HERMES: Persistent spectral graph software

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March 2021, 3(1): 67-97. doi: 10.3934/fods.2021006

HERMES: Persistent spectral graph software

Rui Wang ^1, , Rundong Zhao ^2, , Emily Ribando-Gros ^2, , Jiahui Chen ^1, , Yiying Tong ^2,, and Guo-Wei Wei ^3,,

1.	Department of Mathematics, Michigan State University, MI 48824, USA
2.	Department of Computer Science and Engineering, Michigan State University, MI 48824, USA
3.	Department of Mathematics, Department of Electrical and Computer Engineering, Department of Biochemistry and Molecular Biology, Michigan State University, MI 48824, USA

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

HERMES is available online here and via GitHub.

Received December 2020 Revised February 2021 Published March 2021 Early access March 2021

Fund Project: 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

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.

Keywords: Persistent homology, persistent Laplacian, spectral graph theory, topological data analysis, spectral data analysis, simultaneous geometric and topological analyses.

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

Citation: Rui Wang, Rundong Zhao, Emily Ribando-Gros, Jiahui Chen, Yiying Tong, Guo-Wei Wei. HERMES: Persistent spectral graph software. Foundations of Data Science, 2021, 3 (1) : 67-97. doi: 10.3934/fods.2021006

References:

