Higher order trade-offs in hypergraph community detection

TL;DR

This paper develops a unified framework for community detection in non-uniform hypergraphs, analyzing higher-order trade-offs and proposing spectral clustering methods with theoretical guarantees.

Contribution

It introduces a signal-to-noise ratio for hypergraph community detection, derives a Bethe Hessian operator, and characterizes detectability thresholds in non-uniform hypergraphs.

Findings

Spectral methods match belief propagation in uniform hypergraphs.

Hyperedge splitting strategies affect community detection accuracy.

Empirical data confirms higher-order trade-offs are practically relevant.

Abstract

Extending community detection from pairwise networks to hypergraphs introduces fundamental theoretical challenges. Hypergraphs exhibit structural heterogeneity with no direct graph analogue: hyperedges of varying orders can connect nodes across communities in diverse configurations, introducing new trade-offs in defining and detecting community structure. We address these challenges by developing a unified framework for community detection in non-uniform hypergraphs under the Hypergraph Stochastic Block Model. We introduce a general signal-to-noise ratio that enables a quantitative analysis of trade-offs unique to higher-order networks, such as which hypergedges we choose to split across communities and how we choose to split them. Building on this framework, we derive a Bethe Hessian operator for non-uniform hypergraphs that provides efficient spectral clustering with principled model selection. We characterize the resulting spectral detectability threshold and compare it to belief propagation limits, showing the methods coincide for uniform hypergraphs but diverge in non-uniform settings. Synthetic experiments confirm our analytical predictions and reveal systematic biases toward preserving higher-order and balanced-shape hyperedges. Application to empirical data demonstrates the practical relevance of these higher-order detectability trade-offs in real-world systems.