Discrete Mathematics and its Applications Lecture 6 – PA models Miniconference on the Mathematics of Computation AM8002 Discrete Mathematics and its Applications Lecture 6 – PA models Dr. Anthony Bonato Ryerson University

Key properties of complex networks Large scale. Evolving over time. Power law degree distributions. Small world properties. in this lecture, we consider various models simulating these properties

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

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

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 1 2 3 4 5

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 7.1: A.a.s. the degree of each vertex of G in G(n,p) equals concentration: binomial distribution Theorem 7.2: If p is constant, then a.a.s. diam(G(n,p)) = 2.

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

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

Preferential attachment model Albert-László Barabási Réka Albert

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

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

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.

Properties of the PA model Theorem 7.3 (BRST,01) A.a.s. for all k satisfying 0 ≤ k ≤ t1/15 Theorem 7.4 (Bollobás, Riordan, 04) A.a.s. the diameter of the graph at time t is

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

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.

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.

Power law for expected degree distribution in ACL PA model Theorem 7.5 (ACL,02). 1) 2) For k sufficiently large,