[1]	H. Adams, A. Tausz and M. Vejdemo-Johansson, JavaPlex: A research software package for persistent (co) homology, in International Congress on Mathematical Software, Lecture Notes in Computer Science, 8592, Springer, 2014, 129-136. doi: 10.1007/978-3-662-44199-2_23.
[2]	S. G. Aksoy, C. Joslyn, C. O. Marrero, B. Praggastis and E. Purvine, Hypernetwork science via high-order hypergraph walks, EPJ Data Science, 9 (2020). doi: 10.1140/epjds/s13688-020-00231-0.
[3]	F. Aurenhammer, R. Klein and D.-T. Lee, Voronoi Diagrams and Delaunay Triangulations, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013. doi: 10.1142/8685.
[4]	U. Bauer, Ripser: A lean C++ code for the computation of Vietoris-Rips persistence barcodes, 2017. Software available from: https://github.com/Ripser/ripser.
[5]	U. Bauer, M. Kerber and J. Reininghaus, DIPHA (A distributed persistent homology algorithm), 2014. Software available from: https://github.com/DIPHA/dipha.
[6]	S. Bressan, J. Li, S. Ren and J. Wu, The embedded homology of hypergraphs and applications, Asian J. Math, 23 (2019), 479-500. doi: 10.4310/AJM.2019.v23.n3.a6.
[7]	P. Bubenik and P. T. Kim, A statistical approach to persistent homology, Homology Homotopy Appl., 9 (2007), 337-362. doi: 10.4310/HHA.2007.v9.n2.a12.
[8]	Z. Cang and G.-W. Wei, TopologyNet: Topology based deep convolutional and multi-task neural networks for biomolecular property predictions, PLoS Computational Biology, 13 (2017). doi: 10.1371/journal.pcbi.1005690.
[9]	G. Carlsson, V. De Silva and D. Morozov, Zigzag persistent homology and real-valued functions, in Proceedings of the Twenty-Fifth Annual Symposium on Computational Geometry, ACM, 2009, 247-256. doi: 10.1145/1542362.1542408.
[10]	G. Carlsson, A. Zomorodian, A. Collins and L. Guibas, Persistence barcodes for shapes, International J. Shape Modeling, 11 (2005), 149-187. doi: 10.1142/S0218654305000761.
[11]	J. Cheeger, A lower bound for the smallest eigenvalue of the Laplacian, in Problems in Analysis, Princeton Univ. Press, Princeton, NJ, 1970, 195-199. doi: 10.1515/9781400869312-013.
[12]	J. Chen, R. Zhao, Y. Tong and G.-W. Wei, Evolutionary de Rham-Hodge method, Discrete Contin. Dyn. Syst. Ser. B, (2020). doi: 10.3934/dcdsb.2020257.
[13]	F. R. Chung, Spectral Graph Theory, CBMS Regional Conference Series in Mathematics, 92, American Mathematical Society, Providence, RI, 1997.
[14]	M.-V. Ciocanel, R. Juenemann, A. T. Dawes and S. A. McKinley, Topological data analysis approaches to uncovering the timing of ring structure onset in filamentous networks, Bull. Math. Biol., 83 (2021), 21pp. doi: 10.1007/s11538-020-00847-3.
[15]	V. de Silva and R. Ghrist, Coverage in sensor networks via persistent homology, Algebr. Geom. Topol., 7 (2007), 339-358. doi: 10.2140/agt.2007.7.339.
[16]	B. Delaunay, Sur la sphère vide, Izv. Akad. Nauk SSSR, Otdelenie Matematicheskii i Estestvennyka Nauk, 7 (1934), 793-800.
[17]	T. K. Dey, F. Fan and Y. Wang, Computing topological persistence for simplicial maps, in Computational Geometry (SoCG'14), ACM, New York, 2014, 345-354. doi: 10.1145/2582112.2582165.
[18]	B. Eckmann, Harmonische funktionen und Randwertaufgaben in einem Komplex, Comment. Math. Helv., 17 (1945), 240-255.
[19]	H. Edelsbrunner, Alpha shapes - A survey, Tessellations in the Sciences, 27 (2010), 1-25. Available from: https://pub.ist.ac.at/edels/Papers/2011-B-03-AlphaShapes.pdf.
[20]	H. Edelsbrunner and J. Harer, Persistent homology - A survey, in Surveys on Discrete and Computational Geometry, Contemp. Math., 453, Amer. Math. Soc., Providence, RI, 2008, 257-282.
[21]	B. T. Fasy, J. Kim, F. Lecci, C. Maria, D. L. Millman and M. J. Kim, Package (TDA), 2019.
[22]	J. Friedman, Computing Betti numbers via combinatorial Laplacians, Algorithmica, 21 (1998), 331-346. doi: 10.1007/PL00009218.
[23]	C. Giusti, E. Pastalkova, C. Curto and V. Itskov, Clique topology reveals intrinsic geometric structure in neural correlations, Proc. Natl. Acad. Sci. USA, 112 (2015), 13455-13460. doi: 10.1073/pnas.1506407112.
[24]	D. Hernández Serrano, J. Hernaández-Serrano and D. Sánchez Gómez, Simplicial degree in complex networks. Applications of topological data analysis to network science, Chaos Solitons Fractals, 137 (2020), 21pp. doi: 10.1016/j.chaos.2020.109839.
