# Local Versus Global Distances for Zigzag Persistence Modules

## TL;DR

This paper explores the relationship between local and global distances in zigzag persistence modules, showing bounds on bottleneck distances and discussing implications for metric graph distances and multiparameter modules.

## Contribution

It establishes explicit bounds connecting local and global persistence distances, with applications to metric graphs and multiparameter persistence modules.

## Key findings

- Bottleneck distance between restricted and unrestricted modules is bounded.
- Results have practical implications for metric graph analysis.
- Extension to matching distance in multiparameter persistence modules.

## Abstract

This short note establishes explicit and broadly applicable relationships between persistence-based distances computed locally and globally. In particular, we show that the bottleneck distance between two zigzag persistence modules restricted to an interval is always bounded above by the distance between the unrestricted versions. While this result is not surprising, it could have different practical implications. We give two related applications for metric graph distances, as well as an extension for the matching distance between multiparameter persistence modules.