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Modeling the Functional Network for Spatial Navigation in the Human Brain
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Tight quasi-universality of Reeb graph distances.

Ulrich Bauer1, Håvard Bakke Bjerkevik2,3, Benedikt Fluhr4

  • 1Department of Mathematics and Munich Data Science Institute, Technical University of Munich (TUM), Munich, Germany.

Journal of Applied and Computational Topology
|February 17, 2025
PubMed
Summary
This summary is machine-generated.

This study proves quasi-universality for Reeb graph distances, including a novel functional contortion distance. This finding holds for contour trees and merge trees, demonstrating broad applicability in topological data analysis.

Keywords:
Contour treesFunctional contortion distanceMerge treesReeb graph distancesReeb graphsUniversal distance

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Area of Science:

  • Topological Data Analysis
  • Computational Topology

Background:

  • Reeb graphs are crucial for analyzing scalar fields.
  • Understanding distances between Reeb graphs is key for comparing topological structures.
  • Existing distances like interleaving and functional distortion have limitations.

Purpose of the Study:

  • To establish tight bi-Lipschitz bounds for Reeb graph distances.
  • To introduce and analyze a novel functional contortion distance.
  • To investigate universality properties of these distances for contour and merge trees.

Main Methods:

  • Establishing bi-Lipschitz bounds for graph distances.
  • Defining and analyzing the functional contortion distance.
  • Proving universality for contour trees and merge trees.

Main Results:

  • Tight bi-Lipschitz bounds for interleaving, functional distortion, and functional contortion distances.
  • The functional contortion distance is a novel contribution.
  • Strict universality is proven for the functional contortion distance in contour trees.
  • The functional contortion distance coincides with the interleaving distance for merge trees.

Conclusions:

  • Quasi-universality is certified for multiple Reeb graph distances.
  • The functional contortion distance offers a new tool for topological analysis.
  • Universality is achieved for contour and merge trees under specific distances.