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TAMC’12. Vertex-pursuit in heirarchical social networks. Anthony Bonato Ryerson University. 很高 兴来北京. Friendship networks. network of friends (some real, some virtual) form a large web of interconnected links. 6 degrees of separation.
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TAMC’12 Vertex-pursuit in heirarchical social networks Anthony Bonato Ryerson University Complex Networks
很高兴来北京 Complex Networks
Friendship networks • network of friends (some real, some virtual) form a large web of interconnected links Complex Networks
6 degrees of separation • (Stanley Milgram, 67): famous chain letter experiment Complex Networks
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
Complex Networks • web graph, social networks, biological networks, internet networks, … Complex Networks
nodes: web pages edges: links over 1 trillion nodes, with billions of nodes added each day The web graph Complex Networks
nodes: proteins edges: biochemical interactions Biological networks: proteomics Yeast: 2401 nodes 11000 edges Complex Networks
On-line Social Networks (OSNs)Facebook, RenRen, Twitter, LinkedIn,… Complex Networks
Key parameters • degree distribution: • average distance: • clustering coefficient: Complex Networks
Properties of Complex Networks • power law degree distribution (Broder et al, 01) Complex Networks
Power laws in OSNs (Mislove et al,07): Complex Networks
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
Sample data: Flickr, YouTube, LiveJournal, Orkut • (Mislove et al,07): short average distances and high clustering coefficients Complex Networks
Community structure • (Zachary, 72) • (Mason et al, 09) • (Fortunato, 10) • (Li, Peng, 11): • small community • property Complex Networks
(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
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
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
Many different models Complex Networks
“All models are wrong, but some are more useful.” – G.P.E. Box Complex Networks
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
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
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
Simulation with 5000 nodes Complex Networks
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
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
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
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
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
Simulation with 5000 nodes Complex Networks
Simulation with 5000 nodes random geometric GEO-P Complex Networks
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
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
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
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
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
6 Dimensions of Separation Complex Networks
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
Good guys vs bad guys games in graphs bad good Complex Networks
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
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
Example: G S S G G x Complex Networks
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
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
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
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
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
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
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