# The Relationship Between the Intrinsic Cech and Persistence Distortion Distances for Metric Graphs

## TL;DR

This paper compares two topology-based distances for metric graphs, showing how the intrinsic Cech distance can be explicitly computed and establishing inequalities relating it to the persistence distortion distance in specific graph cases.

## Contribution

The paper introduces a method to compute the intrinsic Cech distance from shortest loops and establishes a key inequality relating it to the persistence distortion distance for certain classes of metric graphs.

## Key findings

- Intrinsic Cech distance can be computed from shortest loops.
- An inequality relates intrinsic Cech and persistence distortion distances.
- The relationship holds for bouquet graphs and wedge sums of cycles and edges.

## Abstract

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.