CS 787 : Advanced Algorithms

For this lecture, consider a graph G = (V,E), with n = |V | and m = |E|. Let du denote the degree of vertex u. Recall some of the quantities we were interested in from last time: Definition 28.1.1 The transition matrix is a matrix P such that Puv denotes the probability of moving from u to v: Puv = Pr[random walk moves from u to v given it is at u]. Definition 28.1.2 Stationary Distribution for the graph starting at v, π∗, is the distribution over nodes such that π∗ = π∗P Definition 28.1.3 The hitting time from u to v, huv, is the expected number of steps to get from u to v. Definition 28.1.4 The commute time between u and v, Cuv, is the expected number of steps to get from u to v and then back to u. Definition 28.1.5 The cover time of a graph, C(G), is the maximum over all nodes in G, of the expected number of steps starting at a node and walking to every other node in G. Last time we showed the following: