Communication and correlation among communities

Given a network and a partition in communities, we consider the issues “how communities influence each other” and “when two given communities do communicate.” Specifically, we address these questions in the context of small-world networks, where an arbitrary quenched graph is given and long-range connections are randomly added. We prove that, among the communities, a superposition principle applies and gives rise to a natural generalization of the effective field theory already presented by M. Ostilli and J. F. F. Mendes Phys. Rev. E 78 031102 (2008) (n=1), which here (n>1) consists in a sort of effective TAP (Thouless, Anderson, and Palmer) equations in which each community plays the role of a microscopic spin. The relative susceptibilities derived from these equations calculated at finite or zero temperature, where the method provides an effective percolation theory, give us the answers to the above issues. Unlike the case n=1, asymmetries among the communities may lead, via the TAP-like structure of the equations, to many metastable states whose number, in the case of negative shortcuts among the communities, may grow exponentially fast with n. As examples we consider the n Viana-Bray communities model and the n one-dimensional small-world communities model. Despite being the simplest ones, the relevance of these models in network theory, as, e.g., in social networks, is crucial and no analytic solution were known until now. Connections between percolation and the fractal dimension of a network are also discussed. Finally, as an inverse problem, we show how, from the relative susceptibilities, a natural and efficient method to detect the community structure of a generic network arises.