A Property of Random Walks on a Cycle Graph

TL;DR

This paper models a hunter and rabbit game on a cycle graph using random walk theory, deriving bounds on capture probability and informing strategy choices based on graph size and rabbit behavior.

Contribution

It formalizes the game with a random walk framework, classifies strategies via Fourier analysis, and provides bounds on capture probabilities depending on parameters and graph size.

Findings

Constant lower bound on capture probability for beta 1

Capture probability order is O(1/ log N) when beta=1

Lower bound is 1/N^{(beta-1)/beta} for beta 1

Abstract

We analyze the Hunter vs Rabbit game on graph, which is a kind of model of communication in an adhoc mobile network. Let $G$ be a cycle graph with $N$ nodes. The hunter can move from a vertex to another vertex on the graph along an edge. The rabbit can move to any vertex on graph at once. We formalized the game using the random walk framework. The strategy of the rabbit is formalized using a one dimensional random walk over $\mathbb{Z}$. We classify strategies using the order $O(k^{-\beta-1})$ of their Fourier transformation. We investigate lower bounds and upper bounds of a probability that the hunter catches the rabbit. We found a constant lower bound if $\beta \in (0,1)$. That is there is not depend on the size $N$ of the graph. We show the order is equivalent to $O(1/\log N)$ if $\beta=1$ and a lower bound is $1/N^{(\beta-1)/\beta}$ if $\beta \in (1,2]$. Those results assist to choose the parameter $\beta$ 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 we also show computing simulation results.