  The `random intersection graph with communities' models networks with
communities, assuming an underlying bipartite structure of groups and
individuals. Each group has its own internal structure described by a (small)
graph, while groups may overlap. The group memberships are generated by a
bipartite configuration model. The model generalizes the classical random
intersection graph model that is included as the special case where each
community is a complete graph (or clique).
  The `random intersection graph with communities' is analytically tractable.
We prove a phase transition in the size of the largest connected component
based on the choice of model parameters. Further, we prove that percolation on
our model produces a graph within the same family, and that percolation also
undergoes a phase transition. Our proofs rely on the connection to the
bipartite configuration model, however, with the arbitrary structure of the
groups, it is not completely straightforward to translate results on the group
structure into results on the graph. Our related results on the bipartite
configuration model are not only instrumental to the study of the random
intersection graph with communities, but are also of independent interest, and
shed light on interesting differences from the unipartite case.
