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HWL-HIN: A Hypergraph-Level Hypergraph Isomorphism Network as Powerful as the Hypergraph Weisfeiler-Lehman Test with Application to Higher-Order Network Robustness

Chengyu Tian, Wenbin Pei

TL;DR

The paper tackles robustness prediction in higher-order networks by modeling systems as hypergraphs and showing that injecting hypergraph-level isomorphism power yields a discriminative, scalable surrogate. It introduces HWL-HIN, a hypergraph-level Hypergraph Isomorphism Network whose expressiveness matches the Hypergraph Weisfeiler-Lehman test and that leverages injective node/hyperedge aggregation with a hypergraph-level readout. Empirically, HWL-HIN outperforms standard HGNNs and graph-based baselines on topology-sensitive tasks while maintaining high efficiency, and its performance advantage grows when topology matters most. The work enables rapid, topology-aware robustness analysis for complex, higher-order systems and lays groundwork for broader hypergraph-level learning tasks.

Abstract

Robustness in complex systems is of significant engineering and economic importance. However, conventional attack-based a posteriori robustness assessments incur prohibitive computational overhead. Recently, deep learning methods, such as Convolutional Neural Networks (CNNs) and Graph Neural Networks (GNNs), have been widely employed as surrogates for rapid robustness prediction. Nevertheless, these methods neglect the complex higher-order correlations prevalent in real-world systems, which are naturally modeled as hypergraphs. Although Hypergraph Neural Networks (HGNNs) have been widely adopted for hypergraph learning, their topological expressive power has not yet reached the theoretical upper bound. To address this limitation, inspired by Graph Isomorphism Networks, this paper proposes a hypergraph-level Hypergraph Isomorphism Network framework. Theoretically, this approach is proven to possess an expressive power strictly equivalent to the Hypergraph Weisfeiler-Lehman test and is applied to predict hypergraph robustness. Experimental results demonstrate that while maintaining superior efficiency in training and prediction, the proposed method not only outperforms existing graph-based models but also significantly surpasses conventional HGNNs in tasks that prioritize topological structure representation.

HWL-HIN: A Hypergraph-Level Hypergraph Isomorphism Network as Powerful as the Hypergraph Weisfeiler-Lehman Test with Application to Higher-Order Network Robustness

TL;DR

The paper tackles robustness prediction in higher-order networks by modeling systems as hypergraphs and showing that injecting hypergraph-level isomorphism power yields a discriminative, scalable surrogate. It introduces HWL-HIN, a hypergraph-level Hypergraph Isomorphism Network whose expressiveness matches the Hypergraph Weisfeiler-Lehman test and that leverages injective node/hyperedge aggregation with a hypergraph-level readout. Empirically, HWL-HIN outperforms standard HGNNs and graph-based baselines on topology-sensitive tasks while maintaining high efficiency, and its performance advantage grows when topology matters most. The work enables rapid, topology-aware robustness analysis for complex, higher-order systems and lays groundwork for broader hypergraph-level learning tasks.

Abstract

Robustness in complex systems is of significant engineering and economic importance. However, conventional attack-based a posteriori robustness assessments incur prohibitive computational overhead. Recently, deep learning methods, such as Convolutional Neural Networks (CNNs) and Graph Neural Networks (GNNs), have been widely employed as surrogates for rapid robustness prediction. Nevertheless, these methods neglect the complex higher-order correlations prevalent in real-world systems, which are naturally modeled as hypergraphs. Although Hypergraph Neural Networks (HGNNs) have been widely adopted for hypergraph learning, their topological expressive power has not yet reached the theoretical upper bound. To address this limitation, inspired by Graph Isomorphism Networks, this paper proposes a hypergraph-level Hypergraph Isomorphism Network framework. Theoretically, this approach is proven to possess an expressive power strictly equivalent to the Hypergraph Weisfeiler-Lehman test and is applied to predict hypergraph robustness. Experimental results demonstrate that while maintaining superior efficiency in training and prediction, the proposed method not only outperforms existing graph-based models but also significantly surpasses conventional HGNNs in tasks that prioritize topological structure representation.
Paper Structure (30 sections, 3 theorems, 31 equations, 2 figures, 2 tables)

This paper contains 30 sections, 3 theorems, 31 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hypergraph
  4. Robustness Measure
  5. Hypergraph-based Load Distribution Model
  6. Data Generation via Adaptive Integration with Relative Precision
  7. Hypergraph Neural Networks and Hypergraph Isomorphism
  8. Hypergraph Neural Networks
  9. Hypergraph Isomorphism and Injectivity
  10. Related Work
  11. Convolutional Neural Networks
  12. Graph Transformers
  13. Graph Isomorphism Networks
  14. Methodology
  15. Input Feature
  16. ...and 15 more sections

Key Result

Lemma 1

Assume $\mathcal{X}$ is a countable set. A function $g$ defined on bounded multisets $X \subset \mathcal{X}$ is injective if and only if it can be decomposed into the form xu2018powerful: where elements $x$ are mapped via the non-linear function $\phi$, and $\rho$ is a post-processing function.

Figures (2)

  • Figure 1: Illustration of Lemma \ref{['lemma:injectivity']}. To ensure injectivity over multisets, element-wise features must be mapped via a non-linear mapping $\phi$ prior to summation.
  • Figure 2: Efficiency comparison of different algorithms.

Theorems & Definitions (3)

  • Lemma 2
  • Theorem 1
  • Lemma 3