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
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
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