A computational transition for detecting correlated stochastic block models by low-degree polynomials

TL;DR

This paper investigates the detectability of correlation in pairs of sparse stochastic block models using low-degree polynomial tests, identifying precise thresholds that separate feasible and infeasible detection regimes.

Contribution

It establishes the exact detection threshold for correlated stochastic block models using low-degree polynomial methods, linking it to known phase transitions and hardness results.

Findings

Detection threshold at s > min{√α, 1/(λϵ²)}

Low-degree tests are optimal above this threshold

Hardness results imply computational limits below the threshold

Abstract

Detection of correlation in a pair of random graphs is a fundamental statistical and computational problem that has been extensively studied in recent years. In this work, we consider a pair of correlated (sparse) stochastic block models $\mathcal{S}(n,\tfrac{\lambda}{n};k,\epsilon;s)$ that are subsampled from a common parent stochastic block model $\mathcal S(n,\tfrac{\lambda}{n};k,\epsilon)$ with $k=O(1)$ symmetric communities, average degree $\lambda=O(1)$, divergence parameter $\epsilon$, and subsampling probability $s$. For the detection problem of distinguishing this model from a pair of independent Erd\H{o}s-R\'enyi graphs with the same edge density $\mathcal{G}(n,\tfrac{\lambda s}{n})$, we focus on tests based on \emph{low-degree polynomials} of the entries of the adjacency matrices, and we determine the threshold that separates the easy and hard regimes. More precisely, we show that this class of tests can distinguish these two models if and only if $s> \min \{ \sqrt{\alpha}, \frac{1}{\lambda \epsilon^2} \}$, where $\alpha\approx 0.338$ is the Otter's constant and $\frac{1}{\lambda \epsilon^2}$ is the Kesten-Stigum threshold. Combining a reduction argument in \cite{Li25+}, our hardness result also implies low-degree hardness for partial recovery and detection (to independent block models) when $s< \min \{ \sqrt{\alpha}, \frac{1}{\lambda \epsilon^2} \}$. Finally, our proof of low-degree hardness is based on a conditional variant of the low-degree likelihood calculation.