Optimal detection of planted stars via a random energy model

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.