The Generalized Skew Spectrum of Graphs

2025-05-29

AI Summary

Traditional skew spectrum methods fail to handle complex graph structures such as attributed graphs, multilayer graphs, and hypergraphs. Method: We propose a family of permutation-invariant generalized graph embedding methods, the first to extend the skew spectrum to a general group-action framework. Leveraging group representation theory and harmonic analysis, our approach integrates generalized Fourier transforms with invariance constraints to enable tunable trade-offs between expressive power and computational efficiency. Contribution/Results: We provide rigorous theoretical guarantees of graph isomorphism invariance. Empirical evaluation demonstrates that, at comparable time complexity, our method significantly outperforms the original skew spectrum in distinguishing non-isomorphic complex graphs, while supporting efficient approximate computation.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

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.

Problem

• Generalizes graph embeddings for richer structures
• Balances computational complexity and expressivity
• Proves invariance and improves expressiveness experimentally

Innovation

• Permutation-invariant graph embeddings generalization
• Group theory and harmonic analysis foundation
• Trade-off between complexity and expressivity