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Persistent hyperdigraph homology and persistent hyperdigraph Laplacians

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  • 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.

    Mathematics Subject Classification: Primary: 55N31; Secondary: 05C65, 18G85.

    Citation:

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  • Figure 1.  a Illustration of hypergraph $ \mathcal{H} $ in Example 2.7. b, c, and d Illustration of 0-hyperedges, 1-hyperedges, and 2-hyperedges, respectively. The solid black vertices indicate 0-hyperedges, purple edges indicate 1-hyperedges, and pink areas indicate 2-hyperedges. The dashed hollow circle indicates a vertex that is not 0-hyperedge

    Figure 2.  a Illustration of hyperdigraph $ \vec{\mathcal{H}} $ in Example 3.16. b, c, and d Illustration of directed 0-hyperedges, directed 1-hyperedges, and directed 2-hyperedges, respectively. The solid black vertices indicate 0-hyperedges and the dashed hollow circle indicates a vertex that is not a directed 0-hyperedge. The yellow background combined with directed edges represents directed 1-hyperedges. The green background combined with 2 consecutive directed edges represents directed 2-hyperedges. Here, the dashed arrows indicate the directed edges that do not exist

    Figure 3.  a Persistent hypergraph of 6-vertex system ($ \vec{\mathcal{H}} $). The distances between the relevant vertices are $ D_{45} = D_{12} = 6 $, $ D_{05} = D_{01} = D_{23} = D_{34} = \sqrt{5} $, and $ D_{24} = D_{15} = 4 $. b Persistent hyperdigraph ($ \mathcal{\vec{H}} $) of 6-vertex system. The distances between the corresponding vertices are the same as a

    Figure 4.  Comparison of persistent hypergraph Laplacians and persistent hyperdigraph Laplacians. The hollow square markers ($ \mathcal{\vec{H}} $) and circle markers ($ \mathcal{H} $) in each subfigure correspond to the five filtration stages in Figure 3, i.e., $ \mathcal{\vec{H}}_1 $, $ \mathcal{\vec{H}}_2 $, ..., $ \mathcal{\vec{H}}_5 $. and $ \mathcal{H}_1 $, $ \mathcal{H}_2 $, ..., $ \mathcal{H}_5 $

    Figure 5.  a Persistent hypergraph Laplacians of a 6-vertex system ($ \mathcal{H} $). All vertices come from the vertices of a regular hexagon with sides of length 2, i.e., $ D_{01} = D_{12} = D_{23} = D_{34} = D_{45} = D_{50} = 2 $. b Comparison of Betti numbers ($ \beta_{0} $, $ \beta_{1} $, and $ \beta_{2} $) and the smallest eigenvalue of the non-harmonic spectra ($ \lambda_{0} $, $ \lambda_{1} $, and $ \lambda_{2} $) of persistent hypergraph Laplacians. c Persistent hyperdigraph Laplacians of a 6-vertex system ($ \vec{\mathcal{H}} $). The distances between the corresponding vertices are the same as a. d Comparison of Betti numbers ($ \beta_{0} $, $ \beta_{1} $, and $ \beta_{2} $) and the smallest eigenvalue of the non-harmonic spectra ($ \lambda_{0} $, $ \lambda_{1} $, and $ \lambda_{2} $) of persistent hyperdigraph Laplacians. The corresponding Betti numbers can not tell the changes from $ \mathcal{H}_3 $ to $ \mathcal{H}_4 $ and $ \vec{\mathcal{H}}_3 $ to $ \vec{\mathcal{H}}_4 $, while the $ \lambda_{0} $, $ \lambda_{1} $, and $ \lambda_{2} $ can capture the difference

    Figure 6.  Illustration of the hyperdigraph Laplacian analysis and hypergraph Laplacian analysis for two B$ _7 $C$ _2 $H$ _9 $ isomers in a and b. The first column shows the structural representation of B$ _7 $C$ _2 $H$ _9 $. The second column shows the structural representations after removing the H atoms. The third column shows the hyperdigraph representations of two structures, and the last column shows the hypergraph representations of two structures. The results in the last two columns indicate that hyperdigraph Laplacians, rather than hypergraph Laplacians, can distinguish two isomers

    Figure 7.  Illustration of the persistent hyperdigraph Laplacian analysis of a protein-ligand complex (PDB ID: 1a99). a Illustration of a filtration-induced topological hyperdigraph. Only C-atoms within 4 Å from the ligand are considered for the protein structure (brown dots). The green line segment indicates the directed 1-hyperedges and the orange line indicates the directed 2-hyperedges. b The three dimensional structure of the protein-ligand complex. c and d Betti numbers $ \beta_n $ ($ n $ = 0, 1, 2) and the smallest eigenvalues of non-harmonic spectra $ \lambda_n $ ($ n $ = 0, 1, 2) of persistent hyperdigraph Laplacian for the protein-ligand complex

