Higher order trade-offs in hypergraph community detection

2026-01-15

AI Summary

This work addresses the unique trade-offs in hypergraph community detection arising from heterogeneous higher-order interactions. The authors propose a unified framework under a non-uniform hypergraph stochastic block model, introducing a generalized signal-to-noise ratio to characterize fundamental detection limits for higher-order structures. They develop a spectral clustering algorithm based on the Bethe Hessian operator that accommodates hyperedges of arbitrary order and provides theoretically grounded model selection. Theoretical analysis yields a spectral detectability threshold: it coincides with the belief propagation limit in the uniform case, while revealing a divergence between the two in non-uniform settings. Experimental results confirm that the algorithm preferentially retains higher-order, structurally balanced hyperedges, aligning closely with theoretical predictions.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated 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

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.

Problem

• hypergraph community detection
• higher-order trade-offs
• structural heterogeneity
• non-uniform hypergraphs
• community structure

Innovation

• hypergraph community detection
• non-uniform hypergraphs
• Bethe Hessian operator
• signal-to-noise ratio
• spectral detectability threshold