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Related Experiment Video

Updated: Jun 11, 2025

Dissection, MicroCT Scanning and Morphometric Analyses of the Baculum
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PERSISTENT HYPERDIGRAPH HOMOLOGY AND PERSISTENT HYPERDIGRAPH LAPLACIANS.

Dong Chen1, Jian Liu2,1, Jie Wu3

  • 1Department of Mathematics, Michigan State University, MI, 48824, USA.

Foundations of Data Science (Springfield, Mo.)
|October 7, 2024
PubMed
Summary

This study introduces hyperdigraph homology and Laplacians for analyzing complex data structures. These new algebraic topology tools enable multiscale analysis of directed relationships and data evolution.

Keywords:
18G85Primary: 55N31Secondary: 05C65Topological hyperdigraphTopological hyperdigraph Laplaciansfiltrationhomologypersistence

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Area of Science:

  • Algebraic Topology
  • Topological Data Analysis
  • Graph Theory

Background:

  • Hypergraphs model complex relationships in structured data.
  • Hyperdigraphs generalize hypergraphs to include asymmetric relationships.
  • Extracting topological information from hyperdigraphs is challenging.

Purpose of the Study:

  • Introduce novel methods for hyperdigraph topological analysis.
  • Develop tools for extracting spectral information from directed data.
  • Enable multiscale topological persistence and shape evolution analysis.

Main Methods:

  • Introduce hyperdigraph homology.
  • Propose topological hyperdigraph Laplacians.
  • Develop persistent hyperdigraph homology and Laplacians via filtration.

Main Results:

  • Novel algebraic topology tools for hyperdigraphs.
  • Methods for extracting harmonic and non-harmonic spectra.
  • Techniques for capturing topological persistence and shape evolution.

Conclusions:

  • The proposed methods provide new multiscale algebraic topology tools.
  • These tools advance topological data analysis for directed and structured data.
  • Enables deeper understanding of complex data relationships across scales.