  This paper studies a statistical network model generated by a large number of
randomly sized overlapping communities, where any pair of nodes sharing a
community is linked with probability $q$ via the community. In the special case
with $q=1$ the model reduces to a random intersection graph which is known to
generate high levels of transitivity also in the sparse context. The parameter
$q$ adds a degree of freedom and leads to a parsimonious and analytically
tractable network model with tunable density, transitivity, and degree
fluctuations. We prove that the parameters of this model can be consistently
estimated in the large and sparse limiting regime using moment estimators based
on partially observed densities of links, 2-stars, and triangles.
