Speeding up random walk mixing by starting from a uniform vertex

TL;DR

This paper introduces a framework to accelerate the mixing of random walks on graphs with bottlenecks by starting from a uniform vertex, significantly reducing mixing times in certain random graph models.

Contribution

The paper presents a general method to achieve logarithmic average mixing time on graphs with bottlenecks, improving upon previous quadratic bounds in specific random graph families.

Findings

Logarithmic average mixing time for perturbed graphs of bounded degeneracy.

Logarithmic average mixing time for supercritically percolated expander graphs.

Alternative proof for logarithmic mixing time of the giant component in supercritical Erdős-Rényi graphs.

Abstract

The theory of rapid mixing random walks plays a fundamental role in the study of modern randomised algorithms. Usually, the mixing time is measured with respect to the worst initial position. It is well known that the presence of bottlenecks in a graph hampers mixing and, in particular, starting inside a small bottleneck significantly slows down the diffusion of the walk in the first steps of the process. The average mixing time is defined to be the mixing time starting at a uniformly random vertex and hence is not sensitive to the slow diffusion caused by these bottlenecks. In this paper we provide a general framework to show logarithmic average mixing time for random walks on graphs with small bottlenecks. The framework is especially effective on certain families of random graphs with heterogeneous properties. We demonstrate its applicability on two random models for which the mixing time was known to be of order $(\log n)^2$, speeding up the mixing to order $\log n$. First, in the context of smoothed analysis on connected graphs, we show logarithmic average mixing time for randomly perturbed graphs of bounded degeneracy. A particular instance is the Newman-Watts small-world model. Second, we show logarithmic average mixing time for supercritically percolated expander graphs. When the host graph is complete, this application gives an alternative proof that the average mixing time of the giant component in the supercritical Erd\H{o}s-R\'enyi graph is logarithmic.