A Complete Characterization of the 1-Dimensional Intrinsic Cech Persistence Diagrams for Metric Graphs

TL;DR

This paper provides a comprehensive topological characterization of 1-dimensional intrinsic Cech persistence diagrams for metric graphs, advancing the understanding of their shape summaries in data analysis.

Contribution

It offers a complete description of 1D intrinsic Cech persistence diagrams for metric graphs, extending previous results to all dimensions and types of graphs.

Findings

Complete characterization of 1D diagrams for metric graphs

Extension of results to all dimensions and graph types

Foundational step for topological data analysis of 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 1-dimensional intrinsic Cech persistence diagrams for metric graphs using persistent homology. Together with complementary results by Adamaszek et. al, which imply results on intrinsic Cech persistence diagrams in all dimensions for a single cycle, our results constitute important steps toward characterizing intrinsic Cech persistence diagrams for arbitrary metric graphs across all dimensions.