Inference via Message Passing on Partially Labeled Stochastic Block Models

TL;DR

This paper introduces a linearized message-passing algorithm for community detection in partially-labeled stochastic block models, achieving near-optimal inference performance depending on the signal-to-noise ratio, and characterizes the fundamental limits of local algorithms.

Contribution

It develops a fast, linearized message-passing method for community detection in partially-labeled SBMs and establishes the fundamental SNR thresholds for successful inference with local algorithms.

Findings

For SNR > 1, the algorithm achieves low misclassification rates exponentially decreasing with SNR.

For SNR < 1 (k=2) and SNR < 1/4 (general k), local algorithms perform only slightly better than random guessing.

The SNR threshold characterizes the fundamental limits of local algorithms in community detection.

Abstract

We study the community detection and recovery problem in partially-labeled stochastic block models (SBM). We develop a fast linearized message-passing algorithm to reconstruct labels for SBM (with $n$ nodes, $k$ blocks, $p,q$ intra and inter block connectivity) when $\delta$ proportion of node labels are revealed. The signal-to-noise ratio ${\sf SNR}(n,k,p,q,\delta)$ is shown to characterize the fundamental limitations of inference via local algorithms. On the one hand, when ${\sf SNR}>1$, the linearized message-passing algorithm provides the statistical inference guarantee with mis-classification rate at most $\exp(-({\sf SNR}-1)/2)$, thus interpolating smoothly between strong and weak consistency. This exponential dependence improves upon the known error rate $({\sf SNR}-1)^{-1}$ in the literature on weak recovery. On the other hand, when ${\sf SNR}<1$ (for $k=2$) and ${\sf SNR}<1/4$ (for general growing $k$), we prove that local algorithms suffer an error rate at least $\frac{1}{2} - \sqrt{\delta \cdot {\sf SNR}}$, which is only slightly better than random guess for small $\delta$.