On topological descriptors for graph products

Mattie Ji, Amauri H Souza, Vikas K Garg

Keywords: topological data analysis, graph neural networks, graph machine learning

TL;DR: We study and give algorithms to compute topological descriptors of color-based filtrations on the Cartesian product of graphs.

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.

Primary Area: Theory (e.g., control theory, learning theory, algorithmic game theory)

Submission Number: 24554