Reeb Graph Metrics from the Ground Up

TL;DR

This paper surveys and analyzes five different metrics for comparing Reeb graphs, aiming to clarify their properties, relationships, and applications in topological data analysis.

Contribution

It provides a comprehensive comparison and clarification of existing Reeb graph metrics, including definitions, properties, and distinctions among them.

Findings

Analysis of stability, universality, and discriminativity of metrics

Clarification of differences between metric variations

Expansion of the taxonomy of the bottleneck distance

Abstract

The Reeb graph has been utilized in various applications including the analysis of scalar fields. Recently, research has been focused on using topological signatures such as the Reeb graph to compare multiple scalar fields by defining distance metrics on the topological signatures themselves. Here we survey five existing metrics that have been defined on Reeb graphs: the bottleneck distance, the interleaving distance, functional distortion distance, the Reeb graph edit distance, and the universal edit distance. Our goal is to (1) provide definitions and concrete examples of these distances in order to develop the intuition of the reader, (2) visit previously proven results of stability, universality, and discriminativity, (3) identify and complete any remaining properties which have only been proven (or disproven) for a subset of these metrics, (4) expand the taxonomy of the bottleneck distance to better distinguish between variations which have been commonly miscited, and (5) reconcile the various definitions and requirements on the underlying spaces for these metrics to be defined and properties to be proven.