Optimal detection of planted stars via a random energy model

TL;DR

This paper investigates the optimal detection of a planted star in a random graph, identifying the critical detection threshold and revealing a phase transition in the likelihood ratio akin to spin glass models.

Contribution

It characterizes the detection threshold and phase transition in the planted star problem using a random energy model analogy.

Findings

Determines the critical scaling window for detection.

Characterizes the total variation distance between hypotheses.

Identifies a condensation phase transition in the likelihood ratio.

Abstract

We study the problem of detecting a planted star in the Erd{\H{o}}s--R{\'e}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.