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PERSISTENT INTERACTION TOPOLOGY IN DATA ANALYSIS.

Jian Liu1, Dong Chen2, Guo-Wei Wei2,3,4

  • 1Mathematical Science Research Center, Chongqing University of Technology, Chongqing 400054, China.

Foundations of Data Science (Springfield, Mo.)
|March 6, 2026
PubMed
Summary
This summary is machine-generated.

We introduce persistent interaction homology and Laplacian to analyze complex systems by focusing on individual elements. These novel topological data analysis methods enhance understanding of interactive data, with applications in molecular science.

Keywords:
Primary: 55N31Secondary: 62R40, 68T09Topological data analysisinteraction Laplacianinteraction Vietoris-Rips complexinteraction homologytopological persistence

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Area of Science:

  • Mathematics
  • Data Science
  • Computational Topology

Background:

  • Topological data analysis (TDA) uses persistent homology and Laplacians to extract features from data.
  • Traditional methods like simplicial complexes lack localized descriptions for individual elements in complex systems.
  • Existing TDA techniques struggle with heterogeneously interactive data and emphasizing specific system components.

Purpose of the Study:

  • Introduce persistent interaction homology and persistent interaction Laplacian for localized topological analysis.
  • Develop methods to emphasize individual interacting elements within complex systems.
  • Extend TDA to better analyze interactive datasets and specific data points.

Main Methods:

  • Developed persistent interaction homology and persistent interaction Laplacian.
  • Demonstrated the stability of persistent interaction homology as a persistent module.
  • Constructed persistent interaction Vietoris-Rips complexes for discrete point sets.
  • Computed interaction homology and interaction Laplacians for these complexes.

Main Results:

  • The new methods provide localized topological descriptions for individual elements.
  • Persistent interaction homology is shown to be a stable persistent module.
  • The framework is applicable to finite discrete point sets in Euclidean space.
  • Demonstrated utility in analyzing molecular data.

Conclusions:

  • Persistent interaction homology and Laplacian offer powerful tools for analyzing complex, interactive data.
  • These methods enable a focus on individual elements, overcoming limitations of classical TDA.
  • The approach shows significant promise for data science applications, particularly in chemistry and materials science.