Wang Rui, Wei Guo-Wei
Department of Mathematics, Michigan State University, MI 48824, USA.
Department of Electrical and Computer Engineering, Michigan State University, MI 48824, USA.
Found Data Sci. 2023 Mar;5(1):26-55. doi: 10.3934/fods.2022015.
Path homology proposed by S.-T.Yau and his co-workers provides a new mathematical model for directed graphs and networks. Persistent path homology (PPH) extends the path homology with filtration to deal with asymmetry structures. However, PPH is constrained to purely topological persistence and cannot track the homotopic shape evolution of data during filtration. To overcome the limitation of PPH, persistent path Laplacian (PPL) is introduced to capture the shape evolution of data. PPL's harmonic spectra fully recover PPH's topological persistence and its non-harmonic spectra reveal the homotopic shape evolution of data during filtration.
丘成桐及其同事提出的路径同调为有向图和网络提供了一种新的数学模型。持久路径同调（PPH）通过过滤扩展了路径同调，以处理不对称结构。然而，PPH仅限于纯粹的拓扑持久性，无法跟踪过滤过程中数据的同伦形状演变。为了克服PPH的局限性，引入了持久路径拉普拉斯算子（PPL）来捕捉数据的形状演变。PPL的调和谱完全恢复了PPH的拓扑持久性，其非调和谱揭示了过滤过程中数据的同伦形状演变。
