Multiple phases in modularity-based community detection

TL;DR

This paper explores the phase transitions in a community detection algorithm called mod-bp, analyzing how parameters like temperature and group number affect its ability to identify meaningful communities in networks.

Contribution

It introduces new order parameters to determine the actual number of communities and reveals multiple phases with varying community counts, enhancing understanding of mod-bp's behavior.

Findings

Existence of phases with any number of communities between 1 and q

Introduction of order parameters to identify the true number of groups

Analysis of phase transitions in synthetic and real networks

Abstract

Detecting communities in a network, based only on the adjacency matrix, is a problem of interest to several scientific disciplines. Recently, Zhang and Moore have introduced an algorithm in [P. Zhang and C. Moore, Proceedings of the National Academy of Sciences 111, 18144 (2014)], called mod-bp, that avoids overfitting the data by optimizing a weighted average of modularity (a popular goodness-of-fit measure in community detection) and entropy (i.e. number of configurations with a given modularity). The adjustment of the relative weight, the "temperature" of the model, is crucial for getting a correct result from mod-bp. In this work we study the many phase transitions that mod-bp may undergo by changing the two parameters of the algorithm: the temperature $T$ and the maximum number of groups $q$. We introduce a new set of order parameters that allow to determine the actual number of groups $\hat{q}$, and we observe on both synthetic and real networks the existence of phases with any $\hat{q} \in \{1,q\}$, which were unknown before. We discuss how to interpret the results of mod-bp and how to make the optimal choice for the problem of detecting significant communities.