Chen Dong, Liu Jian, Wu Jie, Wei Guo-Wei
Department of Mathematics, Michigan State University, MI, 48824, USA.
Mathematical Science Research Center, Chongqing University of Technology, Chongqing 400054, China.
Found Data Sci. 2023 Dec;5(4):558-588. doi: 10.3934/fods.2023010.
Hypergraphs are useful mathematical models for describing complex relationships among members of a structured graph, while hyperdigraphs serve as a generalization that can encode asymmetric relationships in the data. However, obtaining topological information directly from hyperdigraphs remains a challenge. To address this issue, we introduce hyperdigraph homology in this work. We also propose topological hyperdigraph Laplacians, which can extract both harmonic spectra and non-harmonic spectra from directed and internally organized data. Moreover, we introduce persistent hyperdigraph homology and persistent hyperdigraph Laplacians through filtration, enabling the capture of topological persistence and homotopic shape evolution of directed and structured data across multiple scales. The proposed methods offer new multiscale algebraic topology tools for topological data analysis.
超图是用于描述结构化图中成员之间复杂关系的有用数学模型，而超有向图则是一种泛化形式，可对数据中的不对称关系进行编码。然而，直接从超有向图中获取拓扑信息仍然是一个挑战。为了解决这个问题，我们在这项工作中引入了超有向图同调。我们还提出了拓扑超有向图拉普拉斯算子，它可以从有向且内部组织的数据中提取调和谱和非调和谱。此外，我们通过过滤引入了持久超有向图同调与持久超有向图拉普拉斯算子，从而能够在多个尺度上捕捉有向和结构化数据的拓扑持久性和同伦形状演化。所提出的方法为拓扑数据分析提供了新的多尺度代数拓扑工具。
