Meeting times of Markov chains via singular value decomposition

TL;DR

This paper introduces a matrix perturbation approach using singular value decomposition to derive sharp bounds on the expected meeting times of random walks on large graphs, with applications to Erdős-Rényi graphs.

Contribution

It presents a novel non-asymptotic method linking singular value decomposition to meeting times of Markov chains, providing explicit bounds and formulas.

Findings

Derived a formula for expected meeting time using SVD of the killed generator.

Provided sharp bounds for simple random walks on dense Erdős-Rényi graphs.

Demonstrated the effectiveness of the rank-one approximation approach.

Abstract

We suggest a non-asymptotic matrix perturbation-theoretic approach to get sharp bounds on the expected meeting time of random walks on large (possibly random) graphs. We provide a formula for the expected meeting time in terms of the singular value decomposition of the diagonally killed generator of a pair of independent random walks, which we view as a perturbation of the generator. Employing a rank-one approximation of the diagonally killed generator as the proof of concept, we work out sharp bounds on the expected meeting time of simple random walks on sufficiently dense Erd\H{o}s-R\'enyi random graphs.