[25]	T. Kaczynski, K. Mischaikow and M. Mrozek, Computational Homology, Applied Mathematical Sciences, 157, Springer-Verlag, New York, 2004. doi: 10.1007/b97315.
[26]	F. W. Kamber and P. Tondeur, De Rham-Hodge theory for Riemannian foliations, Math. Ann., 277 (1987), 415-431. doi: 10.1007/BF01458323.
[27]	M. Kerber and H. Edelsbrunner, The medusa of spatial sorting: 3D kinetic alpha complexes and implementation, preprint, arXiv:1209.5434.
[28]	Y. Lee, S. D. Barthel, P. Dłotko, S. Mohamad Moosavi, K. Hess and B. Smit, Quantifying similarity of pore-geometry in nanoporous materials, Nature Communications, 8 (2017). doi: 10.1038/ncomms15396.
[29]	V. Maroulas, C. P. Micucci and F. Nasrin, Bayesian topological learning for classifying the structure of biological networks, preprint, arXiv:2009.11974.
[30]	J. May, Multivariate Analysis, Scientific e-Resources, 2018.
[31]	F. Mémoli, Z. Wan and Y. Wang, Persistent Laplacians: Properties, algorithms and implications, preprint, arXiv:2012.02808.
[32]	Z. Meng, D. Vijay Anand, Y. Lu, J. Wu and K. Xia, Weighted persistent homology for biomolecular data analysis, Scientific Reports, 10 (2020), 1-15. doi: 10.1038/s41598-019-55660-3.
[33]	Z. Meng and K. Xia, Persistent spectral based machine learning (PerSpect ML) for drug design, preprint, arXiv:2002.00582.
[34]	K. Mischaikow and V. Nanda, Morse theory for filtrations and efficient computation of persistent homology, Discrete Comput. Geom., 50 (2013), 330-353. doi: 10.1007/s00454-013-9529-6.
[35]	D. Morozov, Dionysus Software, 2012.
[36]	D. Morozov and P. Skraba, DioDe Software, 2017.
[37]	D. Nguyen and G.-W. Wei, AGL-Score: Algebraic graph learning score for protein-ligand binding scoring, ranking, docking, and screening, J. Chemical Information Modeling, 59 (2019), 3291-3304. doi: 10.1021/acs.jcim.9b00334.
[38]	D. D. Nguyen, Z. Cang, K. Wu, M. Wang, Y. Cao and G.-W. Wei, Mathematical deep learning for pose and binding affinity prediction and ranking in D3R Grand Challenges, J. Comput. Aided Mol. Des., 33 (2019), 71-82. doi: 10.1007/s10822-018-0146-6.
[39]	Gudhi Project, GUDHI User and Reference Manual, 2015.
[40]	I. Sgouralis, A. Nebenführ and V. Maroulas, A Bayesian topological framework for the identification and reconstruction of subcellular motion, SIAM J. Imaging Sci., 10 (2017), 871-899. doi: 10.1137/16M1095755.
[41]	D. A. Spielman, Spectral graph theory and its applications, 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07), IEEE, 2007, 29-38. doi: 10.1109/FOCS.2007.56.
[42]	J. Townsend, C. P. Micucci, J. H. Hymel, V. Maroulas and K. D. Vogiatzis, Representation of molecular structures with persistent homology for machine learning applications in chemistry, Nature Communications, 11 (2020). doi: 10.1038/s41467-020-17035-5.
[43]	G. Voronoi, Nouvelles applications des paramètres continus à la théorie des formes quadratiques. Premier mémoire. Sur quelques propriétés des formes quadratiques positives parfaites, J. Reine Angew. Math., 133 (1908), 97-102. doi: 10.1515/crll.1908.133.97.
[44]	R. Wang, D. D. Nguyen and G.-W. Wei, Persistent spectral graph, Int. J. Numer. Methods Biomed. Eng., 36 (2020), 27pp. doi: 10.1002/cnm.3376.
[45]	K. Xia, K. Opron and G.-W. Wei, Multiscale Gaussian network model (mGNM) and multiscale anisotropic network model (mANM), J. Chem. Phys., 143 (2015). doi: 10.1063/1.4936132.
[46]	K. Xia and G.-W. Wei, Persistent homology analysis of protein structure, flexibility, and folding, Int. J. Numer. Methods Biomed. Eng., 30 (2014), 814-844. doi: 10.1002/cnm.2655.
[47]	R. Zhao, M. Desbrun, G.-W. Wei and Y. Tong, 3D hodge decompositions of edge-and face-based vector fields., ACM Transactions on Graphics (TOG), 38 (2019), 1-13. doi: 10.1145/3355089.3356546.
[48]	R. Zhao, M. Wang, J. Chen, Y. Tong and G.-W. Wei, The de Rham-Hodge analysis and modeling of biomolecules, Bull. Math. Biol., 82 (2020), 38pp. doi: 10.1007/s11538-020-00783-2.
[49]	A. Zomorodian and G. Carlsson, Computing persistent homology, Discrete Comput. Geom., 33 (2005), 249-274. doi: 10.1007/s00454-004-1146-y.

