Non-backtracking random walks and a weighted Ihara's theorem

TL;DR

This paper introduces a weighted version of Ihara's Theorem to analyze the spectral properties and mixing rates of non-backtracking random walks on regular and biregular graphs, showing they often mix faster than standard walks.

Contribution

It presents a weighted Ihara's Theorem relating non-backtracking and usual random walk matrices, enabling spectral analysis of non-backtracking walks on specific graph classes.

Findings

Non-backtracking walks often have faster mixing rates than usual walks.

Derived the spectrum of non-backtracking transition matrices for regular and biregular graphs.

Established a weighted Ihara's Theorem linking non-backtracking and standard transition matrices.

Abstract

We study the mixing rate of non-backtracking random walks on graphs by looking at non-backtracking walks as walks on the directed edges of a graph. A result known as Ihara's Theorem relates the adjacency matrix of a graph to a matrix related to non-backtracking walks on the directed edges. We prove a weighted version of Ihara's Theorem which relates the transition probability matrix of a non-backtracking walk to the transition matrix for the usual random walk. This allows us to determine the spectrum of the transition probability matrix of a non-backtracking random walk in the case of regular graphs and biregular graphs. As a corollary, we obtain a result of Alon et. al. that in most cases, a non-backtracking random walk on a regular graph has a faster mixing rate than the usual random walk. In addition, we obtain an analogous result for biregular graphs.