On the Expressivity of Persistent Homology in Graph Learning

Rubén Ballester, Bastian Rieck

Keywords: topology, persistent homology, topological data analysis, tda, graph learning, graphs, expressivity

TL;DR: We prove theoretical and empirical results concerning the expressivity (in terms of the Weisfeiler-Leman hierarchy) of persistent homology for graph learning tasks.

Abstract: Persistent homology, a technique from computational topology, has recently shown strong empirical performance in the context of graph classification. Being able to capture long range graph properties via higher-order topological features, such as cycles of arbitrary length, in combination with multi-scale topological descriptors, has improved predictive performance for data sets with prominent topological structures, such as molecules. At the same time, the theoretical properties of persistent homology have not been formally assessed in this context. This paper intends to bridge the gap between computational topology and graph machine learning by providing a brief introduction to persistent homology in the context of graphs, as well as a theoretical discussion and empirical analysis of its expressivity for graph learning tasks.

Submission Type: Full paper proceedings track submission (max 9 main pages).

Software: https://github.com/aidos-lab/PH_expressivity

Submission Number: 23