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Optimal detection of planted stars via a random energy model

Ijay Narang, Will Perkins, Timothy L. H. Wee

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.

Optimal detection of planted stars via a random energy model

Abstract

We study the problem of detecting a planted star in the Erd{ő}s--R{é}nyi random graph , formulated as a hypothesis test. We determine the scaling window for critical detection in 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.
Paper Structure (27 sections, 20 theorems, 170 equations)

This paper contains 27 sections, 20 theorems, 170 equations.

Table of Contents

  1. Introduction
  2. Summary of contributions
  3. Precise threshold fundamental limits of detection
  4. Likelihood ratio and REM
  5. Simple optimal test and asymmetrical testing errors
  6. Problem formulation
  7. Notation
  8. Main results
  9. Related work
  10. Planted subgraph detection
  11. Error curves in the critical regime
  12. Connection to the random energy model
  13. Simplifying the likelihood ratio
  14. Condensation phase transitions for the likelihood ratio
  15. Case $c>\tfrac{1}{2}$
  16. ...and 12 more sections

Key Result

Theorem 1.1

Suppose the star size $k$ satisfies eqAsmpt_on_k. For $\gamma \in \mathbb{R}$, set Then, as $n \rightarrow \infty$,

Theorems & Definitions (40)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Proposition 1.5
  • Remark 1.6
  • Lemma 1.7
  • Corollary 1.8
  • Lemma 2.1
  • Lemma 2.2: BovierKurkovaLowe2002Fluctuations, Bovier2006Disordered Theorem 9.2.1
  • ...and 30 more