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Week 5 - Models of Complex Networks I

AM8002 Fall 2014. Week 5 - Models of Complex Networks I. Dr. Anthony Bonato Ryerson University. Key properties of complex networks. Large scale. Evolving over time. Power law degree distributions. Small world properties.

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Week 5 - Models of Complex Networks I

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  1. AM8002 Fall 2014 Week 5 - Models of Complex Networks I Dr. Anthony Bonato Ryerson University

  2. Key properties of complex networks • Large scale. • Evolving over time. • Power law degree distributions. • Small world properties. • in the next two lectures, we consider various models simulating these properties

  3. 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

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

  5. G(n,p) random graph model(Erdős, Rényi, 63) • p = p(n) a real number in (0,1), n a positive integer • G(n,p): probability space on graphs with nodes {1,…,n}, two nodes joined independently and with probability p 4 1 2 3 5

  6. Degrees and diameter • an event An happens asymptotically almost surely (a.a.s.) in G(n,p) if it holds there with probability tending to 1 as n→∞ Theorem 5.1: A.a.s. the degree of each vertex of G in G(n,p) equals • concentration: binomial distribution Theorem 5.2: If p is constant, then a.a.s diam(G(n,p)) = 2.

  7. Aside: evolution of G(n,p) • think of G(n,p) as evolving from a co-clique to clique as p increases from 0 to 1 • at p=1/n, Erdős and Rényi observed something interesting happens a.a.s.: • with p = c/n, with c < 1, the graph is disconnected with all components trees, the largest of order Θ(log(n)) • as p = c/n, with c > 1, the graph becomes connected with a giant component of order Θ(n) • Erdős and Rényi called this the double jump • physicists call it the phase transition: it is similar to phenomena like freezing or boiling

  8. degree distribution is binomial low diameter, rich but uniform substructures G(n,p) is not a model forcomplex networks

  9. Preferential attachment model Albert-László Barabási Réka Albert

  10. Preferential attachment • say there are n nodes xi in G, and we add in a new node z • z is joined to the xi by preferential attachment if the probability zxi is an edge is proportional to degrees: • the larger deg(xi), the higher the probability that z is joined to xi

  11. Preferential attachment (PA) model(Barabási, Albert, 99), (Bollobás,Riordan,Spencer,Tusnady,01) • parameter: m a positive integer • at time 0, add a single edge • at time t+1, add m edges from a new node vt+1 to existing nodes forming the graph Gt • the edge vt+1 vs is added with probability

  12. Preferential Attachment Model(Barabási, Albert, 99), (Bollobás,Riordan,Spencer,Tusnady,01) Wilensky, U. (2005). NetLogo Preferential Attachment model. http://ccl.northwestern.edu/netlogo/models/PreferentialAttachment.

  13. Properties of the PA model • Theorem 5.3 (BRST,01) A.a.s. for all k satisfying 0 ≤ k ≤ t1/15 • Theorem 5.4 (Bollobás, Riordan, 04) A.a.s. the diameter of the graph at time t is

  14. Idea of proof of power law degree distribution • Derive an asymptotic expression for E(Nk,t) via a recurrence relation. • Prove that Nk,t concentrates around E(Nk,t). • this is accomplished via martingales or using variance

  15. Azuma-Hoeffding inequality If (Xi:0 ≤ i ≤ t) is a martingale satisfying the c-Lipschitz condition, then for all real λ > 0,

  16. Sketch of proof of (2), when m=1 • let A = Nk,t and Zi = Gi • define Xi = E[A| Z1,…, Zi] • It can be shown that (Xi) is a martingale (ie a Doob martingale) • a new vertex can affect the degrees of at most two existing nodes, so we have that |Xi – Xi-1| ≤ 2 • now apply Azuma-Hoeffdinginequality with

  17. ACL PA model • (Aeillo,Chung,Lu,2002) introduced a preferential attachment model where the parameters allow exponents to range over (2,∞) • Fix p in (0,1). This is the sole parameter of the model. • At t=0, G0 is a single vertex with a loop. • A vertex-step adds a new vertex v and an edge uv, where u is chosen from existing vertices by preferential attachment. • An edge-step adds an edge uv, where both endpoints are chosen by preferential attachment. • To form Gt+1, with probability p take a vertex-step, and with probability 1-p, an edge-step.

  18. ACL PA, continued • note that the number of vertices is a random variable; but it concentrates on 1+pt. • to give a flavour of estimating the expectations of random variables Nk,t we derive the following result. The case (2) for general k>1 follows by an induction.

  19. Power law for expected degree distribution in ACL PA model Theorem 5.5 (ACL,02). 1) 2) For k sufficiently large,

  20. Copying models • new nodes copy some of the link structure of an existing node Motivation: • web page generation (Kumar et al, 00) • mutation in biology (Chung et al, 03)

  21. N(v) v N(u) y u x

  22. Copying model (Kumar et al,00) • Parameters: p in (0,1), d > 0 an integer, and a fixed digraph G0 = H with constant out-degree d • Assume Gt has out-degree d. • At time t+1, an existing vertex, ut, is chosen u.a.r. The vertex ut is called the copying vertex. • To form Gt+1a new vertex vt+1 is added. For each of the d-out-neighbours z of ut, add a directed edge (vt+1,z) with probability 1-p, and with probability p add a directed edge (vt+1,z), where z is chosen u.a.r. from Gt

  23. Properties of the copying model • power laws: • Kumar et al: exponent in interval (2,∞) • Chung, Lu: (1,2) • bipartite subgraphs: • Kumar et al: larger expected number of bicliques than in PA models • simplified model of community structure

  24. Properties of the copying model Theorem 5.6 (Kumar et al, 00) If k > 0, then the copying model with parameter p satisfies a.a.s. In particular, the in-degree distribution follows a power law with exponent (2-p)/(1-p)

  25. Properties of the copying model Theorem 5.7 (Kumar et al, 00) A.a.s. with parameter d >0 and for i ≤ log t, where Nt,i,d is the expected number of Ki,i which are subgraphs of Gt. • indicates strong community structure in copying model

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