Contraction and Hourglass Persistence for Learning on Graphs, Simplices, and Cells
For researchers in topological deep learning, this work provides a novel topological descriptor that enhances graph representation learning, offering a principled alternative to inclusion-based PH.
The paper identifies limitations of standard persistent homology (PH) methods in graph neural networks and proposes Contraction Homology (CH) and Hourglass Persistence, which interleave inclusions and contractions to improve expressivity, learnability, and stability. The methods achieve consistent empirical improvements over existing PH methods on standard graph datasets.
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}.