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

TL;DR

This paper develops a theoretical framework using TAP equations to analyze communication and correlation among communities in small-world networks, revealing complex behaviors like metastability and providing a method to detect community structures.

Contribution

It introduces a novel application of TAP equations to community interactions in small-world networks, highlighting metastable states and a new community detection method.

Findings

TAP equations effectively describe community interactions in small-world networks.

Metastable states can emerge due to asymmetries and negative shortcuts.

A new method for community detection based on susceptibilities is proposed.

Abstract

Given a network and a partition in n communities, we address the issues ``how communities influence each other'' and ``when two given communities do communicate''. We prove that, for a small-world network, among communities, a simple superposition principle applies and each community plays the role of a microscopic spin governed by a sort of effective TAP (Thouless, Anderson and Palmer) equations. 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 grow exponentially fast with n and glassy scenarios with a remarkable presence of abrupt jumps take place. Furthermore, as a byproduct, from the relative susceptibilities a natural and efficient method to detect the community structure of a generic network emerges.