  How can we approximate sparse graphs and sequences of sparse graphs (with
unbounded average degree)? We consider convergence in the first $k$ moments of
the graph spectrum (equivalent to the numbers of closed $k$-walks)
appropriately normalized. We introduce a simple, easy to sample, random graph
model that captures the limiting spectra of many sequences of interest,
including the sequence of hypercube graphs. The Random Overlapping Communities
(ROC) model is specified by a distribution on pairs $(s,q)$, $s \in
\mathbb{Z}_+, q \in (0,1]$. A graph on $n$ vertices with average degree $d$ is
generated by repeatedly picking pairs $(s,q)$ from the distribution, adding an
Erd\H{o}s-R\'{e}nyi random graph of edge density $q$ on a subset of vertices
chosen by including each vertex with probability $s/n$, and repeating this
process so that the expected degree is $d$. Our proof of convergence to a ROC
random graph is based on the Stieltjes moment condition. We also show that the
model is an effective approximation for individual graphs. For almost all
possible triangle-to-edge and four-cycle-to-edge ratios, there exists a pair
$(s,q)$ such that the ROC model with this single community type produces graphs
with both desired ratios, a property that cannot be achieved by stochastic
block models of bounded description size. Moreover, ROC graphs exhibit an
inverse relationship between degree and clustering coefficient, a
characteristic of many real-world networks.
