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

Manuel Fernandez V; Ludovic Stephan; Yizhe Zhu

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arXiv · 2604.20907

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

Manuel Fernandez V·Ludovic Stephan·Yizhe Zhu

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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.

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Manuel Fernandez V, Ludovic Stephan, Yizhe Zhu. 2026-04-21. Achieving the Kesten-Stigum bound in the non-uniform hypergraph stochastic block model. https://arxiv.org/abs/2604.20907

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