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Graph Learning in 4D: a Quaternion-valued Laplacian to Enhance Spectral GCNs

Stefano Fiorini, Stefano Coniglio, Michele Ciavotta, Enza Messina

TL;DR

This work addresses learning on general directed graphs with signed weights and digons by introducing the Quaternionic Laplacian $L^{\text{\coppa}}$, a quaternion-valued, Hermitian operator that preserves full digraph topology. It then builds QuaterGCN, a spectral GCN that uses quaternion-valued weights and a convolution defined via the normalized quaternionic Laplacian, enabling expressive interactions among node features. The authors prove that $L^{\text{\coppa}}$ generalizes the classical Laplacian and the Sign-Magnetic Laplacian, ensures positive semidefiniteness and real eigenvalues, and satisfies key spectral properties for stable convolution with a bound $\lambda_{max}(L^{\text{\coppa}}_{norm}) \le 2$. Empirically, QuaterGCN achieves superior performance across node classification and multiple edge-prediction tasks on real-world and synthetic datasets, particularly when digon information is crucial, highlighting the practical impact of preserving digraph topology with quaternion-valued operators.

Abstract

We introduce QuaterGCN, a spectral Graph Convolutional Network (GCN) with quaternion-valued weights at whose core lies the Quaternionic Laplacian, a quaternion-valued Laplacian matrix by whose proposal we generalize two widely-used Laplacian matrices: the classical Laplacian (defined for undirected graphs) and the complex-valued Sign-Magnetic Laplacian (proposed to handle digraphs with weights of arbitrary sign). In addition to its generality, our Quaternionic Laplacian is the only Laplacian to completely preserve the topology of a digraph, as it can handle graphs and digraphs containing antiparallel pairs of edges (digons) of different weights without reducing them to a single (directed or undirected) edge as done with other Laplacians. Experimental results show the superior performance of QuaterGCN compared to other state-of-the-art GCNs, particularly in scenarios where the information the digons carry is crucial to successfully address the task at hand.

Graph Learning in 4D: a Quaternion-valued Laplacian to Enhance Spectral GCNs

TL;DR

This work addresses learning on general directed graphs with signed weights and digons by introducing the Quaternionic Laplacian , a quaternion-valued, Hermitian operator that preserves full digraph topology. It then builds QuaterGCN, a spectral GCN that uses quaternion-valued weights and a convolution defined via the normalized quaternionic Laplacian, enabling expressive interactions among node features. The authors prove that generalizes the classical Laplacian and the Sign-Magnetic Laplacian, ensures positive semidefiniteness and real eigenvalues, and satisfies key spectral properties for stable convolution with a bound . Empirically, QuaterGCN achieves superior performance across node classification and multiple edge-prediction tasks on real-world and synthetic datasets, particularly when digon information is crucial, highlighting the practical impact of preserving digraph topology with quaternion-valued operators.

Abstract

We introduce QuaterGCN, a spectral Graph Convolutional Network (GCN) with quaternion-valued weights at whose core lies the Quaternionic Laplacian, a quaternion-valued Laplacian matrix by whose proposal we generalize two widely-used Laplacian matrices: the classical Laplacian (defined for undirected graphs) and the complex-valued Sign-Magnetic Laplacian (proposed to handle digraphs with weights of arbitrary sign). In addition to its generality, our Quaternionic Laplacian is the only Laplacian to completely preserve the topology of a digraph, as it can handle graphs and digraphs containing antiparallel pairs of edges (digons) of different weights without reducing them to a single (directed or undirected) edge as done with other Laplacians. Experimental results show the superior performance of QuaterGCN compared to other state-of-the-art GCNs, particularly in scenarios where the information the digons carry is crucial to successfully address the task at hand.
Paper Structure (30 sections, 11 theorems, 19 equations, 2 figures, 8 tables)

This paper contains 30 sections, 11 theorems, 19 equations, 2 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Main Contributions of The Work
  3. Preliminaries and Previous Works
  4. A Quaternion-Valued Laplacian
  5. The Quaternion Number System
  6. The Quaternionic Laplacian
  7. On the Nature of the Elements of $H^\text{\coppa}$
  8. From a Graph to its Quaternionic Laplacian
  9. Relationship between $L^{\text{\coppa}}$ and other Laplacians
  10. Spectral properties of $L^{\text{\coppa}}$
  11. QuaterNet: a spectral GCN based on $L^{\text{\coppa}}$
  12. Complexity of QuaterGCN
  13. Numerical Experiments
  14. Node Classification Task (NC)
  15. Three-Class Edge Prediction Task (3CEP)
  16. ...and 15 more sections

Key Result

Theorem 1

$L^{\text{\coppa}} = L$ for every graph with $A$ symmetric and nonnegative and $D_{vv} > 0$ for all $v \in V$.

Figures (2)

  • Figure 1: Graph and Adjacency Matrix
  • Figure 2: Overview of QuaterGCN

Theorems & Definitions (16)

  • Theorem 1
  • Theorem 2
  • Corollary 1
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • ...and 6 more