On topological descriptors for graph products

TL;DR

This paper investigates how topological descriptors like Euler characteristic and persistent homology can be used to analyze graph products, revealing their expressive power and providing algorithms for computation, with applications in graph classification.

Contribution

It characterizes the expressive power of Euler characteristic on graph products, shows persistent homology captures more information than individual graphs, and provides algorithms for computing these descriptors.

Findings

PH descriptors of graph products contain more information than individual graphs.

EC does not increase in expressive power for graph products.

Algorithms for computing PH diagrams of graph products are developed.

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.