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The Generalized Skew Spectrum of Graphs

Armando Bellante, Martin Plávala, Alessandro Luongo

TL;DR

This work generalizes the Skew Spectrum to handle richer graph structures by introducing Multi-Orbit Skew Spectrum and $k$-Spectrum invariants grounded in Fourier analysis on the symmetric group $\mathbb{S}_n$. It develops efficiency-preserving techniques (reduction and double reduction) to manage higher-order correlations while preserving translation invariance, enabling near-cost-parity with the original Skew Spectrum. The authors demonstrate that these invariants enhance expressivity across attributed graphs, multilayer graphs, and hypergraphs, with empirical gains on synthetic benchmarks and molecular data (QM7), and discuss how these spectra relate to WL tests and GNNs. The framework offers a principled, tunable approach to balancing completeness and computational cost, with potential for integration into deep learning pipelines and broader group-theoretic invariants. Note: All mathematical expressions are presented with appropriate LaTeX-style delimiters for clarity in downstream parsing.

Abstract

This paper proposes a family of permutation-invariant graph embeddings, generalizing the Skew Spectrum of graphs of Kondor & Borgwardt (2008). Grounded in group theory and harmonic analysis, our method introduces a new class of graph invariants that are isomorphism-invariant and capable of embedding richer graph structures - including attributed graphs, multilayer graphs, and hypergraphs - which the Skew Spectrum could not handle. Our generalization further defines a family of functions that enables a trade-off between computational complexity and expressivity. By applying generalization-preserving heuristics to this family, we improve the Skew Spectrum's expressivity at the same computational cost. We formally prove the invariance of our generalization, demonstrate its improved expressiveness through experiments, and discuss its efficient computation.

The Generalized Skew Spectrum of Graphs

TL;DR

This work generalizes the Skew Spectrum to handle richer graph structures by introducing Multi-Orbit Skew Spectrum and -Spectrum invariants grounded in Fourier analysis on the symmetric group . It develops efficiency-preserving techniques (reduction and double reduction) to manage higher-order correlations while preserving translation invariance, enabling near-cost-parity with the original Skew Spectrum. The authors demonstrate that these invariants enhance expressivity across attributed graphs, multilayer graphs, and hypergraphs, with empirical gains on synthetic benchmarks and molecular data (QM7), and discuss how these spectra relate to WL tests and GNNs. The framework offers a principled, tunable approach to balancing completeness and computational cost, with potential for integration into deep learning pipelines and broader group-theoretic invariants. Note: All mathematical expressions are presented with appropriate LaTeX-style delimiters for clarity in downstream parsing.

Abstract

This paper proposes a family of permutation-invariant graph embeddings, generalizing the Skew Spectrum of graphs of Kondor & Borgwardt (2008). Grounded in group theory and harmonic analysis, our method introduces a new class of graph invariants that are isomorphism-invariant and capable of embedding richer graph structures - including attributed graphs, multilayer graphs, and hypergraphs - which the Skew Spectrum could not handle. Our generalization further defines a family of functions that enables a trade-off between computational complexity and expressivity. By applying generalization-preserving heuristics to this family, we improve the Skew Spectrum's expressivity at the same computational cost. We formally prove the invariance of our generalization, demonstrate its improved expressiveness through experiments, and discuss its efficient computation.

Paper Structure

This paper contains 40 sections, 12 theorems, 43 equations, 7 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Notation.
  4. The Symmetric Group
  5. Graph isomorphism and functions on $\mathbb{S}_n$
  6. Fourier transform on $\mathbb{S}_n$
  7. The Skew Spectrum of Graphs
  8. Richer graph structures and multiple orbits
  9. Self-loops.
  10. Node features.
  11. Edge features.
  12. Multilayer graphs.
  13. Hypergraphs.
  14. Spectra of higher correlations
  15. Intuition.
  16. ...and 25 more sections

Key Result

Lemma 2.1

Let $f: \mathbb{S}_n/\mathbb{S}_{n-2} \rightarrow \mathbb{C}$. Its Fourier transform using YOR is a collection of sparse square matrices indexed by specific partitions of $n$. The sparsity pattern is independent of $f$: We use $\Lambda_n$ to denote this set of partitions.

Figures (7)

  • Figure 1: Three graphs and a permutation-invariant embedding.
  • Figure 2: A weighted, directed graph as a function on $\mathbb{S}_4$.
  • Figure 3: Synthetic dataset with four graphs families. Each family contains $1000$ isomorphic graphs. Families are not isomorphic.
  • Figure 4: Two directed, weighted, labeled non-isomorphic graphs.
  • Figure 5: Shattering non-isomorphic graphs with $k$-Spectra ($k \in [3,9]$), their concatenation (C), Laplacian's eigenvalues (L), and 1-WL tests with different iterations (1-WL-ITER).
  • ...and 2 more figures

Theorems & Definitions (24)

  • Lemma 2.1: Fourier sparsity
  • Definition 2.2: Skew Spectrum
  • Definition 2.3: Reduced Skew Spectrum
  • Theorem 2.4: Reduced Skew Spectrum computation
  • Definition 3.1: Multi-Orbit Skew Spectrum
  • Theorem 3.2: Reduced Multi-Orbit Skew Spectrum
  • Definition 4.1: $k$-correlation
  • Definition 4.2: $k$-Spectrum
  • Theorem 4.3: Invariance
  • proof
  • ...and 14 more