Organization of modular networks

We examine the global organization of heterogeneous equilibrium networks consisting of a number of well-distinguished interconnected parts—“communities” or modules. We develop an analytical approach allowing us to obtain the statistics of connected components and the intervertex distance distribution in these modular networks, and to describe their global organization and structure. In particular, we study the evolution of the intervertex distance distribution with an increasing number of interlinks connecting two infinitely large uncorrelated networks. We demonstrate that even a relatively small number of shortcuts unite the networks into one. In more precise terms, if the number of interlinks is any finite fraction of the total number of connections, then the intervertex distance distribution approaches a δ-function peaked form, and so the network is united.