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

2. 2 很高兴来北京

3. Friendship networks network of friends (some real, some virtual) form a large web of interconnected links Complex Networks3

4. 6 degrees of separation Complex Networks4 (Stanley Milgram, 67): famous chain letter experiment

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 Networks5

6. 6 web graph, social networks, biological networks, internet networks, …

7. Complex Networks7 The web graph nodes: web pages edges: links over 1 trillion nodes, with billions of nodes added each day

8. Complex Networks8 Biological networks: proteomics nodes: proteins edges: biochemical interactions Yeast: 2401 nodes 11000 edges

9. Complex Networks9 On-line Social Networks (OSNs) Facebook, RenRen, Twitter, LinkedIn,…

10. Complex Networks10 Key parameters degree distribution: average distance: clustering coefficient:

11. Complex Networks11 Properties of Complex Networks power law degree distribution (Broder et al, 01)

12. Power laws in OSNs (Mislove et al,07): Complex Networks12

13. Complex Networks13 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

14. Complex Networks14 Sample data: Flickr, YouTube, LiveJournal, Orkut (Mislove et al,07): short average distances and high clustering coefficients

15. Complex Networks15 (Zachary, 72) (Mason et al, 09) (Fortunato, 10) (Li, Peng, 11): small community property Community structure

16. Complex Networks16 (Leskovec, Kleinberg, Faloutsos,05): densification power law: average degree is increasing with time decreasing distances (Kumar et al, 06): observed in Flickr, Yahoo! 360

17. Models of complex networks We focus on 1.Geometric models of OSNs. –Logarithmic dimension hypothesis 2.Vertex pursuit in hierarchical social networks –spread of gossip and news; disrupting terrorist networks Complex Networks17

18. Complex Networks18 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

19. Many different models Complex Networks19

20. Complex Networks20 “All models are wrong, but some are more useful.” – G.P.E. Box

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 Networks21

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 Networks22

23. Complex Networks23 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

24. Simulation with 5000 nodes Complex Networks24

25. Spatially Preferred Attachment (SPA) model (Aiello, Bonato, Cooper, Janssen, Prałat, 08), (Cooper, Frieze, Prałat,12) Complex Networks25 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

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 Networks26

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 Networks27

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 Networks28

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 Networks29

30. Simulation with 5000 nodes Complex Networks30

31. Simulation with 5000 nodes Complex Networks31 random geometricGEO-P

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) /2 1-α densification –diameter D = O(n β/(1-α)m log 2α/(1-α)m n) small world: constant order if m = Clog n –clustering coefficient larger than in comparable random graph Complex Networks32

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 log 2 n eminent nodes –sphere of influence of each eminent node covers each hypercube and all neighbouring hypercubes choose eminent node in each hypercube: backbone show all nodes in hypercube distance at most 2 from backbone Complex Networks33

34. Complex Networks34 Spectral properties the spectral gap λ of G is defined by the difference between the two largest eigenvalues of the adjacency matrix of G for G(n,p) random graphs, λ tends to 0 as order grows in the GEO-P model, λ is close to 1 (Estrada, 06): bad spectral expansion in real OSN data

35. Dimension of OSNs 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: Complex Networks35

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 Networks36

37. 6 Dimensions of Separation OSNDimension Facebook7 YouTube6 Twitter4 Flickr4 Cyworld7 Complex Networks37

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 Complex Networks38 How to disrupt this flow? What is a model?

39. Good guys vs bad guys games in graphs 39 slowmediumfasthelicopter slowtraps, tandem-win mediumrobot vacuumCops and Robbersedge searchingeternal security fastcleaningdistance k Cops and Robbers Cops and Robbers on disjoint edge sets The Angel and Devil helicopterseepageHelicopter Cops and Robbers, Marshals, The Angel and Devil, Firefighter Hex bad good Complex Networks

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41. Seepage Complex Networks41 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

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 Networks42

43. Example: G Complex Networks43 S G G S x

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 Networks44

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 Networks45

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 Networks46

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 Networks47

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 Networks48

49. General setting for the model given a DAG G with levels L j, source v, c > 0 game G (G,v,j,c): –nodes in L j are sinks –sequence of discrete time-steps t – nodes protected at time-step t gr j (G,v) = inf{c R + : greens win G (G,v,j,c)} Complex Networks49

