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