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HERMES: Persistent spectral graph software

  • * Corresponding authors: Yiying Tong and Guo-Wei Wei

    * Corresponding authors: Yiying Tong and Guo-Wei Wei 

HERMES is available online here and via GitHub.

This work was supported in part by NIH grant GM126189, NSF grants DMS-2052983, DMS-1761320, and IIS-1900473, NASA grant 80NSSC21M0023, Michigan Economic Development Corporation, George Mason University award PD45722, Bristol-Myers Squibb 65109, and Pfizer

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  • Persistent homology (PH) is one of the most popular tools in topological data analysis (TDA), while graph theory has had a significant impact on data science. Our earlier work introduced the persistent spectral graph (PSG) theory as a unified multiscale paradigm to encompass TDA and geometric analysis. In PSG theory, families of persistent Laplacian matrices (PLMs) corresponding to various topological dimensions are constructed via a filtration to sample a given dataset at multiple scales. The harmonic spectra from the null spaces of PLMs offer the same topological invariants, namely persistent Betti numbers, at various dimensions as those provided by PH, while the non-harmonic spectra of PLMs give rise to additional geometric analysis of the shape of the data. In this work, we develop an open-source software package, called highly efficient robust multidimensional evolutionary spectra (HERMES), to enable broad applications of PSGs in science, engineering, and technology. To ensure the reliability and robustness of HERMES, we have validated the software with simple geometric shapes and complex datasets from three-dimensional (3D) protein structures. We found that the smallest non-zero eigenvalues are very sensitive to data abnormality.

    Mathematics Subject Classification: Primary: 55-04; Secondary: 92-08.

    Citation:

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  • Figure 1.  Illustration of Voronoi diagram, Delaunay triangulation, and Non-Delaunay triangulation. Left chart: The Voronoi diagram and its dual Delaunay triangulation. The points set is $ P $ = {A, B, C, D, E} and the Delaunay is defined as $ \text{DT}(P) $. The blue lines tessellate the plane into Voronoi cells. The red circle are the circumcircles of triangles in $ \text{DT}(P) $. Right chart: A Non-Delaunay triangulation. Vertices E and D are in the green circumcircles, implying the right chart is an example of Non-Delaunay triangulation

    Figure 2.  Illustration of 2D Delaunay triangulation, alpha shapes, and alpha complexes for a set of 6 points A, B, C, D, E, and F. Top left: The 2D Delaunay triangulation. Top right: The alpha shape and alpha complex at filtration value $ \alpha = 0.2 $. Bottom left: The alpha shape and alpha complex at filtration value $ \alpha = 0.6 $. Bottom right: The alpha shape and alpha complex at filtration value $ \alpha = 1.0 $. Here, we use dark blue color to fill the alpha shape

    Figure 3.  The persistent barcode for a set of points as illustrated in Figure 2 that are generated from Gudhi and DioDe

    Figure 4.  The 3D structures of C$ _{20} $ and C$ _{60} $. (a) C$ _{20} $ molecule. A total of 12 pentagon rings can be found in C$ _{20} $. (b) C$ _{60} $ molecule. 12 pentagon rings and 20 hexoagon rings form the structure of C$ _{60} $

    Figure 5.  Illustration of the harmonic spectra (for Rips complex) $ \beta_0^{r, 0.05} $, $ \beta_0^{r, 0.05} $, and $ \beta_2^{r, 0.05} $ (green curves from top chart to bottom chart) and the smallest non-zero eigenvalue $ \lambda_0^{r, 0.05} $, $ \lambda_1^{r, 0.05} $, and $ \lambda_2^{r, 0.05} $ (yellow curves from top chart to bottom chart) of C$ _{20} $ molecule (the bottom left chart in Fig. 9) at different filtration values $ \alpha $ calculated from HERMES. Here, the $ x $-axis represents the radius filtration value $ r $ (unit: Å), the left-$ y $-axes represents the number of zero eigenvalues of $ \mathcal{L}_0^{r, 0.05} $, $ \mathcal{L}_1^{r, 0.05} $, and $ \mathcal{L}_1^{r, 0.05} $ from top to bottom, and the right-$ y $-axes represents the first non-zero eigenvalue of $ \mathcal{L}_0^{r, 0.05} $, $ \mathcal{L}_1^{r, 0.05} $, and $ \mathcal{L}_2^{r, 0.05} $ from top to bottom.

