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

2026-04-24 · Source: stat.ML updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Data Science & Analytics, Mathematics & Computational Sciences · Depth: Expert, extended

Summary

This research investigates community detection in the non-uniform hypergraph stochastic block model (HSBM), which incorporates hyperedges of varying sizes to capture higher-order interactions. The study establishes a Kesten–Stigum-type bound for weak recovery in general non-uniform HSBMs with $r$ blocks. For the $r=2$ case, it demonstrates that weak recovery is achievable if the sum of signal-to-noise ratios across all uniform hypergraph layers exceeds one, confirming a conjecture from [CEH23]. The authors propose a polynomial-time spectral algorithm that leverages an optimally weighted non-backtracking operator to reach this threshold. The approach develops a spectral theory for weighted non-backtracking operators on non-uniform hypergraphs, including a precise characterization of outlier eigenvalues and eigenvector overlaps, and introduces a novel Ihara–Bass formula for efficient low-dimensional representation.

Key takeaway

For AI Scientists and Research Scientists working on community detection in complex, multi-modal datasets, this work provides a principled method to achieve optimal weak recovery. By applying the proposed optimally weighted non-backtracking spectral algorithm, you can combine signals from multiple hypergraph layers, even if individual layers are below their detection thresholds. This approach offers a computationally efficient way to cluster non-uniform hypergraphs and resolve previous conjectures on detection limits.

Key insights

Combining subcritical hypergraph layers can enable weak recovery if their aggregate signal-to-noise ratio surpasses a Kesten–Stigum-type bound.

Principles

• Optimal weighting aggregates heterogeneous higher-order interactions.
• Non-backtracking operators enable spectral theory for non-uniform hypergraphs.

Method

A polynomial-time spectral algorithm using an optimally weighted non-backtracking operator achieves the Kesten–Stigum bound for weak recovery in non-uniform HSBMs.

In practice

• Use optimal hyperedge reweighting for improved spectral clustering.
• Apply the Ihara–Bass formula for efficient reduced operator computation.

Topics

• Non-uniform Hypergraph SBM
• Community Detection
• Kesten-Stigum Bound
• Spectral Algorithms
• Weighted Non-backtracking Operator

Best for: AI Scientist, Research Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.