Organization of modular networks

TL;DR

This paper analyzes the structure of modular networks, developing an analytical approach to understand their global organization, especially how interlinks influence connectivity and the transition to a united network.

Contribution

It introduces an analytical method to characterize the global structure of heterogeneous modular networks and describes how interlinks affect network unification.

Findings

Small numbers of interlinks can unify separate networks.

Intervertex distances decrease as interlinks increase.

Networks become fully connected with finite interlink fractions.

Abstract

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 an 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 the interlinks is any finite fraction of the total number of connections, then the intervertex distance distribution approaches a delta-function peaked form, and so the network is united.