Comparing mixing times on sparse random graphs

TL;DR

This paper analyzes the mixing times of simple and nonbacktracking random walks on irregular sparse random graphs, establishing precise cutoff times and demonstrating the faster mixing of nonbacktracking walks.

Contribution

It provides the first precise analysis of worst-case mixing times for irregular graphs and compares the efficiency of simple versus nonbacktracking walks.

Findings

Simple random walk exhibits cutoff at time proportional to log n.

Nonbacktracking walk mixes faster than simple random walk.

The asymptotic mixing time is larger for simple random walk than for nonbacktracking walk.

Abstract

It is natural to expect that nonbacktracking random walk will mix faster than simple random walks, but so far this has only been proved in regular graphs. To analyze typical irregular graphs, let $G$ be a random graph on $n$ vertices with minimum degree 3 and a degree distribution that has exponential tails. We determine the precise worst-case mixing time for simple random walk on $G$, and show that, with high probability, it exhibits cutoff at time $\mathbf{h}^{-1} \log n$, where $\mathbf{h}$ is the asymptotic entropy for simple random walk on a Galton--Watson tree that approximates $G$ locally. (Previously this was only known for typical starting points.) Furthermore, we show that this asymptotic mixing time is strictly larger than the mixing time of nonbacktracking walk, via a delicate comparison of entropies on the Galton-Watson tree.