Approximating sparse graphs: The random overlapping communities model

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 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 is an element of DOUBLE-STRUCK CAPITAL Z+,q is an element of(0,1]. A graph on n vertices with average degree d is generated by repeatedly picking pairs (s,q) from the distribution, adding an Erdos-Renyi 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. We also show that ROC graphs exhibit an inverse relationship between degree and clustering coefficient, a characteristic of many real-world networks.