50. Random DAG model (Bonato, Mitsche, Prałat,12+) parameters: sequence (w i : i > 0), integer n L 0 = {v}; assume L j defined S: set of n new vertices directed edges point from L j to L j+1 a subset of S each v i in L j generates max{w i -deg - (vi),0} randomly chosen edges to S edges generated independently nodes of S chosen at least once form L j+1 parallel edges possible (though rare in sparse case) Complex Networks50

51. d-regular case for all i, w i = d > 2 a constant –call these random d-regular DAGs in this case, |L j | ≤ d(d-1) j-1 we give bounds on gr j (G,v) as a function of the levels j of the sinks Complex Networks51

52. Main results Theorem (BMP,12+) :If G is a random d-regular DAG, then a.a.s. the following hold. 1)If 2 ≤ j ≤ O(1), then gr j (G,v) = d-2+1/j. 2)If ω is any function tending to infinity with n and ω ≤ j ≤ log d-1 n- ωloglog n, then gr j (G,v) ≤ d-2. 3)If log d-1 n- ωloglog n ≤ j ≤ log d-1 n - 5/2klog 2 log n + log d-1 log n-O(1) for some integer k>0, then d-2-1/k ≤ gr j (G,v) ≤ d-2. Complex Networks52

53. gr j (G,v) is smaller for larger j Theorem (BMP,12+) For a random d-regular DAG G, for s ≥ 4 there is a c onstant C s > 0, such that if j ≥ log d-1 n + C s, then a.a.s. gr j (G,v) ≤ d - 2 - 1/s. proof uses a combinatorial-game theory type argument Complex Networks53

54. Sketch of proof greens protect d-2 vertices on some layers; other layers (every si steps, for i ≥ 0) they protect d-3 greens play greedily: protect vertices adjacent to the sludge ≤1 choice for sludge when the greens protect d-2; at most 2, otherwise greens can move sludge to any vertex in the d-2 layers bad vertex: in-degree at least 2 if there is a bad vertex in the d-2 layers, greens can directs sludge there and sludge loses –greens protect all children Complex Networks54 t = si+1 d-3

55. Sketch of proof, continued sludge wins implies that there are no bad vertices in d-2 layers, and all vertices in the d-3 layers either have in-degree 1 and all but at most one child are sludge- win, or in-degree 2 and all children are sludge-win allows for a cut proceeding inductively from the source to a sink: –in a given d-3 layer, if a vertex has in-degree 1, then we cut away any out-neighbour and all vertices not reachable from the source (after the out-neighbour is removed) if sludge wins, then there is cut which gives a (d-1,d-2)-regular graph the probability that there is such a cut is o(1) Complex Networks55 d-3

56. Power law case fix d, exponent β > 2, and maximum degree M = n α for some α in (0,1) w i = ci -1/β-1 for suitable c and range of i –power law sequence with average degree d ideas: –high degree nodes closer to source, decreasing degree from left to right –greens prevent sludge from moving to the highest degree nodes at each time-step Complex Networks56

57. Theorem (BMP,12+) In a random power law DAG a.a.s. Complex Networks57

58. Contrasting the cases hard to compare d-regular and power law random DAGs, as the number of vertices and average degree are difficult to control consider the first case when there is Cn vertices in the d- regular and power law random DAGs –many high degree vertices in power law case –green number higher than in d-regular case interpretation: in random power law DAGs, more difficult to disrupt the network Complex Networks58

59. Problems and directions: GEO-P Logarithmic Dimension Hypothesis (LDH): the dimension of an OSN is best fit by about log n, where n is the number of users OSN –theoretical evidence GEO-P and MAG ( Leskovec, Kim,12) models –empirical evidence? generalize the GEO-P model to other ranking schemes (such as ranking by age or degree). SCP in GEO-P? Complex Networks59

60. Problems and directions: Seepage other sequences empirical analysis on various hierarchical social networks vertex pursuit in geometric networks –G(n,r), SPA, GEO-P Complex Networks60

61. Complex Networks61 preprints, reprints, contact: search: “Anthony Bonato”

62. Complex Networks62 謝謝