The WidthWall: A Strict Expressivity Hierarchy for Hypergraph Neural Networks

2026-05-13

AI Summary

This work addresses the unclear expressive power boundaries of existing hypergraph neural networks (HGNNs). By introducing homomorphism densities to characterize hypergraph invariants and leveraging the completeness of homomorphism counting alongside invariant approximation theory, the authors establish a strict hierarchy of expressive power indexed by hypertree width. They propose the notion of a “width barrier,” revealing a fundamental limitation: fixed-depth HGNNs cannot surpass the representational capacity of patterns bounded by a specific width. This framework uniformly characterizes the expressive power of 15 distinct HGNN architectures. Experiments demonstrate that the width barrier accurately predicts failure cases of graph simplification methods, while density-aware models effectively overcome the expressiveness bottleneck of bounded-width message passing, as validated on real-world hypergraph node classification tasks.

Abstract

Hypergraphs provide a natural framework to model higher-order interactions in scientific, social, and biological systems. Hypergraph neural networks (HGNNs) aim to learn from such data, yet it remains unclear which higher-order structures these models can represent. We show that hypergraph expressivity is governed by which small patterns an architecture can detect and count. We formalize this via homomorphism densities, which measure how often a structural motif appears in a hypergraph. Combining classical homomorphism-count completeness with invariant approximation, we show that homomorphism densities generate all continuous hypergraph invariants and organize them into a strict hierarchy indexed by hypertree width. This yields a Width Wall: a fundamental architectural limit beyond which no hidden dimension, training procedure or fixed-depth HGNN can represent invariants requiring wider patterns. Our framework provides a unified characterization of 15 HGNN architectures, precisely identifies information lost by clique expansion, and motivates density-aware models that extend expressivity beyond bounded-width message passing. We experimentally validate this finding on an APPLICATION NODE CLASSIFICATION SUITE of real-world hypergraphs, where the Width Wall predicts when graph-reduction baselines fail and when density features help.

Problem

• hypergraph neural networks
• expressivity
• homomorphism densities
• hypertree width
• higher-order interactions

Innovation

• hypergraph neural networks
• homomorphism densities
• hypertree width
• expressivity hierarchy
• Width Wall