Multiple Random Walks on Graphs: Mixing Few to Cover Many

TL;DR

This paper investigates the efficiency of multiple random walks on graphs, providing new bounds and characterizations for cover times, especially in worst-case scenarios, and introduces a novel partial mixing time concept.

Contribution

It offers improved bounds on stationary cover times, characterizes worst-case cover times using a new partial mixing time, and applies these results to fundamental network types.

Findings

Unconditional lower bound of ((n/k) \, log n) on stationary cover time.

Determined stationary cover times for key network classes up to constant factors.

Introduced the partial mixing time to analyze worst-case cover times.

Abstract

Random walks on graphs are an essential primitive for many randomised algorithms and stochastic processes. It is natural to ask how much can be gained by running $k$ multiple random walks independently and in parallel. Although the cover time of multiple walks has been investigated for many natural networks, the problem of finding a general characterisation of multiple cover times for worst-case start vertices (posed by Alon, Avin, Kouck\'y, Kozma, Lotker, and Tuttle~in 2008) remains an open problem. First, we improve and tighten various bounds on the stationary cover time when $k$ random walks start from vertices sampled from the stationary distribution. For example, we prove an unconditional lower bound of $\Omega((n/k) \log n)$ on the stationary cover time, holding for any $n$-vertex graph $G$ and any $1 \leq k =o(n\log n )$. Secondly, we establish the stationary cover times of multiple walks on several fundamental networks up to constant factors. Thirdly, we present a framework characterising worst-case cover times in terms of stationary cover times and a novel, relaxed notion of mixing time for multiple walks called the partial mixing time. Roughly speaking, the partial mixing time only requires a specific portion of all random walks to be mixed. Using these new concepts, we can establish (or recover) the worst-case cover times for many networks including expanders, preferential attachment graphs, grids, binary trees and hypercubes.