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图和超图上路径的持久狄拉克（指标）

PERSISTENT DIRAC OF PATHS ON DIGRAPHS AND HYPERGRAPHS.

作者信息

Suwayyid Faisal, Wei Guo-Wei

机构信息

Department of Mathematics, King Fahd University of Petroleum and Minerals, Dhahran 31261, KSA.

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

出版信息

Found Data Sci. 2024 Jun;6(2):124-153. doi: 10.3934/fods.2024001.

DOI:10.3934/fods.2024001
PMID:39640928
原文链接:https://pmc.ncbi.nlm.nih.gov/articles/PMC11620167/
Abstract

This work introduces the development of path Dirac and hypergraph Dirac operators, along with an exploration of their persistence. These operators excel in distinguishing between harmonic and non-harmonic spectra, offering valuable insights into the subcomplexes within these structures. The paper showcases the functionality of these operators through a series of examples in various contexts. An essential facet of this research involves examining the operators' sensitivity to filtration, emphasizing their capacity to adapt to topological changes. The paper also explores a significant application of persistent path Dirac and persistent hypergraph Dirac in molecular science, specifically in analyzing molecular structures. The study introduces strict preorders derived from molecular structures, which generate graphs and digraphs with intricate path structures. The depth of information within these path complexes reflects the complexity of different preorder classes influenced by molecular structures. This characteristic underscores the effectiveness of these tools in the realm of topological data analysis.

摘要

这项工作介绍了路径狄拉克算子和超图狄拉克算子的发展，以及对它们持久性的探索。这些算子在区分调和谱和非调和谱方面表现出色，为这些结构中的子复形提供了有价值的见解。本文通过一系列不同背景下的例子展示了这些算子的功能。这项研究的一个重要方面是考察算子对过滤的敏感性，强调它们适应拓扑变化的能力。本文还探讨了持久路径狄拉克算子和持久超图狄拉克算子在分子科学中的一个重要应用，特别是在分析分子结构方面。该研究引入了从分子结构导出的严格预序，这些预序生成了具有复杂路径结构的图和有向图。这些路径复形中的信息深度反映了受分子结构影响的不同预序类别的复杂性。这一特性突出了这些工具在拓扑数据分析领域的有效性。

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