• Corpus ID: 238634284

Reeb Graph Metrics from the Ground Up

  title={Reeb Graph Metrics from the Ground Up},
  author={Brian Bollen and Erin W. Chambers and Joshua A. Levine and E. Munch},
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… 
1 Citations
Adaptive Covers for Mapper Graphs Using Information Criteria
  • N. Chalapathi, Youjia Zhou, Bei Wang
  • 2021 IEEE International Conference on Big Data (Big Data)
  • 2021
The mapper construction is a widely used tool from topological data analysis in obtaining topological summaries of large, high-dimensional point cloud data. It has enjoyed great success in data


Local Equivalence and Intrinsic Metrics between Reeb Graphs
It is shown that the bottleneck distance is in fact good enough {\em locally}, in the sense that it is able to discriminate a Reeb graph from any other Reeb graphs in a small enough neighborhood, as efficiently as the other metrics do.
Strong Equivalence of the Interleaving and Functional Distortion Metrics for Reeb Graphs
This paper shows that the two metrics are strongly equivalent on the space of Reeb graphs, and gives an immediate proof of bottleneck stability for persistence diagrams in terms of the Reeb graph interleaving distance.
Measuring Distance between Reeb Graphs
The main result is that the functional distortion distance between two Reeb graphs is bounded from below by the bottleneck distance between both the ordinary and extended persistence diagrams for appropriate dimensions.
The Reeb Graph Edit Distance is Universal
An edit distance for Reeb graphs is defined and it is proved that it is stable and universal, meaning that it provides an upper bound to any other stable distance.
Convergence between Categorical Representations of Reeb Space and Mapper
Using tools from category theory, it is formally proved that the convergence between the Reeb space and mapper is proved in terms of an interleaving distance between their categorical representations.
Comparing Stars: On Approximating Graph Edit Distance
Three novel methods to compute the upper and lower bounds for the edit distance between two graphs in polynomial time are introduced and result shows that these methods achieve good scalability in terms of both the number of graphs and the size of graphs.
The Edit Distance for Reeb Graphs of Surfaces
A combinatorial distance for Reeb graphs of orientable surfaces in terms of the cost necessary to transform one graph into another by edit operations is defined in order to determine the stability property of these graphs.
Intrinsic Interleaving Distance for Merge Trees
It is shown that the interleaving distance is intrinsic on the space of labeled merge trees and provided an algorithm to construct metric 1-centers for collections of labeling merge trees, and it is proved that the intrinsic property of the Interleaving Distance also holds for thespace of unlabeled merge trees.
A survey of graph edit distance
The research advance of G ED is surveyed in order to provide a review of the existing literatures and offer some insights into the studies of GED.
Categorified Reeb Graphs
A natural construction for smoothing a Reeb graph to reduce its topological complexity is obtained and an ‘interleaving’ distance is defined which is stable under the perturbation of a function.