New Bounds for Edge-Cover by Random Walk

TL;DR

This paper establishes new tight upper bounds on the expected time for a random walk to traverse all edges in a graph and return to the start, with implications for Brownian motion on networks.

Contribution

It introduces novel bounds on edge-cover times for random walks, extending to graphs with arbitrary edge lengths and connecting to Brownian motion.

Findings

Expected traversal time is at most 2m^2 for unidirectional edges.

Expected traversal time is at most 3m^2 if edges are traversed in both directions.

Bounds are tight and applicable to networks with arbitrary edge lengths.

Abstract

We show that the expected time for a random walk on a (multi-)graph $G$ to traverse all $m$ edges of $G$, and return to its starting point, is at most $2m^2$; if each edge must be traversed in both directions, the bound is $3m^2$. Both bounds are tight and may be applied to graphs with arbitrary edge lengths, with implications for Brownian motion on a finite or infinite network of total edge-length $m$.