Achieving the Kesten-Stigum bound in the non-uniform hypergraph stochastic block model

2026-04-21

AI Summary

This study addresses the problem of weak community detection in non-uniform hypergraph stochastic block models: specifically, whether recovery is possible by integrating multi-order information when each individual uniform hypergraph layer lies below its respective detectability threshold. We establish, for the first time, a Kesten–Stigum-type threshold for weak recovery on non-uniform hypergraphs, proving that weak recovery is feasible whenever the sum of signal-to-noise ratios across layers exceeds one. To achieve this, we introduce a tailored weighted non-backtracking operator and a novel Ihara–Bass formula, enabling a polynomial-time spectral algorithm. In the two-block setting, this algorithm provably achieves the theoretical threshold and efficiently recovers communities via low-dimensional spectral embedding and clustering, with applicability extending to unweighted hypergraphs.

Technology Category

• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

We study the community detection problem in the non-uniform hypergraph stochastic block model (HSBM), where hyperedges of varying sizes coexist. This setting captures higher-order and multi-view interactions and raises a fundamental question: can multiple uniform hypergraph layers below the detection threshold be combined to enable weak recovery? We answer this question by establishing a Kesten--Stigum-type bound for weak recovery in a general class of non-uniform HSBMs with $r$ blocks, generated according to multiple symmetric probability tensors. In the case $r=2$, we show that weak recovery is possible whenever the sum of the signal-to-noise ratios across all uniform hypergraph layers exceeds one, thereby confirming the positive part of a conjecture in (Chodrow et al., 2023). Moreover, we provide a polynomial-time spectral algorithm that achieves this threshold via an optimally weighted non-backtracking operator. For the unweighted non-backtracking matrix, our spectral method attains a different algorithmic threshold, also conjectured in (Chodrow et al., 2023). Our approach develops a spectral theory for weighted non-backtracking operators on non-uniform hypergraphs, including a precise characterization of outlier eigenvalues and eigenvector overlaps. We introduce a novel Ihara--Bass formula tailored to weighted non-uniform hypergraphs, which yields an efficient low-dimensional representation and leads to a provable spectral reconstruction algorithm. Taken together, these results provide a principled and computationally efficient approach to clustering in non-uniform hypergraphs, and highlight the role of optimal weighting in aggregating heterogeneous higher-order interactions.

Problem

• community detection
• non-uniform hypergraph
• stochastic block model
• weak recovery
• Kesten-Stigum bound

Innovation

• non-uniform hypergraph
• Kesten-Stigum bound
• weighted non-backtracking operator
• spectral algorithm
• Ihara-Bass formula