Faster random walks via infrequent steering

TL;DR

This paper introduces a method to accelerate random walks on graphs by infrequent steering, enabling visits to all vertices in near-linear time for bounded degree graphs.

Contribution

It presents a novel approach to steer random walks efficiently, leveraging graph decomposition into small-diameter components.

Findings

Random walks can be significantly sped up with infrequent steering.

Graphs can be decomposed into small-diameter pieces to facilitate faster visits.

Visit all vertices in near-linear time for bounded degree graphs.

Abstract

Random walks on graphs can be slow. To speed them up, imagine that at each step instead of choosing the neighbor at random, there is a small probability $\varepsilon>0$ that we can choose it. We show that in this case, at least for graphs of bounded degree, there is a way to steer the walk so that it visits every vertex in $n^{1+o(1)}$ steps with high probability. The key to this result is a way to decompose arbitrary graphs into small-diameter pieces.