Going beyond persistent homology using persistent homology

TL;DR

This paper introduces a theoretical framework to understand the limitations of persistent homology in graph analysis, proposes a new method RePHINE that combines vertex- and edge-level features, and demonstrates improved graph classification performance.

Contribution

It provides necessary and sufficient conditions for graph distinguishability using persistent homology and introduces RePHINE, a method that surpasses existing topological feature approaches in graph neural networks.

Findings

RePHINE outperforms standard persistent homology in graph classification benchmarks.

Vertex- and edge-level persistent homology are shown to have distinct, non-overlapping capabilities.

Theoretical conditions for graph distinguishability using persistent homology are established.

Abstract

Representational limits of message-passing graph neural networks (MP-GNNs), e.g., in terms of the Weisfeiler-Leman (WL) test for isomorphism, are well understood. Augmenting these graph models with topological features via persistent homology (PH) has gained prominence, but identifying the class of attributed graphs that PH can recognize remains open. We introduce a novel concept of color-separating sets to provide a complete resolution to this important problem. Specifically, we establish the necessary and sufficient conditions for distinguishing graphs based on the persistence of their connected components, obtained from filter functions on vertex and edge colors. Our constructions expose the limits of vertex- and edge-level PH, proving that neither category subsumes the other. Leveraging these theoretical insights, we propose RePHINE for learning topological features on graphs. RePHINE efficiently combines vertex- and edge-level PH, achieving a scheme that is provably more powerful than both. Integrating RePHINE into MP-GNNs boosts their expressive power, resulting in gains over standard PH on several benchmarks for graph classification.