A threshold for cutoff in two-community random graphs

TL;DR

This paper investigates how a two-community structure in sparse random graphs affects the mixing time of non-backtracking random walks, revealing a phase transition at a critical community connectivity threshold.

Contribution

It establishes a precise cutoff threshold for the mixing behavior of non-backtracking random walks in two-community sparse graphs, depending on the inter-community edge fraction.

Findings

For ig rac{1}{\u221a{ ext{log} N}}, the walk exhibits cutoff with a Gaussian tail profile.

When ig rac{1}{\u221a{ ext{log} N}} or smaller, the mixing time scales as 1/ig rac{1}{\u221a{ ext{log} N}}, with no cutoff.

The cutoff window broadens with increased community bottleneck strength.

Abstract

In this paper, we are interested in the impact of communities on the mixing behavior of the non-backtracking random walk. We consider sequences of sparse random graphs of size $N$ generated according to a variant of the classical configuration model which incorporates a two-community structure. The strength of the bottleneck is measured by a parameter $\alpha$ which roughly corresponds to the fraction of edges that go from one community to the other. We show that if $\alpha\gg \frac{1}{\log N}$, then the non-backtracking random walk exhibits cutoff at the same time as in the one-community case, but with a larger cutoff window, and that the distance profile inside this window converges to the Gaussian tail function. On the other hand, if $\alpha \ll \frac{1}{\log N}$ or $\alpha \asymp \frac{1}{\log N}$, then the mixing time is of order $1/\alpha$ and there is no cutoff.