  We propose a novel statistical model for sparse networks with overlapping
community structure. The model is based on representing the graph as an
exchangeable point process, and naturally generalizes existing probabilistic
models with overlapping block-structure to the sparse regime. Our construction
builds on vectors of completely random measures, and has interpretable
parameters, each node being assigned a vector representing its level of
affiliation to some latent communities. We develop methods for simulating this
class of random graphs, as well as to perform posterior inference. We show that
the proposed approach can recover interpretable structure from two real-world
networks and can handle graphs with thousands of nodes and tens of thousands of
edges.
