Optimal detection of planted stars via a random energy model

2026-02-17

AI Summary

This work investigates the statistical detectability of an implanted star subgraph in an Erdős–Rényi random graph \( G(n,m) \), formulated as a binary hypothesis test between a null model and an alternative containing the planted structure. By analyzing the total variation distance between the distributions under the two hypotheses and leveraging tools from random graph theory and the Random Energy Model (REM) from spin glass theory, the study reveals a condensation-type phase transition in the likelihood ratio, thereby establishing a deep connection between graph detection problems and statistical physics. The main contribution lies in precisely characterizing the critical scaling window for detectability of the star and fully describing the asymptotic behavior of the total variation distance within this window.

Technology Category

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

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: Social media analysis through the lenses of networks

Abstract

We study the problem of detecting a planted star in the Erd{ő}s--R{é}nyi random graph $G(n,m)$, formulated as a hypothesis test. We determine the scaling window for critical detection in $m$ in terms of the star size, and characterize the asymptotic total variation distance between the null and alternative hypotheses in this window. In the course of the proofs we show a condensation phase transition in the likelihood ratio that closely resembles that of the random energy model from spin glass theory.

Problem

• planted star
• random graph
• hypothesis testing
• detection
• Erdős–Rényi

Innovation

• planted star detection
• random energy model
• condensation phase transition
• total variation distance
• Erdős–Rényi random graph