Homomorphism Expressivity of Spectral Invariant Graph Neural Networks

2025-03-01

AI Summary

Existing theoretical understanding of spectral invariance in Graph Neural Networks (GNNs) remains incomplete, particularly regarding their expressive power. Method: We introduce the first systematic characterization of GNN expressivity from a homomorphism-based perspective, integrating homomorphism counting theory, spectral graph theory, and higher-order GNN analysis to establish a rigorous framework for the homomorphism expressivity of spectral-invariant GNNs. Contributions/Results: We formally characterize the precise expressive boundaries across spectral-invariant GNN architectures, proving that such models can exactly count a class of tree-like subgraphs—parallel trees—and establishing an explicit quantitative relationship between model depth and subgraph counting capacity. Our results extend to higher-order GNNs and resolve multiple open problems posed by Arvind et al. (2024) and Zhang et al. (2024), thereby filling a critical gap in the theoretical foundations of spectral invariance for GNN expressivity.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Graph spectra are an important class of structural features on graphs that have shown promising results in enhancing Graph Neural Networks (GNNs). Despite their widespread practical use, the theoretical understanding of the power of spectral invariants -- particularly their contribution to GNNs -- remains incomplete. In this paper, we address this fundamental question through the lens of homomorphism expressivity, providing a comprehensive and quantitative analysis of the expressive power of spectral invariants. Specifically, we prove that spectral invariant GNNs can homomorphism-count exactly a class of specific tree-like graphs which we refer to as parallel trees. We highlight the significance of this result in various contexts, including establishing a quantitative expressiveness hierarchy across different architectural variants, offering insights into the impact of GNN depth, and understanding the subgraph counting capabilities of spectral invariant GNNs. In particular, our results significantly extend Arvind et al. (2024) and settle their open questions. Finally, we generalize our analysis to higher-order GNNs and answer an open question raised by Zhang et al. (2024).

Problem

• Analyzes expressive power of spectral invariants in GNNs.
• Proves spectral GNNs can count homomorphisms in parallel trees.
• Extends and resolves open questions in GNN expressivity.

Innovation

• Analyzes spectral invariants' expressivity in GNNs.
• Proves GNNs count homomorphisms in parallel trees.
• Extends and answers open questions in GNN theory.