Strong Equivalence of the Interleaving and Functional Distortion Metrics for Reeb Graphs

TL;DR

This paper proves that the interleaving and functional distortion metrics are strongly equivalent for Reeb graphs, simplifying comparisons and stability analysis in topological data analysis.

Contribution

It establishes the strong equivalence of two key metrics on Reeb graphs, linking their theoretical foundations and practical applications.

Findings

The two metrics are strongly equivalent on Reeb graphs.

Immediate proof of bottleneck stability for persistence diagrams.

Simplifies comparison and analysis of Reeb graphs.

Abstract

The Reeb graph is a construction that studies a topological space through the lens of a real valued function. It has widely been used in applications, however its use on real data means that it is desirable and increasingly necessary to have methods for comparison of Reeb graphs. Recently, several methods to define metrics on the space of Reeb graphs have been presented. In this paper, we focus on two: the functional distortion distance and the interleaving distance. The former is based on the Gromov--Hausdorff distance, while the latter utilizes the equivalence between Reeb graphs and a particular class of cosheaves. However, both are defined by constructing a near-isomorphism between the two graphs of study. In this paper, we show that the two metrics are strongly equivalent on the space of Reeb graphs. In particular, this gives an immediate proof of bottleneck stability for persistence diagrams in terms of the Reeb graph interleaving distance.