Cluster expansion of the log-likelihood ratio: Optimal detection of planted matchings

TL;DR

This paper uses cluster expansion from statistical physics to precisely analyze the detection of hidden matchings in random graphs, showing optimal detection thresholds and no computational gap.

Contribution

It introduces the cluster expansion technique to analyze the log-likelihood ratio in planted matching detection, providing non-asymptotic characterizations and revealing the absence of a statistical-to-computational gap.

Findings

Identified the critical regime for planted matching detection.

Proved the asymptotic normality of the log-likelihood ratio.

Designed efficient tests that achieve optimal detection limits.

Abstract

To understand how hidden information can be extracted from statistical networks, planted models in random graphs have been the focus of intensive study in recent years. In this work, we consider the detection of a planted matching, i.e., an independent edge set, hidden in an Erd\H{o}s-R\'enyi random graph, which is formulated as a hypothesis testing problem. We identify the critical regime for this testing problem and prove that the log-likelihood ratio is asymptotically normal. Via analyses of computationally efficient edge or wedge count test statistics that attain the optimal limits of detection, our results also reveal the absence of a statistical-to-computational gap. Our main technical tool is the cluster expansion from statistical physics, which allows us to prove a precise, non-asymptotic characterization of the log-likelihood ratio. Our analyses rely on a careful reorganization and cancellation of terms that occur in the difference between monomer-dimer log partition functions on the complete and Erd\H{o}s-R\'enyi graphs. This combinatorial and statistical physics approach represents a significant departure from the more established methods such as orthogonal decompositions, and positions the cluster expansion as a viable technique in the study of log-likelihood ratios for planted models in general.