Detectability threshold in weighted modular networks

TL;DR

This paper analytically investigates the detectability threshold in weighted modular networks, revealing how different edge weight distributions influence the ability of spectral modularity optimization to identify true community structures.

Contribution

It derives a general expression for the detectability threshold considering various weight distributions, highlighting the impact of weight variability on community detectability.

Findings

Dirac weights yield the lowest detectability threshold

Exponential weights increase the threshold by a factor of √2

Higher variability in weights can hinder community detection

Abstract

We study the necessary condition to detect, by means of spectral modularity optimization, the ground-truth partition in networks generated according to the weighted planted-partition model with two equally sized communities. We analytically derive a general expression for the maximum level of mixing tolerated by the algorithm to retrieve community structure, showing that the value of this detectability threshold depends on the first two moments of the distributions of node degree and edge weight. We focus on the standard case of Poisson-distributed node degrees and compare the detectability thresholds of five edge-weight distributions: Dirac, Poisson, exponential, geometric, and signed Bernoulli. We show that Dirac distributed weights yield the smallest detectability threshold, while exponentially distributed weights increase the threshold by a factor $\sqrt{2}$, with other distributions exhibiting distinct behaviors that depend, either or both, on the average values of the degree and weight distributions. Our results indicate that larger variability in edge weights can make communities less detectable. In cases where edge weights carry no information about community structure, incorporating weights in community detection is detrimental.