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Updated: Oct 21, 2025

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HERMES: PERSISTENT SPECTRAL GRAPH SOFTWARE.

Rui Wang1, Rundong Zhao2, Emily Ribando-Gros2

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

Foundations of Data Science (Springfield, Mo.)
|September 6, 2021
PubMed
Summary
This summary is machine-generated.

Persistent spectral graph (PSG) theory unifies topological data analysis and geometric analysis. The new HERMES software package enables broad applications, revealing data abnormalities through eigenvalue sensitivity.

Keywords:
Persistent homologyPrimary: 55-04Secondary: 92-08persistent Laplaciansimultaneous geometricspectral data analysisspectral graph theorytopological analysestopological data analysis

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Area of Science:

  • Topological Data Analysis
  • Graph Theory
  • Data Science
  • Geometric Analysis

Background:

  • Persistent homology (PH) is a key tool in topological data analysis (TDA).
  • Graph theory significantly impacts data science.
  • Persistent spectral graph (PSG) theory unifies TDA and geometric analysis.
  • Families of persistent Laplacian matrices (PLMs) are constructed via filtration for multiscale data sampling.

Purpose of the Study:

  • Develop an open-source software package, HERMES, for broad applications of PSG theory.
  • Enable scientists, engineers, and technologists to utilize PSG theory.
  • Validate the reliability and robustness of the HERMES software.

Main Methods:

  • Constructing families of persistent Laplacian matrices (PLMs) across topological dimensions.
  • Utilizing filtrations to sample datasets at multiple scales.
  • Analyzing harmonic and non-harmonic spectra of PLMs.
  • Developing the HERMES open-source software package.
  • Validating HERMES with simple geometric shapes and 3D protein structures.

Main Results:

  • PSG theory, via PLMs, provides persistent Betti numbers (topological invariants) and additional geometric insights.
  • The HERMES software package is developed and validated for reliability and robustness.
  • Smallest non-zero eigenvalues derived from PLMs are highly sensitive to data abnormalities.
  • HERMES enables broad applications of PSG theory in science, engineering, and technology.

Conclusions:

  • The HERMES software package facilitates the application of PSG theory, unifying TDA and geometric analysis.
  • HERMES offers a robust tool for multiscale data analysis, revealing both topological and geometric features.
  • The sensitivity of eigenvalues to data abnormalities highlights HERMES' potential for anomaly detection.