Small-world of communities: communication and correlation of the meta-network

Given a network and a partition in n communities, we address the issues of 'how communities influence each other' and 'when do two given communities communicate'. We prove that, for a small-world network, a simple superposition principle applies among communities and each community plays the role of a microscopic spin governed by a sort of effective TAP (Thouless, Anderson and Palmer) equation. 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. As for the already studied case n = 1, these equations are exact in the paramagnetic regions (at T = 0 this means below the percolation threshold) and provide effective approximations in the other regions. However, 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 short-cuts among the communities, may increase exponentially fast with n and glassy scenarios with a remarkable number of abrupt jumps take place. Furthermore, as a by-product, a natural and efficient method for detecting the community structure of a generic network emerges from the relative susceptibilities. (letter)