    Figure 6.  Illustration of the harmonic spectra (for alpha complex) $ \beta_0^{\alpha, 0.05} $, $ \beta_0^{\alpha, 0.05} $, and $ \beta_2^{\alpha, 0.05} $ (green curves from top chart to bottom chart) and the smallest non-zero eigenvalue $ \lambda_0^{\alpha, 0.05} $, $ \lambda_1^{\alpha, 0.05} $, and $ \lambda_2^{\alpha, 0.05} $ (yellow curves from top chart to bottom chart) of C$ _{20} $ molecule (the bottom left chart in Fig. 9) at different filtration value $ \alpha $ calculated from HERMES. Here, the $ x $-axis represents the radius filtration value $ \alpha $ (unit: Å), the left-$ y $-axes represents the number of zero eigenvalues of $ \mathcal{L}_0^{\alpha, 0.05} $, $ \mathcal{L}_1^{\alpha, 0.05} $, and $ \mathcal{L}_1^{\alpha, 0.05} $ from top to bottom, and the right-$ y $-axes represents the first non-zero eigenvalue of $ \mathcal{L}_0^{\alpha, 0.05} $, $ \mathcal{L}_1^{\alpha, 0.05} $, and $ \mathcal{L}_2^{\alpha, 0.05} $ from top to bottom.

    Figure 7.  Illustration of the harmonic spectra $ \beta_0^{r, 0.05} $, $ \beta_0^{r, 0.05} $, and $ \beta_2^{r, 0.05} $ (blue curves from top chart to bottom chart) and the smallest non-zero eigenvalue $ \lambda_0^{r, 0.05} $, $ \lambda_1^{r, 0.05} $, and $ \lambda_2^{r, 0.05} $ (red curves from top chart to bottom chart) of C$ _{60} $ molecule (the bottom left chart in Fig. 9) at different filtration value $ \alpha $ calculated from HERMES. Here, the $ x $-axis represents the radius filtration value $ \alpha $ (unit: $Å $), the left-$ y $-axes represents the number of zero eigenvalues of $ \mathcal{L}_0^{r, 0.05} $, $ \mathcal{L}_1^{r, 0.05} $, and $ \mathcal{L}_1^{r, 0.05} $ from top to bottom, and the right-$ y $-axes represents the first non-zero eigenvalue of $ \mathcal{L}_0^{r, 0.05} $, $ \mathcal{L}_1^{r, 0.05} $, and $ \mathcal{L}_2^{r, 0.05} $ from top to bottom.

    Figure 8.  Illustration of the harmonic spectra $ \beta_0^{\alpha, 0.05} $, $ \beta_0^{\alpha, 0.05} $, and $ \beta_2^{\alpha, 0.05} $ (green curves from top chart to bottom chart) and the smallest non-zero eigenvalue $ \lambda_0^{\alpha, 0.05} $, $ \lambda_1^{\alpha, 0.05} $, and $ \lambda_2^{\alpha, 0.05} $ (yellow curves from top chart to bottom chart) of C$ _{60} $ molecule (the bottom left chart in Fig. 9) at different filtration value $ \alpha $ calculated from HERMES. Here, the $ x $-axis represents the radius filtration value $ \alpha $ (unit: $Å $), the left-$ y $-axes represents the number of zero eigenvalues of $ \mathcal{L}_0^{\alpha, 0.05} $, $ \mathcal{L}_1^{\alpha, 0.05} $, and $ \mathcal{L}_1^{\alpha, 0.05} $ from top to bottom, and the right-$ y $-axes represents the first non-zero eigenvalue of $ \mathcal{L}_0^{\alpha, 0.05} $, $ \mathcal{L}_1^{\alpha, 0.05} $, and $ \mathcal{L}_2^{\alpha, 0.05} $ from top to bottom.

