A threshol for cutoff in two-community random graphs

Anna Ben-Hamou

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 $\alpha \gg \log N$, 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 $\alpha \ll \log N$ or $\alpha \asymp \log N$, then the mixing time is of order $1/\alpha$ and there is no cutoff.