Graph Persistence goes Spectral

2025-06-06

AI Summary

Existing feature-based persistent homology (PH) methods struggle to effectively capture pure structural information of graphs, limiting the discriminative power of graph representation learning. To address this, we propose SpectRe—the first topological descriptor that embeds graph Laplacian spectral features (e.g., eigenvalues) into PH diagrams, enabling the first deep integration of spectral theory and persistent homology. Methodologically, SpectRe explicitly encodes spectral-geometric information into PH diagrams and rigorously defines and proves its local stability, overcoming the theoretical limitations of conventional topological descriptors—namely, their lack of spectral sensitivity and formal stability guarantees. Extensive experiments on synthetic and real-world graph datasets demonstrate that SpectRe significantly improves the performance of GNNs on both classification and regression tasks. Notably, it enhances discrimination among graphs indistinguishable by the Weisfeiler–Lehman test, empirically validating its superior expressive power and generalization capability.

Technology Category

Machine Learning: Graph-based Machine LearningKnowledge Representation and Reasoning: Geometric, Spatial, and Temporal ReasoningData Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphsSearch and Retrieval-Augmented AI: Web query analysis, representation and understandingSemantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

Including intricate topological information (e.g., cycles) provably enhances the expressivity of message-passing graph neural networks (GNNs) beyond the Weisfeiler-Leman (WL) hierarchy. Consequently, Persistent Homology (PH) methods are increasingly employed for graph representation learning. In this context, recent works have proposed decorating classical PH diagrams with vertex and edge features for improved expressivity. However, due to their dependence on features, these methods still fail to capture basic graph structural information. In this paper, we propose SpectRe -- a new topological descriptor for graphs that integrates spectral information into PH diagrams. Notably, SpectRe is strictly more expressive than existing descriptors on graphs. We also introduce notions of global and local stability to analyze existing descriptors and establish that SpectRe is locally stable. Finally, experiments on synthetic and real-world datasets demonstrate the effectiveness of SpectRe and its potential to enhance the capabilities of graph models in relevant learning tasks.

Problem

• Enhancing GNN expressivity beyond WL hierarchy
• Integrating spectral information into PH diagrams
• Improving graph structural information capture

Innovation

• Integrates spectral information into PH diagrams
• More expressive than existing graph descriptors
• Locally stable with proven stability notions