How can we approximate sparse graphs and sequences of sparse graphs (with average degree unbounded and $o(n)$)? 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 查看全文>>