The Annals of Applied Probability

Mixing times of random walks on dynamic configuration models

Luca Avena, Hakan Güldaş, Remco van der Hofstad, and Frank den Hollander

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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

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

Received: June 2016
Revised: January 2017
First available in Project Euclid: 9 August 2018

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Zentralblatt MATH identifier

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

Random graph random walk mixing time coupling dynamic configuration model


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.

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