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.

翻译：持续同调是一种来自计算拓扑的技术，近年来在图分类任务中展现出强大的实证性能。通过高阶拓扑特征（如任意长度的环）结合多尺度拓扑描述符，该方法能够捕捉图的远距离屬性，从而提升具有显著拓扑结构（如分子）的数据集的预测性能。然而，持续同调在此背景下的理论特性尚未得到正式评估。本文旨在通过简要介绍图环境中的持续同调概念，并从理论探讨与实证分析两方面研究其图学习任务的表达性，从而弥合计算拓扑与图机器学习之间的差距。