    Figure 9.  The alpha carbon network plots of 15 proteins: PDB IDs 1CCR, 1NKO, 1O08, 1OPD, 1QTO, 1R7J, 1V70, 1W2L, 1WHI, 2CG7, 2FQ3, 2HQK, 2PKT, 2VIM, and 5CYT from left to right and top to bottom. The color represents the normalized diagonal element of the accumulated Laplacian at each alpha carbon atom

    Figure 10.  Illustration of the harmonic spectra $ \beta_q^{\alpha, 0} $ (blue curve) and the smallest non-zero eigenvalue $ \lambda_q^{\alpha, 0} $ (red curve) of PDB ID 5CYT (the bottom left chart in Fig. 9) at different filtration value $ \alpha $ when $ q = 0, 1, 2 $. The $ \beta_q^{\alpha, 0} $ are calculated from Gudhi, DioDe, and HERMES, and $ \lambda_q^{\alpha, 0} $ are obtained only from HERMES. Here, the $ x $-axis represents the radius filtration value $ \alpha $ (unit: $Å $), the left-$ y $-axis represents the number of zero eigenvalues of $ \mathcal{L}_q^{\alpha, 0} $, and the right-$ y $-axis represents the first non-zero eigenvalue of $ \mathcal{L}_q^{\alpha, 0} $. Note that the harmonic spectra from three methods are indistinguishable

    Figure 11.  Illustration of the harmonic spectra $ \beta_0^{\alpha, 0.5} $, $ \beta_0^{\alpha, 0.5} $, and $ \beta_2^{\alpha, 0.5} $ (green curves from top chart to bottom chart) and the smallest non-zero eigenvalue $ \lambda_0^{\alpha, 0.5} $, $ \lambda_1^{\alpha, 0.5} $, and $ \lambda_2^{\alpha, 0.5} $ (yellow curves from top chart to bottom chart) of PDB ID 5CYT (the bottom left chart in Fig. 9) at different filtration value $ \alpha $ calculated from HERMES. Here, the $ x $-axis represents the radius filtration value $ \alpha $ (unit: $Å $), the left-$ y $-axes represents the number of zero eigenvalues of $ \mathcal{L}_0^{\alpha, 0.5} $, $ \mathcal{L}_1^{\alpha, 0.5} $, and $ \mathcal{L}_1^{\alpha, 0.5} $ from top to bottom, and the right-$ y $-axes represents the first non-zero eigenvalue of $ \mathcal{L}_0^{\alpha, 0.5} $, $ \mathcal{L}_1^{\alpha, 0.5} $, and $ \mathcal{L}_2^{\alpha, 0.5} $ from top to bottom

    Figure 12.  (a) The 3D secondary structure of PDB ID 1O08. The blue, purple, and orange colors represent helix, sheet, and random coils of PDB ID 1O08. The ball represents the alpha carbon of PDB ID 1O08. (b) Illustration of the harmonic spectra $ \beta_q^{\alpha, 0} $ (blue curve) and the smallest non-zero eigenvalue $ \lambda_q^{\alpha, 0} $ (red curve) of PDB ID 1O08 at different filtration value $ \alpha $ when $ q = 0, 1, 2 $. The $ \beta_q^{\alpha, 0} $ are calculated from Gudhi, DioDe, and HERMES, and $ \lambda_q^{\alpha, 0} $ are calculated only from HERMES. Here, the $ x $-axis represents the radius filtration value $ \alpha $ (unit: $Å $), the left-$ y $-axis represents for the number of zero eigenvalue of $ \mathcal{L}_q^{\alpha, 0} $, and the right-$ y $-axis represents for the non-zero eigenvalues of $ \mathcal{L}_q^{\alpha, 0} $. Note that the harmonic spectra from three methods are indistinguishable

    Figure 13.  Illustration of the harmonic spectra $ \beta_0^{r, 0} $, $ \beta_0^{r, 0} $, and $ \beta_2^{r, 0} $ (blue curves from top chart to bottom chart) and the smallest non-zero eigenvalue $ \lambda_0^{r, 0} $, $ \lambda_1^{r, 0} $, and $ \lambda_2^{r, 0} $ (red curves from top chart to bottom chart) of C$ _{60} $ molecule with one atom shifted (the bottom left chart in Fig. 9) at different filtration value $ \alpha $ calculated from HERMES. Here, the $ x $-axis represents the radius filtration value $ \alpha $ (unit: $Å $), the left-$ y $-axes represents the number of zero eigenvalues of $ \mathcal{L}_0^{r, 0} $, $ \mathcal{L}_1^{r, 0} $, and $ \mathcal{L}_1^{r, 0} $ from top to bottom, and the right-$ y $-axes represents the first non-zero eigenvalue of $ \mathcal{L}_0^{r, 0} $, $ \mathcal{L}_1^{r, 0} $, and $ \mathcal{L}_2^{r, 0} $ from top to bottom.

