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Persistent Topological Laplacians—A Survey

Author

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  • Xiaoqi Wei

    (Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA)

  • Guo-Wei Wei

    (Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA
    Department of Electrical and Computer Engineering, Michigan State University, East Lansing, MI 48824, USA
    Department of Biochemistry and Molecular Biology, Michigan State University, East Lansing, MI 48824, USA)

Abstract

Persistent topological Laplacians constitute a new class of tools in topological data analysis (TDA). They are motivated by the necessity to address challenges encountered in persistent homology when handling complex data. These Laplacians combine multiscale analysis with topological techniques to characterize the topological and geometrical features of functions and data. Their kernels fully retrieve the topological invariants of corresponding persistent homology, while their non-harmonic spectra provide supplementary information. Persistent topological Laplacians have demonstrated superior performance over persistent homology in the analysis of large-scale protein engineering datasets. In this survey, we offer a pedagogical review of persistent topological Laplacians formulated in various mathematical settings, including simplicial complexes, path complexes, flag complexes, digraphs, hypergraphs, hyperdigraphs, cellular sheaves, and N -chain complexes.

Suggested Citation

  • Xiaoqi Wei & Guo-Wei Wei, 2025. "Persistent Topological Laplacians—A Survey," Mathematics, MDPI, vol. 13(2), pages 1-29, January.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:2:p:208-:d:1563702
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    References listed on IDEAS

    as
    1. Shenglong Hu & Liqun Qi, 2015. "The Laplacian of a uniform hypergraph," Journal of Combinatorial Optimization, Springer, vol. 29(2), pages 331-366, February.
    2. Ozan Candogan & Ishai Menache & Asuman Ozdaglar & Pablo A. Parrilo, 2011. "Flows and Decompositions of Games: Harmonic and Potential Games," Mathematics of Operations Research, INFORMS, vol. 36(3), pages 474-503, August.
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