Multiple Random Walks on Graphs: Mixing Few to Cover Many

TL;DR

The paper develops a comprehensive theory for the cover time of $k$ parallel random walks on graphs, bridging stationary-start analyses with worst-case start scenarios via a novel partial mixing time notion. It proves three tight stationary bounds and a universal lower bound, and introduces a geometric-reset coupling to derive lower bounds for stationary and worst-case settings. The central contribution is a min–max framework that relates $t_{\mathsf{cov}}^{(k)}$ to both $t_{\mathsf{mix}}^{(\tilde{k},k)}$ and $t_{\mathsf{cov}}^{(\tilde{k})}(\pi)$, enabling tight results across cycles, grids, trees, tori, hypercubes, expanders, and PA graphs. This framework clarifies when speed-ups from parallelism are achieved and highlights how graph structure governs the optimal balance between mixing fewer walks and covering efficiently, with potential implications for distributed sampling and network exploration.

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, Koucký, 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 $Ω((n/k) \log n)$ 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.

Theorems & Definitions (93)

• Theorem 3.1
• Theorem 3.2
• Corollary 3.3
• proof
• Lemma 3.4
• Lemma 3.5
• Theorem 3.6
• Definition 3.7: The Geometric Reset Graph $\widehat{G}(x)$
• Lemma 3.8
• Lemma 3.9