show all references

References:

[1]	H. Adams, A. Tausz and M. Vejdemo-Johansson, JavaPlex: A research software package for persistent (co) homology, in International Congress on Mathematical Software, Lecture Notes in Computer Science, 8592, Springer, 2014, 129-136. doi: 10.1007/978-3-662-44199-2_23.
[2]	S. G. Aksoy, C. Joslyn, C. O. Marrero, B. Praggastis and E. Purvine, Hypernetwork science via high-order hypergraph walks, EPJ Data Science, 9 (2020). doi: 10.1140/epjds/s13688-020-00231-0.
[3]	F. Aurenhammer, R. Klein and D.-T. Lee, Voronoi Diagrams and Delaunay Triangulations, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013. doi: 10.1142/8685.
[4]	U. Bauer, Ripser: A lean C++ code for the computation of Vietoris-Rips persistence barcodes, 2017. Software available from: https://github.com/Ripser/ripser.
[5]	U. Bauer, M. Kerber and J. Reininghaus, DIPHA (A distributed persistent homology algorithm), 2014. Software available from: https://github.com/DIPHA/dipha.
[6]	S. Bressan, J. Li, S. Ren and J. Wu, The embedded homology of hypergraphs and applications, Asian J. Math, 23 (2019), 479-500. doi: 10.4310/AJM.2019.v23.n3.a6.
[7]	P. Bubenik and P. T. Kim, A statistical approach to persistent homology, Homology Homotopy Appl., 9 (2007), 337-362. doi: 10.4310/HHA.2007.v9.n2.a12.
[8]	Z. Cang and G.-W. Wei, TopologyNet: Topology based deep convolutional and multi-task neural networks for biomolecular property predictions, PLoS Computational Biology, 13 (2017). doi: 10.1371/journal.pcbi.1005690.
[9]	G. Carlsson, V. De Silva and D. Morozov, Zigzag persistent homology and real-valued functions, in Proceedings of the Twenty-Fifth Annual Symposium on Computational Geometry, ACM, 2009, 247-256. doi: 10.1145/1542362.1542408.
[10]	G. Carlsson, A. Zomorodian, A. Collins and L. Guibas, Persistence barcodes for shapes, International J. Shape Modeling, 11 (2005), 149-187. doi: 10.1142/S0218654305000761.
[11]	J. Cheeger, A lower bound for the smallest eigenvalue of the Laplacian, in Problems in Analysis, Princeton Univ. Press, Princeton, NJ, 1970, 195-199. doi: 10.1515/9781400869312-013.
[12]	J. Chen, R. Zhao, Y. Tong and G.-W. Wei, Evolutionary de Rham-Hodge method, Discrete Contin. Dyn. Syst. Ser. B, (2020). doi: 10.3934/dcdsb.2020257.
[13]	F. R. Chung, Spectral Graph Theory, CBMS Regional Conference Series in Mathematics, 92, American Mathematical Society, Providence, RI, 1997.
[14]	M.-V. Ciocanel, R. Juenemann, A. T. Dawes and S. A. McKinley, Topological data analysis approaches to uncovering the timing of ring structure onset in filamentous networks, Bull. Math. Biol., 83 (2021), 21pp. doi: 10.1007/s11538-020-00847-3.
[15]	V. de Silva and R. Ghrist, Coverage in sensor networks via persistent homology, Algebr. Geom. Topol., 7 (2007), 339-358. doi: 10.2140/agt.2007.7.339.
[16]	B. Delaunay, Sur la sphère vide, Izv. Akad. Nauk SSSR, Otdelenie Matematicheskii i Estestvennyka Nauk, 7 (1934), 793-800.
[17]	T. K. Dey, F. Fan and Y. Wang, Computing topological persistence for simplicial maps, in Computational Geometry (SoCG'14), ACM, New York, 2014, 345-354. doi: 10.1145/2582112.2582165.
[18]	B. Eckmann, Harmonische funktionen und Randwertaufgaben in einem Komplex, Comment. Math. Helv., 17 (1945), 240-255.
[19]	H. Edelsbrunner, Alpha shapes - A survey, Tessellations in the Sciences, 27 (2010), 1-25. Available from: https://pub.ist.ac.at/edels/Papers/2011-B-03-AlphaShapes.pdf.
[20]	H. Edelsbrunner and J. Harer, Persistent homology - A survey, in Surveys on Discrete and Computational Geometry, Contemp. Math., 453, Amer. Math. Soc., Providence, RI, 2008, 257-282.
[21]	B. T. Fasy, J. Kim, F. Lecci, C. Maria, D. L. Millman and M. J. Kim, Package (TDA), 2019.
[22]	J. Friedman, Computing Betti numbers via combinatorial Laplacians, Algorithmica, 21 (1998), 331-346. doi: 10.1007/PL00009218.