    Figure 14.  Illustration of the harmonic spectra $ \beta_q^{\alpha, 0} $ (blue curve) and the smallest non-zero eigenvalue $ \lambda_q^{\alpha, 0} $ (red curve) of PDB ID 1CCR at different filtration value $ \alpha $ when $ q = 0, 1, 2 $. The $ \beta_q^{\alpha, 0} $ are calculated from Gudhi, DioDe, and HERMES, and $ \lambda_q^{\alpha, 0} $ are obtained only from HERMES. Here, the $ x $-axis represents the radius filtration value $ \alpha $ (unit: $Å $), the left-$ y $-axis represents the number of zero eigenvalues of $ \mathcal{L}_q^{\alpha, 0} $, and the right-$ y $-axis represents the first non-zero eigenvalue of $ \mathcal{L}_q^{\alpha, 0} $. Note that the harmonic spectra from three methods are indistinguishable

    Figure 15.  Illustration of the harmonic spectra $ \beta_q^{\alpha, 0} $ (blue curve) and the smallest non-zero eigenvalue $ \lambda_q^{\alpha, 0} $ (red curve) of PDB ID 1NKO at different filtration value $ \alpha $ when $ q = 0, 1, 2 $. The $ \beta_q^{\alpha, 0} $ are calculated from Gudhi, DioDe, and HERMES, and $ \lambda_q^{\alpha, 0} $ are obtained only from HERMES. Here, the $ x $-axis represents the radius filtration value $ \alpha $ (unit: $Å $), the left-$ y $-axis represents the number of zero eigenvalues of $ \mathcal{L}_q^{\alpha, 0} $, and the right-$ y $-axis represents the first non-zero eigenvalue of $ \mathcal{L}_q^{\alpha, 0} $. Note that the harmonic spectra from three methods are indistinguishable

    Figure 16.  Illustration of the harmonic spectra $ \beta_q^{\alpha, 0} $ (blue curve) and the smallest non-zero eigenvalue $ \lambda_q^{\alpha, 0} $ (red curve) of PDB ID 1OPD at different filtration value $ \alpha $ when $ q = 0, 1, 2 $. The $ \beta_q^{\alpha, 0} $ are calculated from Gudhi, DioDe, and HERMES, and $ \lambda_q^{\alpha, 0} $ are obtained only from HERMES. Here, the $ x $-axis represents the radius filtration value $ \alpha $ (unit: $Å $), the left-$ y $-axis represents the number of zero eigenvalues of $ \mathcal{L}_q^{\alpha, 0} $, and the right-$ y $-axis represents the first non-zero eigenvalue of $ \mathcal{L}_q^{\alpha, 0} $. Note that the harmonic spectra from three methods are indistinguishable

    Figure 17.  Illustration of the harmonic spectra $ \beta_q^{\alpha, 0} $ (blue curve) and the smallest non-zero eigenvalue $ \lambda_q^{\alpha, 0} $ (red curve) of PDB ID 1QTO at different filtration value $ \alpha $ when $ q = 0, 1, 2 $. The $ \beta_q^{\alpha, 0} $ are calculated from Gudhi, DioDe, and HERMES, and $ \lambda_q^{\alpha, 0} $ are obtained only from HERMES. Here, the $ x $-axis represents the radius filtration value $ \alpha $ (unit: $Å $), the left-$ y $-axis represents the number of zero eigenvalues of $ \mathcal{L}_q^{\alpha, 0} $, and the right-$ y $-axis represents the first non-zero eigenvalue of $ \mathcal{L}_q^{\alpha, 0} $. Note that the harmonic spectra from three methods are indistinguishable

