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arXiv · 2501.03553

Persistent Homology with Path-Representable Distances on Graph Data

Eunwoo Heo·Byeongchan Choi·Jae-Hun Jung
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Abstract

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.

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BibTeXRIS

Eunwoo Heo, Byeongchan Choi, Jae-Hun Jung. 2026-02-16. Persistent Homology with Path-Representable Distances on Graph Data. https://arxiv.org/abs/2501.03553

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