A Complete Characterization of the One-Dimensional Intrinsic ÄŒech Persistence Diagrams for Metric Graphs

Abstract

Metric graphs are special types of metric spaces used to model and represent simple, ubiquitous, geometric relations in data such as biological networks, social networks, and road networks. We are interested in giving a qualitative description of metric graphs using topological summaries. In particular, we provide a complete characterization of the one-dimensional intrinsic ÄŒech persistence diagrams for finite metric graphs using persistent homology. Together with complementary results by Adamaszek et al., which imply the results on intrinsic ÄŒech persistence diagrams in all dimensions for a single cycle, our results constitute the important steps toward characterizing intrinsic ÄŒech persistence diagrams for arbitrary finite metric graphs across all dimensions.