Modelling and Searching Networks Lecture 5 – Random graphs Miniconference on the Mathematics of Computation MTH 707 Modelling and Searching Networks Lecture 5 – Random graphs Dr. Anthony Bonato Ryerson University

Random graphs Paul Erdős Alfred Rényi Complex Networks

Complex Networks

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 Complex Networks

Formal definition n a positive integer p a real number in [0,1] G(n,p) is a probability space on labelled graphs with vertex set V = [n] = {1,2,…,n} such that NB: p can be a function of n today, p is a constant

Properties of G(n,p) consider some graph G in G(n,p) the graph G could be any n-vertex graph, so not much can be said about G with certainty some properties of G, however, are likely to hold we are interested in properties that occur with high probability when n is large

A.a.s. 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 6.1. A.a.s. G in G(n,p) is diameter 2. just say: A.a.s. G(n,p) has diameter 2.

First moment method in G(n,p), all graph parameters: |E(G)|, γ(G), ω(G), … become random variables we focus on computing the averages of these parameters or expectation

Exercise 6.1 Calculate the expected number of edges in G(n,p). use of expectation when studying random graphs is sometimes referred to as the first moment method

Degrees and diameter Theorem 6.2: A.a.s. the degree of each vertex of G in G(n,p) equals concentration: binomial distribution

Markov’s inequality Theorem 6.3 (Markov’s inequality) For any non-negative random variable X and t > 0, we have that

Chernoff bound Theorem 6.4 (Chernoff bound) Let X be a binomially distributed random variable on G(n,p) with E[x] = np. Then for ε ≤ 3/2 we have that

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 Complex Networks

Complex Networks