked: initial ranking greater than some fixed R • partition hypercube into small hypercubes • choose size of hypercubes and R so that • each hypercube contains at least log2n eminent nodes • sphere of influence of each eminent node covers each hypercube and all neighbouringhypercubes • choose eminent node in each hypercube: backbone • show all nodes in hypercube distance at most 2 from backbone Complex Networks

34. Spectral properties • the spectral gapλ of G is defined by the difference between the two largest eigenvalues of the adjacency matrix of G • for G(n,p) random graphs, λtends to0 as order grows • in the GEO-P model, λis close to 1 • (Estrada, 06): bad spectral expansion in real OSN data Complex Networks

35. Dimension of OSNs • given the order of the network n, power law exponentb, average degree d, and diameterD, we can calculate m • gives formula for dimension of OSN: Complex Networks

36. 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 Complex Networks

37. 6 Dimensions of Separation Complex Networks

38. Hierarchical social networks • Twitter is highly directed: can view a user and followers as a directed acyclic graph (DAG) • flow of information is top-down • such hierarchical social networks also appear in the social organization of companies and in terrorist networks • How to disrupt this flow? What is a model? Complex Networks

39. Good guys vs bad guys games in graphs bad good Complex Networks

40. Complex Networks

41. Seepage • motivated by the 1973 eruption of the Eldfell volcano in Iceland • to protect the harbour, the inhabitants poured water on the lava in order to solidify and halt it Complex Networks

42. Seepage (Clarke,Finbow,Fitzpatrick,Messinger,Nowakowski,2009) • greens and sludge, played on a directed acylic graph (DAG) with one source s • the players take turns, with the sludge going first by contaminating s • on subsequent moves sludge contaminates a non-protected vertex that is adjacent to a contaminated vertex • the greens, on their turn, choose some non-protected, non-contaminated vertex to protect • once protected or contaminated, a vertex stays in that state to the end of the game • sludge wins if some sink is contaminated; otherwise, the greens win Complex Networks

43. Example: G S S G G x Complex Networks

44. Green number • green number of a DAG G, gr(G), is the minimum number of greens needed to win • gr(G) = 1: G is green-win; as in previous example • (CFFMN,2009): • characterized green-win trees • bounds given on green number of truncated Cartesian products of paths Complex Networks

45. Mathematical counter-terrorism • (Farley et al. 2003-):ordered sets as simplified models of terrorist networks • the maximal elements of the poset are the leaders • submit plans down via the edges to the foot soldiers or minimal nodes • only one messenger needs to receive the message for the plan to be executed. • considered finding minimum order cuts: neutralize operatives in the network Complex Networks

46. Seepage as a counter-terrorism model? • seepage has a similar paradigm to model of (Farley et al) • main difference: seepage is dynamic • as messages move down the network towards foot soldiers, operatives are neutralized over time Complex Networks

47. Structure of terrorist networks • competing views; for eg (Xu et al, 06), (Memon, Hicks, Larsen, 07), (Medina,Hepner,08): • complex network: power law degree distribution • some members more influential and have high out-degree • regular network: members have constant out-degree • members are all about equally influential Complex Networks

48. Our model • we consider a stochastic DAG model • total expected degrees of vertices are specified • directed analogue of the G(w) model of Chung and Lu Complex Networks

49. General setting for the model • given a DAG G with levels Lj, source v, c > 0 • game G(G,v,j,c): • nodes in Lj are sinks • sequence of discrete time-steps t • nodes protected at time-step t • grj(G,v) = inf{c ϵR+: greens win G(G,v,j,c)} Complex Networks

50. Random DAG model (Bonato, Mitsche, Prałat,12+) • parameters: sequence (wi : i > 0), integer n • L0 = {v}; assume Lj defined • S: set of n new vertices • directed edges point from Lj to Lj+1 a subset of S • each vi in Lj generates max{wi -deg-(vi),0} randomly chosen edges to S • edges generated independently • nodes of S chosen at least once form Lj+1 • parallel edges possible (though rare in sparse case) Complex Networks