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August 2020 A threshold for cutoff in two-community random graphs
Anna Ben-Hamou
Ann. Appl. Probab. 30(4): 1824-1846 (August 2020). DOI: 10.1214/19-AAP1544

Abstract

In this paper, we are interested in the impact of communities on the mixing behavior of the nonbacktracking 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 nonbacktracking 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.

Citation

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Anna Ben-Hamou. "A threshold for cutoff in two-community random graphs." Ann. Appl. Probab. 30 (4) 1824 - 1846, August 2020. https://doi.org/10.1214/19-AAP1544

Information

Received: 1 September 2018; Revised: 1 June 2019; Published: August 2020
First available in Project Euclid: 4 August 2020

MathSciNet: MR4132638
Digital Object Identifier: 10.1214/19-AAP1544

Subjects:
Primary: 60J10
Secondary: 05C80, 05C81

Rights: Copyright © 2020 Institute of Mathematical Statistics

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Vol.30 • No. 4 • August 2020
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