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

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

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

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.

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

10 Preferential attachment model Albert-László Barabási Réka Albert

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

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

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

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

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

Azuma-Hoeffding inequality If (X i :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 = 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

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

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

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

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)

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

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

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

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 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