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

• URL: http://arxiv.org/abs/2409.00966v1
• Date: Mon, 2 Sep 2024 06:14:05 GMT
• Title: A computational transition for detecting correlated stochastic block models by low-degree polynomials
• Authors: Guanyi Chen, Jian Ding, Shuyang Gong, Zhangsong Li,
• Abstract summary: Detection of correlation in a pair of random graphs is a fundamental statistical and computational problem that has been extensively studied in recent years. We consider a pair of correlated block models $mathcalS(n,tfraclambdan;k,epsilon;s)$ that are subsampled from a common parent block model $mathcal S(n,tfraclambdan;k,epsilon;s)$ We focus on tests based on emphlow-degrees of the entries of the adjacency
• Score: 13.396246336911842
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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. Our proof of low-degree hardness is based on a conditional variant of the low-degree likelihood calculation.

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