Discrete Mathematics and its Applications Lecture 3 – ILT model Miniconference on the Mathematics of Computation AM8002 Discrete Mathematics and its Applications Lecture 3 – ILT model Dr. Anthony Bonato Ryerson University

Key properties of complex networks Large scale. Evolving over time. Power law degree distributions. Small world properties. Other properties are also important: densification power law, shrinking distances,…

Transitivity Anthony Bonato

Iterated Local Transitivity (ILT) model (Bonato, Hadi, Horn, Prałat, Wang, 08) key paradigm is transitivity: friends of friends are more likely friends nodes often only have local influence evolves over time, but retains memory of initial graph

ILT model start with a graph of order n to form the graph Gt+1 for each node x from time t, add a node x’, the clone of x, so that xx’ is an edge, and x’ is joined to each node joined to x order of Gt is n2t

G0 = C4 Anthony Bonato

Exercise 4.1 Why would ILT be an appropriate model for a social network such as Facebook? 4.2 What are some drawbacks of the ILT model?

“All models are wrong, but some are more useful.” - George Box

Properties of ILT model average degree increasing to with time average distance bounded by constant and converging, and in many cases decreasing with time; diameter does not change clustering coefficient higher than in a random generated graph with same average degree bad expansion: small gaps between 1st and 2nd eigenvalues in adjacency and normalized Laplacian matrices of Gt

et ≈ nta, where a = log(3)/log(2). Densification nt = order of Gt, et = size of Gt Lemma 4.1: For t > 0, nt = 2tn0, et = 3t(e0+n0) - nt. → densification power law: et ≈ nta, where a = log(3)/log(2).

Exercise 4.3 Prove Lemma 4.1. (Hint: use induction and draw a picture explaining how new edges are added.)

Average distance Theorem 4.2: If t > 0, then 2 average distance bounded by a constant, and converges; for many initial graphs (large cycles) it decreases diameter does not change from time 0

Clustering Coefficient Theorem 4.3: If t > 0, then c(Gt) = ntlog(7/8)+o(1). higher clustering than in a random graph G(nt,p) with same order and average degree as Gt, which satisfies c(G(nt,p)) = ntlog(3/4)+o(1)

…Degree distribution generate power law graphs from ILT? ILT model gives a binomial-type distribution

Exercise 4.7 Let degt(z) be the degree of z at time t. a) Show that if x is in Gt, then degt+1(x) = 2degt(x)+1. b) Show that degt+1(x’) = degt(x) +1. 4.8 Assuming you start with a 3-clique, find the degrees of all vertices after two additional time-steps of the ILT model (Hint: you don’t have to draw this graph!).