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

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Key properties of complex networks 1.Large scale. 2.Evolving over time. 3.Power law degree distributions. 4.Small world properties. in the next two lectures, we consider various models simulating these properties 2

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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

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

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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

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6 Degrees and diameter an event A n 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.

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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

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9 G(n,p) is not a model for complex networks degree distribution is binomial low diameter, rich but uniform substructures

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10 Preferential attachment model Albert-László Barabási Réka Albert

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

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12 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 v t+1 to existing nodes forming the graph G t –the edge v t+1 v s is added with probability

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13 Wilensky, U. (2005). NetLogo Preferential Attachment model. Preferential Attachment Model (Barabási, Albert, 99), (Bollobás,Riordan,Spencer,Tusnady,01)

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14 Theorem 5.3 (BRST,01) A.a.s. for all k satisfying 0 ≤ k ≤ t 1/15 Theorem 5.4 (Bollobás, Riordan, 04) A.a.s. the diameter of the graph at time t is Properties of the PA model

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Idea of proof of power law degree distribution 15 1.Derive an asymptotic expression for E(N k,t ) via a recurrence relation. 2.Prove that N k,t concentrates around E(N k,t ). –this is accomplished via martingales or using variance

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Azuma-Hoeffding inequality If (X i :0 ≤ i ≤ t) is a martingale satisfying the c-Lipschitz condition, then for all real λ > 0, 16

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Sketch of proof of (2), when m=1 let A = N k,t and Z i = G i define X i = E[A| Z 1,…, Z i ] It can be shown that (X i ) is a martingale (ie a Doob martingale) a new vertex can affect the degrees of at most two existing nodes, so we have that |X i – X i-1 | ≤ 2 now apply Azuma-Hoeffding inequality with 17

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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, G 0 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 G t+1, with probability p take a vertex-step, and with probability 1-p, an edge-step. 18

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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 N k,t we derive the following result. The case (2) for general k>1 follows by an induction. 19

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Power law for expected degree distribution in ACL PA model Theorem 5.5 (ACL,02). 1) 2) For k sufficiently large, 20

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21 Copying models new nodes copy some of the link structure of an existing node Motivation: 1.web page generation (Kumar et al, 00) 2.mutation in biology (Chung et al, 03)

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22 u v x y N(u)N(u) N(v)N(v)

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Copying model (Kumar et al,00) Parameters: p in (0,1), d > 0 an integer, and a fixed digraph G 0 = H with constant out-degree d Assume G t has out-degree d. At time t+1, an existing vertex, u t, is chosen u.a.r. The vertex u t is called the copying vertex. To form G t+1 a new vertex v t+1 is added. For each of the d-out-neighbours z of u t, add a directed edge (v t+1,z) with probability 1-p, and with probability p add a directed edge (v t+1,z), where z is chosen u.a.r. from G t 23

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24 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

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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

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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 N t,i,d is the expected number of K i,i which are subgraphs of G t. indicates strong community structure in copying model 26

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