  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.
