Metric graphs are meaningful objects for modeling complex structures that arise in many real-world applications, such as road networks, river systems, earthquake faults, blood vessels, and filamentary structures in galaxies. To study metric graphs in the context of comparison, we are interested in determining the relative discriminative capabilities of two topology-based distances between a pair of arbitrary finite metric graphs: the persistence distortion distance and the intrinsic Cech distance. We explicitly show how to compute the intrinsic Cech distance between two metric graphs based solely on knowledge of the shortest systems of loops for the graphs. Our main theorem establishes an inequality between the intrinsic Cech and persistence distortion distances in the case when one of the graphs is a bouquet graph and the other is arbitrary. The relationship also holds when both graphs are constructed via wedge sums of cycles and edges.
A Closed-Form Persistence-Landmark Pipeline for Certified Point-Cloud and Graph Classification
Data Science, George Washington University, USA（乔治华盛顿大学数据科学系）；Department of Mathematical Sciences, Montana Technological University, USA（蒙塔纳技术大学数学科学系）；Faculty of Computer and Information Science, University of Ljubljana, Institute IMFM, Slovenia（卢布尔雅那大学计算机与信息科学系，IMFM研究所，斯洛文尼亚）；Biostatistics and Bioinformatics, George Washington University, USA（乔治华盛顿大学生物统计学与生物信息学系）
