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Enhancing the Utility of Higher-Order Information in Relational Learning

Raphael Pellegrin, Lukas Fesser, Melanie Weber

TL;DR

This work investigates how to leverage higher-order information in relational learning by comparing hypergraph-level and graph-level GNNs, and by introducing hypergraph-level encodings. It demonstrates that graph-level architectures applied to hypergraph clique expansions often match or exceed hypergraph-specific methods, challenging the assumption that hypergraph architectures are always superior for multi-way relations. The authors propose hypergraph-level encodings (including $H$-$k$-LAPE, $H$-$k$-RWPE, HCP, and H-LDP) and prove they can exceed the expressivity of graph-level counterparts, with empirical results showing substantial gains when these encodings are used with graph-level models. The findings provide practical guidance for model design on multi-way relational data and point to promising directions for true hypergraph benchmarks and further expressivity analyses.

Abstract

Higher-order information is crucial for relational learning in many domains where relationships extend beyond pairwise interactions. Hypergraphs provide a natural framework for modeling such relationships, which has motivated recent extensions of graph neural network architectures to hypergraphs. However, comparisons between hypergraph architectures and standard graph-level models remain limited. In this work, we systematically evaluate a selection of hypergraph-level and graph-level architectures, to determine their effectiveness in leveraging higher-order information in relational learning. Our results show that graph-level architectures applied to hypergraph expansions often outperform hypergraph-level ones, even on inputs that are naturally parametrized as hypergraphs. As an alternative approach for leveraging higher-order information, we propose hypergraph-level encodings based on classical hypergraph characteristics. While these encodings do not significantly improve hypergraph architectures, they yield substantial performance gains when combined with graph-level models. Our theoretical analysis shows that hypergraph-level encodings provably increase the representational power of message-passing graph neural networks beyond that of their graph-level counterparts.

Enhancing the Utility of Higher-Order Information in Relational Learning

TL;DR

This work investigates how to leverage higher-order information in relational learning by comparing hypergraph-level and graph-level GNNs, and by introducing hypergraph-level encodings. It demonstrates that graph-level architectures applied to hypergraph clique expansions often match or exceed hypergraph-specific methods, challenging the assumption that hypergraph architectures are always superior for multi-way relations. The authors propose hypergraph-level encodings (including --LAPE, --RWPE, HCP, and H-LDP) and prove they can exceed the expressivity of graph-level counterparts, with empirical results showing substantial gains when these encodings are used with graph-level models. The findings provide practical guidance for model design on multi-way relational data and point to promising directions for true hypergraph benchmarks and further expressivity analyses.

Abstract

Higher-order information is crucial for relational learning in many domains where relationships extend beyond pairwise interactions. Hypergraphs provide a natural framework for modeling such relationships, which has motivated recent extensions of graph neural network architectures to hypergraphs. However, comparisons between hypergraph architectures and standard graph-level models remain limited. In this work, we systematically evaluate a selection of hypergraph-level and graph-level architectures, to determine their effectiveness in leveraging higher-order information in relational learning. Our results show that graph-level architectures applied to hypergraph expansions often outperform hypergraph-level ones, even on inputs that are naturally parametrized as hypergraphs. As an alternative approach for leveraging higher-order information, we propose hypergraph-level encodings based on classical hypergraph characteristics. While these encodings do not significantly improve hypergraph architectures, they yield substantial performance gains when combined with graph-level models. Our theoretical analysis shows that hypergraph-level encodings provably increase the representational power of message-passing graph neural networks beyond that of their graph-level counterparts.

Paper Structure

This paper contains 58 sections, 5 theorems, 36 equations, 11 figures, 23 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Summary of Contributions
  4. Background
  5. Architectures
  6. Message-passing GNN
  7. Transformer-based GNN
  8. Graph-level architectures
  9. Hypergraph-level architectures
  10. Encodings
  11. Representational Power
  12. Hypergraph-level Encodings
  13. Laplacian Eigenvectors
  14. Random Walk Transition Probabilities
  15. Local Curvature Profiles
  16. ...and 43 more sections

Key Result

Theorem 3.2

(H-$k$-LAPE Expressivity). For any $k$ MPGNNs with H-$k$-LAPE are strictly more expressive than the 1-WL test and hence MPGNNs without encodings. Furthermore, there exist graphs which can be distinguished using H-$k$-LAPE, but not using LAPE.

Figures (11)

  • Figure 1: A pair of graph from the BREC "Basic" category (top left), the graphs' liftings (top right), the hyperedge sizes (bottom left) and node degrees (bottom right).
  • Figure 2: Example of a clique expansion of a hypergraph to a graph. The plots are created using NetworkX hagberg2008exploring and HyperNetX praggastis2023hypernetx.
  • Figure 3: Lifting of a graph to a hypergraph.
  • Figure 4: Histogram of the number of nodes in the graphs in BREC. The bars are stacked. We exclude pairs containing non-connected graphs from the original 800 graphs to arrive at 780 graphs. Best seen in color.
  • Figure 5: Histogram of the number of edges in the graphs in BREC. The bars are stacked. We exclude pairs containing non-connected graphs from the original 800 graphs to arrive at 780 graphs. Best seen in color.
  • ...and 6 more figures

Theorems & Definitions (20)

  • Definition 3.1
  • Theorem 3.2
  • proof
  • Remark 3.3
  • Definition 3.4
  • Theorem 3.5
  • proof
  • Remark 3.6
  • Remark 3.7
  • Definition 3.8
  • ...and 10 more