Persistent Homology with Path-Representable Distances on Graph Data

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.