Contraction and Hourglass Persistence for Learning on Graphs, Simplices, and Cells

TL;DR

This paper introduces Hourglass Persistence, a novel topological descriptor that interleaves inclusions and contractions to enhance the expressivity and stability of graph neural networks using persistent homology.

Contribution

It analyzes contractions as a topological operation, introduces Hourglass Persistence, and provides algorithms that improve GNN performance on real-world datasets.

Findings

Hourglass Persistence boosts expressivity and stability in GNNs.

Algorithms are efficient and integrate into end-to-end pipelines.

Empirical results show consistent improvements over existing PH methods.

Abstract

Persistent homology (PH) encodes global information, such as cycles, and is thus increasingly integrated into graph neural networks (GNNs). PH methods in GNNs typically traverse an increasing sequence of subgraphs. In this work, we first expose limitations of this inclusion procedure. To remedy these shortcomings, we analyze contractions as a principled topological operation, in particular, for graph representation learning. We study the persistence of contraction sequences, which we call Contraction Homology (CH). We establish that forward PH and CH differ in expressivity. We then introduce Hourglass Persistence, a class of topological descriptors that interleave a sequence of inclusions and contractions to boost expressivity, learnability, and stability. We also study related families parametrized by two paradigms. We also discuss how our framework extends to simplicial and cellular networks. We further design efficient algorithms that are pluggable into end-to-end differentiable GNN pipelines, enabling consistent empirical improvements over many PH methods across standard real-world graph datasets. Code is available at \href{https://github.com/Aalto-QuML/Hourglass}{this https URL}.