Random intersection graphs with communities

Random intersection graphs model networks with communities, assuming an underlying bipartite structure of groups and individuals, where these may overlap. Group memberships are generated through the configuration model. Conditionally on group memberships, classical random graph is obtained by connecting individuals when they together in at least one group. We generalize this definition, allowing for arbitrary community structures within groups. In our new model, might overlap have their own internal described a graph, setting corresponding to being complete graphs. Our turns out be tractable. analyze overlapping derive asymptotic degree distribution local clustering coefficient. These proofs rely weak convergence, which also implies that subgraph counts converge. further exploit connection we prove interesting its right.