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Computing Persistent Homology by Spanning Trees and Critical Simplices.

Dinghua Shi1, Zhifeng Chen2, Chuang Ma3

  • 1Department of Mathematics, College of Science, Shanghai University, Shanghai, China.

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Summary
This summary is machine-generated.

This study introduces a new method using simplicial networks and spanning trees to analyze complex data. It efficiently calculates topological features and cavities, offering a powerful tool for data analysis.

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Area of Science:

  • Data Science
  • Computational Topology
  • Network Analysis

Background:

  • Topological data analysis (TDA) extracts information from high-dimensional data using persistent homology.
  • Persistent homology identifies topological features across various scales, revealing relationships within datasets.
  • Simplicial networks, comprising all-order simplices, are crucial for this analysis.

Purpose of the Study:

  • To develop a novel method for analyzing simplicial networks based on discrete Morse functions and critical simplices.
  • To efficiently compute topological invariants like Betti numbers and cavity compositions.
  • To demonstrate the effectiveness and feasibility of the proposed method through examples.

Main Methods:

  • Representing nested simplicial subnetworks as a discrete Morse function.
  • Developing a method based on critical simplices and searching all-order spanning trees.
  • Calculating Morse function values, Betti numbers, and cavity structures.

Main Results:

  • The new method achieves the theoretical minimum number of critical simplices.
  • It enables rapid calculation of Betti numbers and the composition of all-order cavities.
  • Comparative analysis with existing methods confirms its effectiveness and feasibility.

Conclusions:

  • The proposed method offers an efficient and feasible approach to topological data analysis.
  • It provides a robust way to extract topological features and understand data structure.
  • This technique enhances the application of persistent homology in complex datasets.