Fast Random Walks on Finite Graphs and Graph Topological Information

A random walk on a graph is a process in which a particle on a vertex repeatedly moves to its adjacent vertex according to transition probability, which is given in advance. The behavior of random walks depend on its transition probability, and the "speed'' of random walks also can be measured from several viewpoints. Among the several measures, the hitting time and the cover time are two popular ones and often used for evaluation. In this paper, we consider the speed of random walks from the viewpoint of topological information of graphs and its use. For example, it is known that a simple random walk, in which a particle moves to its adjacent vertex uniformly at random, visits all the vertices in O(n3) expected steps (which is the cover time), while a random walk utilizing all the topological information on a graph can visit all the vertices in O(n2) expected steps, where n is the number of vertices. In this paper, we briefly survey work focusing on the relationship between the speed of random walks on a graph and its usage of topological information.