Graph energy as a measure of community detectability in networks

TL;DR

This paper demonstrates that the difference in graph energy between planted partition model networks and Erdős–Rényi networks exhibits a clear transition at the community detectability threshold, indicating spectral methods can distinguish detectable communities.

Contribution

The study reveals that graph energy based on the full spectrum of adjacency matrices can identify the detectability threshold in community detection, even where spectral methods typically fail.

Findings

Graph energy shows a transition at the detectability limit.

Full spectrum analysis can distinguish between detectable and undetectable communities.

Standard adjacency matrices retain detectability information.

Abstract

A key challenge in network science is the detection of communities, which are sets of nodes in a network that are densely connected internally but sparsely connected to the rest of the network. A fundamental result in community detection is the existence of a nontrivial threshold for community detectability on sparse graphs that are generated by the planted partition model (PPM). Below this so-called ``detectability limit'', no community-detection method can perform better than random chance. Spectral methods for community detection fail before this detectability limit because the eigenvalues corresponding to the eigenvectors that are relevant for community detection can be absorbed by the bulk of the spectrum. One can bypass the detectability problem by using special matrices, like the non-backtracking matrix, but this requires one to consider higher-dimensional matrices. In this paper, we show that the difference in graph energy between a PPM and an Erd\H{o}s--R\'enyi (ER) network has a distinct transition at the detectability threshold even for the adjacency matrices of the underlying networks. The graph energy is based on the full spectrum of an adjacency matrix, so our result suggests that standard graph matrices still allow one to separate the parameter regions with detectable and undetectable communities.