    Figure 18.  Illustration of the harmonic spectra $ \beta_q^{\alpha, 0} $ (blue curve) and the smallest non-zero eigenvalue $ \lambda_q^{\alpha, 0} $ (red curve) of PDB ID 1R7J at different filtration value $ \alpha $ when $ q = 0, 1, 2 $. The $ \beta_q^{\alpha, 0} $ are calculated from Gudhi, DioDe, and HERMES, and $ \lambda_q^{\alpha, 0} $ are obtained only from HERMES. Here, the $ x $-axis represents the radius filtration value $ \alpha $ (unit: $Å $), the left-$ y $-axis represents the number of zero eigenvalues of $ \mathcal{L}_q^{\alpha, 0} $, and the right-$ y $-axis represents the first non-zero eigenvalue of $ \mathcal{L}_q^{\alpha, 0} $. Note that the harmonic spectra from three methods are indistinguishable

    Figure 19.  Illustration of the harmonic spectra $ \beta_q^{\alpha, 0} $ (blue curve) and the smallest non-zero eigenvalue $ \lambda_q^{\alpha, 0} $ (red curve) of PDB ID 1V70 at different filtration value $ \alpha $ when $ q = 0, 1, 2 $. The $ \beta_q^{\alpha, 0} $ are calculated from Gudhi, DioDe, and HERMES, and $ \lambda_q^{\alpha, 0} $ are obtained only from HERMES. Here, the $ x $-axis represents the radius filtration value $ \alpha $ (unit: $Å $), the left-$ y $-axis represents the number of zero eigenvalues of $ \mathcal{L}_q^{\alpha, 0} $, and the right-$ y $-axis represents the first non-zero eigenvalue of $ \mathcal{L}_q^{\alpha, 0} $. Note that the harmonic spectra from three methods are indistinguishable

    Figure 20.  Illustration of the harmonic spectra $ \beta_q^{\alpha, 0} $ (blue curve) and the smallest non-zero eigenvalue $ \lambda_q^{\alpha, 0} $ (red curve) of PDB ID 1W2L at different filtration value $ \alpha $ when $ q = 0, 1, 2 $. The $ \beta_q^{\alpha, 0} $ are calculated from Gudhi, DioDe, and HERMES, and $ \lambda_q^{\alpha, 0} $ are obtained only from HERMES. Here, the $ x $-axis represents the radius filtration value $ \alpha $ (unit: $Å $), the left-$ y $-axis represents the number of zero eigenvalues of $ \mathcal{L}_q^{\alpha, 0} $, and the right-$ y $-axis represents the first non-zero eigenvalue of $ \mathcal{L}_q^{\alpha, 0} $. Note that the harmonic spectra from three methods are indistinguishable

    Figure 21.  Illustration of the harmonic spectra $ \beta_q^{\alpha, 0} $ (blue curve) and the smallest non-zero eigenvalue $ \lambda_q^{\alpha, 0} $ (red curve) of PDB ID 1WHI at different filtration value $ \alpha $ when $ q = 0, 1, 2 $. The $ \beta_q^{\alpha, 0} $ are calculated from Gudhi, DioDe, and HERMES, and $ \lambda_q^{\alpha, 0} $ are obtained only from HERMES. Here, the $ x $-axis represents the radius filtration value $ \alpha $ (unit: $Å $), the left-$ y $-axis represents the number of zero eigenvalues of $ \mathcal{L}_q^{\alpha, 0} $, and the right-$ y $-axis represents the first non-zero eigenvalue of $ \mathcal{L}_q^{\alpha, 0} $. Note that the harmonic spectra from three methods are indistinguishable

    Figure 22.  Illustration of the harmonic spectra $ \beta_q^{\alpha, 0} $ (blue curve) and the smallest non-zero eigenvalue $ \lambda_q^{\alpha, 0} $ (red curve) of PDB ID 2CG7 at different filtration value $ \alpha $ when $ q = 0, 1, 2 $. The $ \beta_q^{\alpha, 0} $ are calculated from Gudhi, DioDe, and HERMES, and $ \lambda_q^{\alpha, 0} $ are obtained only from HERMES. Here, the $ x $-axis represents the radius filtration value $ \alpha $ (unit: $Å $), the left-$ y $-axis represents the number of zero eigenvalues of $ \mathcal{L}_q^{\alpha, 0} $, and the right-$ y $-axis represents the first non-zero eigenvalue of $ \mathcal{L}_q^{\alpha, 0} $. Note that the harmonic spectra from three methods are indistinguishable

