# Hierarchies and Ranks for Persistence Pairs

## TL;DR

This paper introduces a new hierarchy for zero-dimensional persistence pairs that captures detailed spatial relationships and branching structures, enhancing the analysis of persistence diagrams.

## Contribution

It develops a novel hierarchy for persistence pairs that encodes spatial relations and branching, improving the expressive power of persistence diagram analysis.

## Key findings

- The hierarchy captures differences in branching structures.
- It enables measurement of spatial stability of pairings.
- A rank function for persistence pairs is defined and applied.

## Abstract

We develop a novel hierarchy for zero-dimensional persistence pairs, i.e., connected components, which is capable of capturing more fine-grained spatial relations between persistence pairs. Our work is motivated by a lack of spatial relationships between features in persistence diagrams, leading to a limited expressive power. We build upon a recently-introduced hierarchy of pairs in persistence diagrams that augments the pairing stored in persistence diagrams with information about which components merge. Our proposed hierarchy captures differences in branching structure. Moreover, we show how to use our hierarchy to measure the spatial stability of a pairing and we define a rank function for persistence pairs and demonstrate different applications.