Random Walks on Random Graphs

The aim of this article is to discuss some of the notions and applications of random walks on finite graphs, especially as they apply to random graphs. In this section we give some basic definitions, in Section 2 we review applications of random walks in computer science, and in Section 3 we focus on walks in random graphs. Given a graphG = (V,E), let dG(v) denote the degree of vertex v for all v ∈ V . The random walk Wv = (Wv(t), t = 0, 1, . . .) is defined as follows: Wv(0) = v and given x = Wv(t), Wv(t+ 1) is a randomly chosen neighbour of x. When one thinks of a random walk, one often thinks of Polya’s Classical result for a walk on the d-dimensional lattice Z, d ≥ 1. In this graph two vertices x = (x1, x2, . . . , xd) and y = (y1, y2, . . . , yd) are adjacent iff there is an index i such that (i) xj = yj for j = i and (ii) |xi − yi| = 1. Polya [33] showed that if d ≤ 2 then a walk starting at the origin returns to the origin with probability 1 and that if d ≥ 3 then it returns with probability p(d) < 1. See also Doyle and Snell [22]. A random walk on a graph G defines a Markov chain on the vertices V . If G is a finite, connected and non-bipartite graph, then this chain has a stationary distribution π given by πv = dG(v)/(2|E|). Thus if P (t) v (w) = Pr(Wv(t) = w), then limt→∞ P (t) v (w) = πw, independent of the starting vertex v. In this paper we only consider finite graphs, and we will focus on two aspects of a random walk: The Mixing Time and the Cover Time.