    Figure 23.  Illustration of the harmonic spectra $ \beta_q^{\alpha, 0} $ (blue curve) and the smallest non-zero eigenvalue $ \lambda_q^{\alpha, 0} $ (red curve) of PDB ID 2FQ3 at different filtration value $ \alpha $ when $ q = 0, 1, 2 $. The $ \beta_q^{\alpha, 0} $ are calculated from Gudhi, DioDe, and HERMES, and $ \lambda_q^{\alpha, 0} $ are obtained only from HERMES. Here, the $ x $-axis represents the radius filtration value $ \alpha $ (unit: $Å $), the left-$ y $-axis represents the number of zero eigenvalues of $ \mathcal{L}_q^{\alpha, 0} $, and the right-$ y $-axis represents the first non-zero eigenvalue of $ \mathcal{L}_q^{\alpha, 0} $. Note that the harmonic spectra from three methods are indistinguishable

    Figure 24.  Illustration of the harmonic spectra $ \beta_q^{\alpha, 0} $ (blue curve) and the smallest non-zero eigenvalue $ \lambda_q^{\alpha, 0} $ (red curve) of PDB ID 2HQK at different filtration value $ \alpha $ when $ q = 0, 1, 2 $. The $ \beta_q^{\alpha, 0} $ are calculated from Gudhi, DioDe, and HERMES, and $ \lambda_q^{\alpha, 0} $ are obtained only from HERMES. Here, the $ x $-axis represents the radius filtration value $ \alpha $ (unit: $Å $), the left-$ y $-axis represents the number of zero eigenvalues of $ \mathcal{L}_q^{\alpha, 0} $, and the right-$ y $-axis represents the first non-zero eigenvalue of $ \mathcal{L}_q^{\alpha, 0} $. Note that the harmonic spectra from three methods are indistinguishable

    Figure 25.  Illustration of the harmonic spectra $ \beta_q^{\alpha, 0} $ (blue curve) and the smallest non-zero eigenvalue $ \lambda_q^{\alpha, 0} $ (red curve) of PDB ID 2PKT at different filtration value $ \alpha $ when $ q = 0, 1, 2 $. The $ \beta_q^{\alpha, 0} $ are calculated from Gudhi, DioDe, and HERMES, and $ \lambda_q^{\alpha, 0} $ are obtained only from HERMES. Here, the $ x $-axis represents the radius filtration value $ \alpha $ (unit: $Å $), the left-$ y $-axis represents the number of zero eigenvalues of $ \mathcal{L}_q^{\alpha, 0} $, and the right-$ y $-axis represents the first non-zero eigenvalue of $ \mathcal{L}_q^{\alpha, 0} $. Note that the harmonic spectra from three methods are indistinguishable

    Figure 26.  Illustration of the harmonic spectra $ \beta_q^{\alpha, 0} $ (blue curve) and the smallest non-zero eigenvalue $ \lambda_q^{\alpha, 0} $ (red curve) of PDB ID 2VIM at different filtration value $ \alpha $ when $ q = 0, 1, 2 $. The $ \beta_q^{\alpha, 0} $ are calculated from Gudhi, DioDe, and HERMES, and $ \lambda_q^{\alpha, 0} $ are obtained only from HERMES. Here, the $ x $-axis represents the radius filtration value $ \alpha $ (unit: $Å $), the left-$ y $-axis represents the number of zero eigenvalues of $ \mathcal{L}_q^{\alpha, 0} $, and the right-$ y $-axis represents the first non-zero eigenvalue of $ \mathcal{L}_q^{\alpha, 0} $. Note that the harmonic spectra from three methods are indistinguishable

    Table 1.  The matrix representation of $ q $-boundary operator and its $ q $th-order persistent Laplacian with corresponding dimension, rank, nullity, and spectra from alpha complex $ K_{0.6} \to K_{0.6} $

     | Show Table
    DownLoad: CSV

    Table 2.  The matrix representation of $ q $-boundary operator and its $ q $th-order persistent Laplacian with corresponding dimension, rank, nullity, and spectra from alpha complex $ K_{0.2} \to K_{0.6} $

     | Show Table
    DownLoad: CSV
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