Fundamental Limits of Community Detection in Contextual Multi-Layer Stochastic Block Models

TL;DR

This paper establishes the fundamental limits and efficient algorithms for community detection in multi-layer stochastic block models with high-dimensional covariates, revealing no gap between statistical possibility and computational feasibility.

Contribution

It extends previous work to the constant-degree regime with noisy data and introduces novel inequalities and algorithms that achieve the detection threshold.

Findings

Derived a sharp detection threshold in the high-dimensional, constant-degree regime.

Designed efficient algorithms that match the theoretical detection limits.

Proved there is no statistical-computational gap in this community detection setting.

Abstract

We consider the problem of community detection from the joint observation of a high-dimensional covariate matrix and $L$ sparse networks, all encoding noisy, partial information about the latent community labels of $n$ subjects. In the asymptotic regime where the networks have constant average degree and the number of features $p$ grows proportionally with $n$, we derive a sharp threshold under which detecting and estimating the subject labels is possible. Our results extend the work of \cite{MN23} to the constant-degree regime with noisy measurements, and also resolve a conjecture in \cite{YLS24+} when the number of networks is a constant. Our information-theoretic lower bound is obtained via a novel comparison inequality between Bernoulli and Gaussian moments, as well as a statistical variant of the ``recovery to chi-square divergence reduction'' argument inspired by \cite{DHSS25}. On the algorithmic side, we design efficient algorithms based on counting decorated cycles and decorated paths and prove that they achieve the sharp threshold for both detection and weak recovery. In particular, our results show that there is no statistical-computational gap in this setting.