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}.

翻译：摘要：持续同调（PH）编码了循环等全局信息，因此正日益融入图神经网络（GNN）。GNN中的PH方法通常遍历递增的子图序列。本文首先揭示了这种包含过程的局限性。为弥补这些缺陷，我们将收缩分析为一种原则性拓扑操作，尤其适用于图表示学习。我们研究了收缩序列的持续性，称之为收缩同调（CH）。我们证明正向PH与CH在表达能力上存在差异。随后引入沙漏持续同调（Hourglass Persistence），这是一类通过交错包含与收缩序列来增强表达能力、可学习性与稳定性的拓扑描述符。我们还研究了由两种范式参数化的相关族，并讨论了该框架如何推广至单纯复形与胞腔网络。我们进一步设计了可嵌入端到端可微GNN流水线的高效算法，在标准真实世界图数据集上相比多种PH方法实现了一致的实证性能提升。代码见\href{https://github.com/Aalto-QuML/Hourglass}{此链接}。