Random walks on dynamic configuration models: a trichotomy

TL;DR

This paper studies the mixing times of a random walk on a dynamic configuration model graph, revealing a trichotomy in behavior depending on the rate of edge rewiring, with different cutoff phenomena.

Contribution

It extends previous work by analyzing the mixing time behavior for different regimes of edge rewiring rate, uncovering a new cutoff phenomenon in the dynamic setting.

Findings

Mixing time is of order log n when the rewiring rate parameter is finite.

A one-sided cutoff occurs for intermediate rewiring rates, a rare phenomenon.

A two-sided cutoff occurs when the rewiring rate tends to zero.

Abstract

We consider a dynamic random graph on $n$ vertices that is obtained by starting from a random graph generated according to the configuration model with a prescribed degree sequence and at each unit of time randomly rewiring a fraction $\alpha_n$ of the edges. We are interested in the mixing time of a random walk without backtracking on this dynamic random graph in the limit as $n\to\infty$, when $\alpha_n$ is chosen such that $\lim_{n\to\infty} \alpha_n (\log n)^2 = \beta \in [0,\infty]$. In [1] we found that, under mild regularity conditions on the degree sequence, the mixing time is of order $1/\sqrt{\alpha_n}$ when $\beta=\infty$. In the present paper we investigate what happens when $\beta \in [0,\infty)$. It turns out that the mixing time is of order $\log n$, with the scaled mixing time exhibiting a one-sided cutoff when $\beta \in (0,\infty)$ and a two-sided cutoff when $\beta=0$. The occurrence of a one-sided cutoff is a rare phenomenon. In our setting it comes from a competition between the time scales of mixing on the static graph, as identified by Ben-Hamou and Salez [4], and the regeneration time of first stepping across a rewired edge.