    Table 1.  Topological Laplacians on different objects

    Homologies notation restriction topological Laplacians
    graph $ G=(V,E) $, $ E\subseteq {\bf{P}}_{2}(V) $ none graph Laplacian
    simplicial complex $ \mathcal{K}=(V,K) $, $ K\subseteq {\bf{P}}(V) $ $ \partial_{\ast}K\subseteq K $ combinatorial Laplacian
    digraph $ G=(V,E) $, $ E\subseteq {\bf{S}}_{2}(V) $ none path Laplacian
    path complex $ \mathcal{P}=(V,P) $, $ P\subseteq {\bf{\tilde{S}}}(V) $ $ \partial_{0}P,\partial_{\infty}P\subseteq P $ path Laplacian
    hypergraph $ \mathcal{H}=(V,E) $, $ E\subseteq {\bf{P}}(V) $ none hypergraph Laplacian
    hyperdigraph $ \vec{\mathcal{H}}=(V,\vec{E}) $, $ \vec{E}\subseteq {\bf{S}}(V) $ none hyperdigraph Laplacian
     | Show Table
    DownLoad: CSV

    Table 2.  Illustration of hyperdigraph in Example 3.18

    $ n $ $ n=0 $ $ n=1 $ $ n=2 $
    $ B_{n+1} $ $ \left(\begin{array}{ccc} -1&1&0\\-1&0&1\\1&-1&0\\0&-1&1\\1&0&-1\\0&1&-1\\\end{array} \right) $ $ \left( \begin{array}{cccccc} 1&-1&0&1&0&0\\-1&1&0&0&0&1\\0&1&1&-1&0&0\\0&0&-1&1&1&0\\1&0&0&0&1&-1\\0&0&1&0&-1&1\\\end{array} \right) $ $ 6\times 0 $ empty matrix
    $ L_{n} $ $ \left( \begin{array}{ccc} 4&-2&-2\\-2&4&-2\\-2&-2&4\\\end{array} \right) $ $ \left( \begin{array}{cccccc} 5&-1&-2&0&0&-1\\-1&5&0&-1&-2&0\\-2&0&5&-1&-1&0\\0&-1&-1&5&0&-2\\0&-2&-1&0&5&-1\\-1&0&0&-2&-1&5\\\end{array} \right) $ $ \left( \begin{array}{cccccc} 3&-2&-2&1&1&0\\-2&3&1&0&-2&1\\-2&1&3&-2&0&1\\1&0&-2&3&1&-2\\1&-2&0&1&3&-2\\0&1&1&-2&-2&3\\\end{array} \right) $
    $ \beta_{n} $ 1 0 2
    $ {\bf{Spec}}(L_{n}) $ {0, 6, 6} {1, 4, 4, 6, 6, 9} {0, 0, 1, 4, 4, 9}
     | Show Table
    DownLoad: CSV

    Table 3.  Illustration of hyperdigraph in Example 3.19

    $ n $ $ n=0 $ $ n=1 $ $ n=2 $
    $ B_{n+1} $ $ 0\times 3 $ empty matrix $ \left( \begin{array}{ccc} -\frac{1}{2}&\frac{3}{2\sqrt{5}}&- \frac{\sqrt{13}}{\sqrt{10}}\\\end{array} \right) $ $ 6\times 0 $ empty matrix
    $ L_{n} $ none $ \left( \begin{array}{ccc} \frac{1}{4}&-\frac{3}{4\sqrt{5}}&\frac{\sqrt{13}}{2\sqrt{10}}\\-\frac{3}{4\sqrt{5}}&\frac{9}{20}&-\frac{3\sqrt{13}}{10\sqrt{2}}\\\frac{\sqrt{13}}{2\sqrt{10}}&-\frac{3\sqrt{13}}{10\sqrt{2}}&\frac{13}{10}\\\end{array} \right) $ 2
    $ \beta_{n} $ 0 2 0
    $ {\bf{Spec}}(L_{n}) $ none {0, 0, 2} {2}
     | Show Table
    DownLoad: CSV

    Table 4.  The Betti numbers and spectra for the volume-based filtration of hyperdigraphs in Example 4.4

    $ \mathcal{H}^{f}(t) $ Betti numbers Spectra
    $ \beta_{0} $ $ \beta_{1} $ $ \beta_{2} $ $ p=0 $ $ p=1 $ $ p=2 $
    $ t=0 $ 3 0 0 {0, 0, 0} none none
    $ t=\sqrt{2} $ 2 0 0 {0, 0, 2} {2} none
    $ t=\sqrt{3} $ 2 0 0 {0, 0, 2} {2} none
    $ t=\sqrt{5} $ 1 0 0 {0, 3, 3} {3, 3, 3} {3}
     | Show Table
    DownLoad: CSV
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