Characterization of expansion-related properties of modular graphs

A fundamental organizing principle of real-world complex networked systems is modularity, where networks have interactions at different levels. In this paper we consider a modular graph G having modules with arbitrary intraconnections and random interconnections between activated vertices in different modules. The vertices in different modules are activated with probability r and linked by an interconnecting edge with probability p independently. We present results regarding the Cheeger constant, robustness, algebraic connectivity as well as the smallest eigenvalue for the Dirichlet Laplacian matrix of G with high probability. Our results suggest that r=(lnn)/n is a potential scaling for the recently observed external field-like phenomena of modular networks in statistical mechanics.