Persistent homology (PH) has been widely applied to graph data to extract topological features. However, little attention has been paid to how different distance functions on a graph affect the resulting persistence barcodes and their interpretations. In this paper, we define a class of distances on graphs, called path-representable distances, and investigate structural relationships between their induced persistent homologies. In particular, we identify a nontrivial injection between the 1-dimensional barcodes induced by two commonly used graph distances: the unweighted and weighted shortest-path distances. We formally establish sufficient conditions under which such embeddings arise, focusing on a subclass we call cost-dominated distances. The injection property is shown to hold in 0- and 1-dimensions, while we provide counterexamples for higher-dimensional cases. To make these relationships measurable, we introduce the total persistence difference (TPD), a new topological measure that quantifies changes between filtrations induced by cost-dominated distances on a fixed graph. We prove a stability result for TPD when the distance functions admit a partial order and apply the method to the SNAP EU Research Institution E-Mail dataset. TPD captures both periodic patterns and global trends in the data, and shows stronger alignment with classical graph statistics compared to an existing PH-based measure applied to the same dataset.

翻译：持久同调（PH）已被广泛应用于图数据以提取拓扑特征。然而，不同距离函数对图数据所生成的持续条形码及其解释的影响尚未得到充分关注。本文定义了一类称为路径可表示距离的图距离，并研究其诱导的持久同调之间的结构关系。特别地，我们识别了两种常用图距离——未加权最短路径距离与加权最短路径距离——所诱导的一维条形码之间的非平凡单射关系。通过聚焦于我们称为成本主导距离的子类，我们形式化地建立了此类嵌入关系产生的充分条件。该单射性质在0维和1维情形下被证明成立，同时我们为更高维情形提供了反例。为使这些关系可量化，我们引入了总持久度差异（TPD）——一种新的拓扑度量，用于量化固定图上由成本主导距离诱导的滤链之间的变化。我们证明了当距离函数存在偏序关系时TPD的稳定性定理，并将该方法应用于SNAP欧盟研究机构电子邮件数据集。TPD既能捕捉数据中的周期性模式与全局趋势，与现有应用于同数据集的PH基度量相比，也展现出与传统图统计量更强的关联性。