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HyperMagNet: A Magnetic Laplacian based Hypergraph Neural Network

Tatyana Benko, Martin Buck, Ilya Amburg, Stephen J. Young, Sinan G. Aksoy

TL;DR

Hypergraphs capture multi-way relations that traditional graphs cannot. HyperMagNet adopts a non-reversible EDVW random walk to form a magnetic Laplacian with a learnable charge matrix, enabling direct hypergraph convolution without graph reductions. The approach yields improved node classification across text, citation, and vision datasets and demonstrates the value of edge-dependent weighting and complex-valued spectral learning. The work lays groundwork for directed hypergraph processing and scalable hypergraph neural networks, with potential extensions to link prediction and sparsification-based efficiency improvements.

Abstract

In data science, hypergraphs are natural models for data exhibiting multi-way relations, whereas graphs only capture pairwise. Nonetheless, many proposed hypergraph neural networks effectively reduce hypergraphs to undirected graphs via symmetrized matrix representations, potentially losing important information. We propose an alternative approach to hypergraph neural networks in which the hypergraph is represented as a non-reversible Markov chain. We use this Markov chain to construct a complex Hermitian Laplacian matrix - the magnetic Laplacian - which serves as the input to our proposed hypergraph neural network. We study HyperMagNet for the task of node classification, and demonstrate its effectiveness over graph-reduction based hypergraph neural networks.

HyperMagNet: A Magnetic Laplacian based Hypergraph Neural Network

TL;DR

Hypergraphs capture multi-way relations that traditional graphs cannot. HyperMagNet adopts a non-reversible EDVW random walk to form a magnetic Laplacian with a learnable charge matrix, enabling direct hypergraph convolution without graph reductions. The approach yields improved node classification across text, citation, and vision datasets and demonstrates the value of edge-dependent weighting and complex-valued spectral learning. The work lays groundwork for directed hypergraph processing and scalable hypergraph neural networks, with potential extensions to link prediction and sparsification-based efficiency improvements.

Abstract

In data science, hypergraphs are natural models for data exhibiting multi-way relations, whereas graphs only capture pairwise. Nonetheless, many proposed hypergraph neural networks effectively reduce hypergraphs to undirected graphs via symmetrized matrix representations, potentially losing important information. We propose an alternative approach to hypergraph neural networks in which the hypergraph is represented as a non-reversible Markov chain. We use this Markov chain to construct a complex Hermitian Laplacian matrix - the magnetic Laplacian - which serves as the input to our proposed hypergraph neural network. We study HyperMagNet for the task of node classification, and demonstrate its effectiveness over graph-reduction based hypergraph neural networks.
Paper Structure (19 sections, 12 equations, 2 figures, 5 tables)

This paper contains 19 sections, 12 equations, 2 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Hypergraphs and Random Walks.
  4. The Representative Digraph of a Hypergraph.
  5. The Magnetic Laplacian.
  6. HyperMagNet
  7. A Hypergraph Laplacian and Convolution.
  8. A Learnable Charge Matrix and HyperMagNet Architecture.
  9. Related Work
  10. Hypergraph Neural Networks.
  11. Edge-Dependent Vertex Weights in Hypergraph Neural Networks.
  12. Experiments
  13. Term-Document Data.
  14. Citation and Author-Paper Networks.
  15. Computer Vision.
  16. ...and 4 more sections

Figures (2)

  • Figure 1: HyperMagNet (HMN) uses a non-reversible Markov chain to build a hypergraph Laplacian which avoids Laplacians associated with the star graph or clique expansion.
  • Figure 2: Left: EDVW (top) vs EIVW (bottom) random walks yielding graph vs digraph random walks. Right: hypergraphs (top) with identical, intersection-weighted clique and line graphs (bottom) yet are distinguished by their EDVW random walk properties and magnetic Laplacian spectra.