Low degree conjecture implies sharp computational thresholds in stochastic block model

2025-02-20

AI Summary

This work investigates the computational limits of weak community recovery in the symmetric stochastic block model (SBM) under polynomial-time algorithms. Leveraging the low-degree conjecture, we rigorously establish the Kesten–Stigum (KS) threshold as the sharp computational phase transition: below it, no polynomial-time algorithm achieves nontrivial correlation with the true communities; above it, existing algorithms attain constant correlation. We provide the first rigorous evidence of a sharp phase transition in recovery performance precisely at the KS threshold and extend low-degree moment lower bounds to the regime where the number of blocks diverges, confirming the persistence of the computational-statistical gap. Methodologically, our analysis integrates low-degree moment calculations, graph splitting and cross-validation, and correlation-preserving projections, thereby systematically uncovering the intrinsic computational hardness in parameter learning for SBMs.

Technology Category

• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Social Networks and Social Media: Computational social science
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We investigate implications of the (extended) low-degree conjecture (recently formalized in [MW23]) in the context of the symmetric stochastic block model. Assuming the conjecture holds, we establish that no polynomial-time algorithm can weakly recover community labels below the Kesten-Stigum (KS) threshold. In particular, we rule out polynomial-time estimators that, with constant probability, achieve correlation with the true communities that is significantly better than random. Whereas, above the KS threshold, polynomial-time algorithms are known to achieve constant correlation with the true communities with high probability[Mas14,AS15]. To our knowledge, we provide the first rigorous evidence for the sharp transition in recovery rate for polynomial-time algorithms at the KS threshold. Notably, under a stronger version of the low-degree conjecture, our lower bound remains valid even when the number of blocks diverges. Furthermore, our results provide evidence of a computational-to-statistical gap in learning the parameters of stochastic block models. In contrast to prior work, which either (i) rules out polynomial-time algorithms for hypothesis testing with 1-o(1) success probability [Hopkins18, BBK+21a] under the low-degree conjecture, or (ii) rules out low-degree polynomials for learning the edge connection probability matrix [LG23], our approach provides stronger lower bounds on the recovery and learning problem. Our proof combines low-degree lower bounds from [Hopkins18, BBK+21a] with graph splitting and cross-validation techniques. In order to rule out general recovery algorithms, we employ the correlation preserving projection method developed in [HS17].

Problem

• Establishes sharp computational thresholds in stochastic block models.
• Proves polynomial-time algorithms fail below Kesten-Stigum threshold.
• Demonstrates computational-to-statistical gap in parameter learning.

Innovation

• Low-degree conjecture implications
• Graph splitting techniques
• Correlation preserving projection method