First Hitting times of Simple Random Walks on Graphs with Congestion Points

We derive the explicit formulas of the probability generating functions of the first hitting times of simple random walks on graphs with congestion points using group representations. 1. Introduction. Random walk on a graph is a Markov chain whose state space is the vertex set of the graph and whose transition from a given vertex to an adjacent vertex along an edge is defined according to some probability distribution. The probability distribution might depend on vertices, and the case of the uniform distribution over incident edges is called a simple random walk. Many researches have been done on various aspects of random walks such as transience or recurrence, asymptotic behavior of transition probabilities , convergence rates to its stationary distributions, and convergence to a boundary and harmonic functions [2, 6]. Random walks can describe the structure of graphs, groups, and related objects and the structure of computer networks or electric networks [3, 8]. It is quite useful to devise probabilistic algorithms of random walks on graphs which reflect combinatorial problems when deterministic methods to analyze them are known to be difficult [10]. It is well known that random walks play a crucial role in the design of randomized algorithms (off-or online) [4, 11, 15]. The first hitting time (also called the first passage time) is the time taken to reach a vertex for the first time starting from another vertex. This is one of the classical problems on Markov chains, and one can investigate the probabilistic properties of the first hitting time, the stopping time, or transition probabilities [1, 12]. The expected hitting times for random walks on graphs have been computed using the relation between electrical networks and random walks [9, 13, 14] and for random walks on finite groups using group representations [5]. The group representation approach has been quite powerful in measuring the convergence rate of random walks to its stationary distributions [5, 7]. In this paper, we apply group representation technique for a simple random walk on finite graphs with cutvertices (or congestion points), which can be decomposed into several finite groups. Finite graphs with congestion points may