The hitting and cover times of random walks on finite graphs using local degree information

Abstract

Standard random walks on finite graphs select the vertex visited next to the adjacent vertices at random with the same probability. Despite not using any global topological information, they guarantee O ( n 3 ) hitting and cover times for any graph, where n is the order of the graph. Motivated by network protocol applications, this paper investigates the impact of local topological information on designing “better” random walks. We first show that (a) for any transition probability matrix, the hitting (and hence the cover) time of a path graph is Ω ( n 2 ) . We next investigate for any graph G = ( V , E ) a transition probability matrix P = ( p ( u , v ) ) u , v ∈ V defined by p ( u , v ) = { deg − 1 / 2 ( v ) ∑ w ∈ N ( u ) deg − 1 / 2 ( w ) if v ∈ N ( u ) , 0 otherwise , where N ( u ) and deg ( u ) are respectively the set of adjacent vertices of u and the u ’s degree. Random walks obeying this transition probability matrix are shown to guarantee the following: For any graph, (b) the hitting time is O ( n 2 ) , and (c) the cover time is O ( n 2 log n ) . Facts (a) and (b) show that the degree information on the adjacent vertices is powerful enough for random walks to achieve the optimum hitting time.