The Annals of Applied Probability

Mixing times of random walks on dynamic configuration models

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 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.

Article information

Source
Ann. Appl. Probab., Volume 28, Number 4 (2018), 1977-2002.

Dates
Revised: January 2017
First available in Project Euclid: 9 August 2018

https://projecteuclid.org/euclid.aoap/1533780265

Digital Object Identifier
doi:10.1214/17-AAP1289

Mathematical Reviews number (MathSciNet)
MR3843821

Zentralblatt MATH identifier
06974743

Subjects
Primary: 60C05: Combinatorial probability
Secondary: 37A25: Ergodicity, mixing, rates of mixing 05C81: Random walks on graphs

Citation

Avena, Luca; Güldaş, Hakan; van der Hofstad, Remco; den Hollander, Frank. Mixing times of random walks on dynamic configuration models. Ann. Appl. Probab. 28 (2018), no. 4, 1977--2002. doi:10.1214/17-AAP1289. https://projecteuclid.org/euclid.aoap/1533780265

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