An algebraic-combinatorial framework for finding the average hitting times in graphs with high regularity

TL;DR

This paper introduces an algebraic-combinatorial framework leveraging maximal-entropy random walks and weight-equitable partitions to efficiently compute average hitting times in highly regular finite graphs.

Contribution

It develops a novel method connecting maximal-entropy random walks with weight-equitable partitions, extending existing techniques for calculating hitting times.

Findings

Provides a unified framework for high-regularity graphs

Extends Rao's method for symmetric starting vertices

Enhances computational techniques for average hitting times

Abstract

For any given vertices $u$ and $v$ in a graph, the hitting time of a random walk on a finite graph is the number of steps it takes for a random walk to reach vertex $v$ starting at vertex $u$. The expected value of the hitting time is the average hitting time. In this paper, we present an algebraic-combinatorial method for calculating the average hitting time between vertices of finite graphs exhibiting high regularity, along with its applications to multiple graph classes. Our approach exploits a novel connection between maximal-entropy random walks and weight-equitable partitions, providing a unifying framework that strengthens and extends several known results, including Rao's method [Statistics \& Probability Letters, 2013] for computing the hitting time from a vertex to a neighbor under certain symmetries of the starting vertex.