Statistical Mechanics of Community Detection

TL;DR

This paper presents a statistical mechanics framework for community detection in networks, interpreting communities as ground states of a spin glass model, applicable to weighted and directed networks, with methods for hierarchy, overlap, and significance assessment.

Contribution

It introduces a unified spin glass approach to community detection, encompassing existing quality functions and modularity, and provides efficient algorithms for local optimization and significance testing.

Findings

Community detection as spin glass ground state

Effective local update rules for optimization

Benchmark results for community around a node

Abstract

Starting from a general \textit{ansatz}, we show how community detection can be interpreted as finding the ground state of an infinite range spin glass. Our approach applies to weighted and directed networks alike. It contains the \textit{at hoc} introduced quality function from \cite{ReichardtPRL} and the modularity $Q$ as defined by Newman and Girvan \cite{Girvan03} as special cases. The community structure of the network is interpreted as the spin configuration that minimizes the energy of the spin glass with the spin states being the community indices. We elucidate the properties of the ground state configuration to give a concise definition of communities as cohesive subgroups in networks that is adaptive to the specific class of network under study. Further we show, how hierarchies and overlap in the community structure can be detected. Computationally effective local update rules for optimization procedures to find the ground state are given. We show how the \textit{ansatz} may be used to discover the community around a given node without detecting all communities in the full network and we give benchmarks for the performance of this extension. Finally, we give expectation values for the modularity of random graphs, which can be used in the assessment of statistical significance of community structure.