  A wide variety of complex networks (social, biological, information etc.)
exhibit local clustering with substantial variation in the clustering
coefficient (the probability of neighbors being connected). Existing models of
large graphs capture power law degree distributions (Barab\'asi-Albert) and
small-world properties (Watts-Strogatz), but only limited clustering behavior.
We introduce a generalization of the classical Erd\H{o}s-R\'enyi model of
random graphs which provably achieves a wide range of desired clustering
coefficient, triangle-to-edge and four-cycle-to-edge ratios for any given graph
size and edge density. Rather than choosing edges independently at random, in
the Random Overlapping Communities model, a graph is generated by choosing a
set of random, relatively dense subgraphs ("communities"). We discuss the
explanatory power of the model and some of its consequences.
