Mixing times of random walks on dynamic configuration models

TL;DR

This paper studies how the mixing time of a random walk on a dynamic configuration model graph scales with graph rewiring speed, revealing a square-root logarithmic growth under certain fast dynamics conditions.

Contribution

It introduces a new analysis of mixing times on dynamic graphs, showing how rapid edge rewiring affects convergence to equilibrium.

Findings

Mixing time grows like the square root of a logarithm of inverse epsilon.

Fast graph dynamics (alpha_n) significantly reduce mixing time.

Results hold under mild conditions on degree sequences ensuring local tree-likeness.

Abstract

The mixing time of a random walk, with or without backtracking, on a random graph generated according to the configuration model on $n$ vertices, is known to be of order $\log n$. In this paper we investigate what happens when the random graph becomes {\em dynamic}, namely, at each unit of time a fraction $\alpha_n$ of the edges is randomly rewired. Under mild conditions on the degree sequence, guaranteeing that the graph is locally tree-like, we show that for every $\varepsilon\in(0,1)$ the $\varepsilon$-mixing time of random walk without backtracking grows like $\sqrt{2\log(1/\varepsilon)/\log(1/(1-\alpha_n))}$ as $n \to \infty$, provided that $\lim_{n\to\infty} \alpha_n(\log n)^2=\infty$. The latter condition corresponds to a regime of fast enough graph dynamics. Our proof is based on a randomised stopping time argument, in combination with coupling techniques and combinatorial estimates. The stopping time of interest is the first time that the walk moves along an edge that was rewired before, which turns out to be close to a strong stationary time.