A property of random walks on a cycle graph

Abstract

We analyze the Hunter vs. Rabbit game on a graph, which is a model of communication in adhoc mobile networks. Let G be a cycle graph with N nodes. The hunter can move from a vertex to a vertex along an edge. The rabbit can jump from any vertex to any vertex on the graph. We formalize the game using the random walk framework. The strategy of the rabbit is formalized using a one dimensional random walk over Z. We classify strategies using the order O(k −β−1 ) of their Fourier transformation. We investigate lower bounds and upper bounds of the probability that the hunter catches the rabbit. We found a constant lower bound if β ∈ (0, 1) which does not depend on the size N of the graph. We show the order is equivalent to O(1/ log N) if β = 1 and a lower bound is 1/N (β−1)/β if β ∈ (1, 2]. These results help us to choose the parameter β of a rabbit strategy according to the size N of the given graph. We introduce a formalization of strategies using a random walk, theoretical estimation of bounds of a probability that the hunter catches the rabbit, and also show computing simulation results.