  We present a Bayesian nonparametric Poisson factorization model for modeling
network data with an unknown and potentially growing number of overlapping
communities. The construction is based on completely random measures and allows
the number of communities to either increase with the number of nodes at a
specified logarithmic or polynomial rate, or be bounded. We develop asymptotics
for the number of nodes and the degree distribution of the network and derive a
Markov chain Monte Carlo algorithm for targeting the exact posterior
distribution for this model. The usefulness of the approach is illustrated on
various real networks.
