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：持久同调作为计算拓扑学的一项技术，近年来在图分类任务中展现出卓越的实证性能。它能够通过高阶拓扑特征（如任意长度的环）结合多尺度拓扑描述符，捕捉图的长程性质，从而在具有显著拓扑结构的数据集（如分子数据）上提升了预测性能。然而，持久同调的理论特性在此背景下尚未得到正式评估。本文旨在通过简要介绍图背景下的持久同调方法，并对其在图学习任务中的表达能力进行理论探讨与实证分析，以弥合计算拓扑学与图机器学习之间的研究空白。