Graph energy as a measure of community detectability in networks

2026-01-08

arXiv.org

AI Summary

This work addresses the challenge of identifying the detectability threshold of community structure in sparse networks, where conventional spectral methods often fail. The authors propose leveraging graph energy—a global measure derived from the full spectrum of the adjacency matrix—as a sensitive indicator of detectability. By comparing the stochastic block model with planted community structure against the structureless Erdős–Rényi random graph, they demonstrate that graph energy exhibits a pronounced phase transition precisely at the theoretical detectability threshold. Notably, this approach requires only the standard adjacency matrix without constructing high-dimensional representations, effectively distinguishing between detectable and undetectable regimes. This study establishes graph energy as a novel and powerful tool for both theoretical analysis and practical detection of community structure in sparse networks.

Technology Category

• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Social Networks and Social Media: Computational social science

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.

Problem

• community detection
• detectability limit
• graph energy
• spectral methods
• planted partition model

Innovation

• graph energy
• community detectability
• detectability threshold
• planted partition model
• spectral methods