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Combinatorial and Hodge Laplacians: Similarities and Differences.

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Summary

The boundary-induced graph (BIG) Laplacian bridges the gap between combinatorial and Hodge Laplacians for data analysis. BIG Laplacians enable comparisons of spectral properties for discrete and continuous data, aiding shape and topology characterization.

Keywords:
05C5020G1058A14Hodge Laplaciansalgebraic topologydifferential geometryspectral geometryspectral graph theory

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Area of Science:

  • Spectral geometry
  • Combinatorial graph theory
  • Discrete exterior calculus

Background:

  • The Hodge Laplacian and combinatorial Laplacian are crucial in spectral geometry and graph theory, respectively.
  • Both Laplacians reveal data's topological dimension and geometric shape, and are used in diffusion and harmonic measure minimization.
  • Existing Laplacians have different definitions and data applicability, hindering direct comparison and analysis of vector fields.

Purpose of the Study:

  • To bridge the gap between combinatorial and Hodge Laplacians for discretizing continuous manifolds with boundaries.
  • To introduce boundary-induced graph (BIG) Laplacians for comparing discrete and continuous data.
  • To examine similarities and differences among combinatorial, BIG, and Hodge Laplacians.

Main Methods:

  • Introduction of boundary-induced graph (BIG) Laplacians using discrete exterior calculus (DEC).
  • Definition of BIG Laplacians on discrete domains with boundary conditions for topology and shape characterization.
  • Experimental analysis using Eulerian representation of 3D domains as level-set functions on regular grids.

Main Results:

  • BIG Laplacians facilitate comparison between combinatorial and Hodge Laplacians.
  • Experimental conditions for the convergence of BIG Laplacian eigenvalues to Hodge Laplacian eigenvalues were identified for elementary shapes.
  • The study clarifies the relationship between different Laplacian types in data analysis.

Conclusions:

  • BIG Laplacians offer a unified framework for analyzing discrete and continuous data.
  • The findings advance the application of spectral methods in geometry and graph theory.
  • This work enables more robust characterization of data topology and shape using discrete Laplacians.