  We consider a variant of the configuration model with an embedded community
structure and study the mixing properties of a simple random walk on it. Every
vertex has an internal $\mathrm{deg}^{\text{int}}\geq 3$ and an outgoing
$\mathrm{deg}^{\text{out}}$ number of half-edges. Given a stochastic matrix
$Q$, we pick a random perfect matching of the half-edges subject to the
constraint that each vertex $v$ has $\mathrm{deg}^{\text{int}}(v)$ neighbours
inside its community and the proportion of outgoing half-edges from community
$i$ matched to a half-edge from community $j$ is $Q(i,j)$. Assuming the number
of communities is constant and they all have comparable sizes, we prove the
following dichotomy: simple random walk on the resulting graph exhibits cutoff
if and only if the product of the Cheeger constant of $Q$ times $\log n$ (where
$n$ is the number of vertices) diverges.
  In [4], Ben-Hamou established a dichotomy for cutoff for a non-backtracking
random walk on a similar random graph model with 2 communities. We prove the
same characterisation of cutoff holds for simple random walk.
