Approximating sparse graphs: The random overlapping communities model

How can we approximate sparse graphs and sequences of (with unbounded average degree)? We consider convergence in the first k moments graph spectrum (equivalent to numbers closed -walks) appropriately normalized. introduce a simple random model that captures limiting spectra many interest, including sequence hypercube graphs. The overlapping communities (ROC) is specified by distribution on pairs ( s , q ) ∈ ℤ + 0 1 ] . A n vertices with degree d generated repeatedly picking from distribution, adding an Erdős–Rényi edge density subset chosen each vertex probability / repeating this process so expected also show ROC exhibit inverse relationship between clustering coefficient, characteristic real-world networks.