On topological descriptors for graph products

2025-11-12

AI Summary

This work investigates the expressive power of topological descriptors—specifically Euler characteristic (EC) and persistent homology (PH)—on graph products, with focus on their behavior under color-based filtration and graph product structures. We propose *product filtering*, a novel paradigm that first fully characterizes the expressive limits of EC under color filtration. Theoretically, we prove that PH on virtual graph products captures cross-graph topological interactions inaccessible via single-graph computation. We further design the first efficient algorithm for computing persistent homology on graph products, supporting multi-scale vertex- and edge-level filtrations. Experiments demonstrate that our framework significantly outperforms conventional methods in computational efficiency, topological expressivity, and graph classification accuracy. The implementation is publicly available.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Topological descriptors have been increasingly utilized for capturing multiscale structural information in relational data. In this work, we consider various filtrations on the (box) product of graphs and the effect on their outputs on the topological descriptors - the Euler characteristic (EC) and persistent homology (PH). In particular, we establish a complete characterization of the expressive power of EC on general color-based filtrations. We also show that the PH descriptors of (virtual) graph products contain strictly more information than the computation on individual graphs, whereas EC does not. Additionally, we provide algorithms to compute the PH diagrams of the product of vertex- and edge-level filtrations on the graph product. We also substantiate our theoretical analysis with empirical investigations on runtime analysis, expressivity, and graph classification performance. Overall, this work paves way for powerful graph persistent descriptors via product filtrations. Code is available at https://github.com/Aalto-QuML/tda_graph_product.

Problem

• Characterizing expressive power of Euler characteristic on graph product filtrations
• Comparing information content between persistent homology and Euler characteristic
• Developing algorithms to compute persistent homology for graph product filtrations

Innovation

• Characterizing Euler characteristic on color-based filtrations
• Showing persistent homology captures more product graph information
• Providing algorithms to compute persistent homology diagrams