PERSISTENT HYPERDIGRAPH HOMOLOGY AND PERSISTENT HYPERDIGRAPH LAPLACIANS.

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.