[23]	C. Giusti, E. Pastalkova, C. Curto and V. Itskov, Clique topology reveals intrinsic geometric structure in neural correlations, Proc. Natl. Acad. Sci. USA, 112 (2015), 13455-13460. doi: 10.1073/pnas.1506407112.
[24]	D. Hernández Serrano, J. Hernaández-Serrano and D. Sánchez Gómez, Simplicial degree in complex networks. Applications of topological data analysis to network science, Chaos Solitons Fractals, 137 (2020), 21pp. doi: 10.1016/j.chaos.2020.109839.
[25]	T. Kaczynski, K. Mischaikow and M. Mrozek, Computational Homology, Applied Mathematical Sciences, 157, Springer-Verlag, New York, 2004. doi: 10.1007/b97315.
[26]	F. W. Kamber and P. Tondeur, De Rham-Hodge theory for Riemannian foliations, Math. Ann., 277 (1987), 415-431. doi: 10.1007/BF01458323.
[27]	M. Kerber and H. Edelsbrunner, The medusa of spatial sorting: 3D kinetic alpha complexes and implementation, preprint, arXiv:1209.5434.
[28]	Y. Lee, S. D. Barthel, P. Dłotko, S. Mohamad Moosavi, K. Hess and B. Smit, Quantifying similarity of pore-geometry in nanoporous materials, Nature Communications, 8 (2017). doi: 10.1038/ncomms15396.
[29]	V. Maroulas, C. P. Micucci and F. Nasrin, Bayesian topological learning for classifying the structure of biological networks, preprint, arXiv:2009.11974.
[30]	J. May, Multivariate Analysis, Scientific e-Resources, 2018.
[31]	F. Mémoli, Z. Wan and Y. Wang, Persistent Laplacians: Properties, algorithms and implications, preprint, arXiv:2012.02808.
[32]	Z. Meng, D. Vijay Anand, Y. Lu, J. Wu and K. Xia, Weighted persistent homology for biomolecular data analysis, Scientific Reports, 10 (2020), 1-15. doi: 10.1038/s41598-019-55660-3.
[33]	Z. Meng and K. Xia, Persistent spectral based machine learning (PerSpect ML) for drug design, preprint, arXiv:2002.00582.
[34]	K. Mischaikow and V. Nanda, Morse theory for filtrations and efficient computation of persistent homology, Discrete Comput. Geom., 50 (2013), 330-353. doi: 10.1007/s00454-013-9529-6.
[35]	D. Morozov, Dionysus Software, 2012.
[36]	D. Morozov and P. Skraba, DioDe Software, 2017.
[37]	D. Nguyen and G.-W. Wei, AGL-Score: Algebraic graph learning score for protein-ligand binding scoring, ranking, docking, and screening, J. Chemical Information Modeling, 59 (2019), 3291-3304. doi: 10.1021/acs.jcim.9b00334.
[38]	D. D. Nguyen, Z. Cang, K. Wu, M. Wang, Y. Cao and G.-W. Wei, Mathematical deep learning for pose and binding affinity prediction and ranking in D3R Grand Challenges, J. Comput. Aided Mol. Des., 33 (2019), 71-82. doi: 10.1007/s10822-018-0146-6.
[39]	Gudhi Project, GUDHI User and Reference Manual, 2015.
[40]	I. Sgouralis, A. Nebenführ and V. Maroulas, A Bayesian topological framework for the identification and reconstruction of subcellular motion, SIAM J. Imaging Sci., 10 (2017), 871-899. doi: 10.1137/16M1095755.
[41]	D. A. Spielman, Spectral graph theory and its applications, 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07), IEEE, 2007, 29-38. doi: 10.1109/FOCS.2007.56.
[42]	J. Townsend, C. P. Micucci, J. H. Hymel, V. Maroulas and K. D. Vogiatzis, Representation of molecular structures with persistent homology for machine learning applications in chemistry, Nature Communications, 11 (2020). doi: 10.1038/s41467-020-17035-5.
[43]	G. Voronoi, Nouvelles applications des paramètres continus à la théorie des formes quadratiques. Premier mémoire. Sur quelques propriétés des formes quadratiques positives parfaites, J. Reine Angew. Math., 133 (1908), 97-102. doi: 10.1515/crll.1908.133.97.
[44]	R. Wang, D. D. Nguyen and G.-W. Wei, Persistent spectral graph, Int. J. Numer. Methods Biomed. Eng., 36 (2020), 27pp. doi: 10.1002/cnm.3376.
[45]	K. Xia, K. Opron and G.-W. Wei, Multiscale Gaussian network model (mGNM) and multiscale anisotropic network model (mANM), J. Chem. Phys., 143 (2015). doi: 10.1063/1.4936132.
[46]	K. Xia and G.-W. Wei, Persistent homology analysis of protein structure, flexibility, and folding, Int. J. Numer. Methods Biomed. Eng., 30 (2014), 814-844. doi: 10.1002/cnm.2655.
[47]	R. Zhao, M. Desbrun, G.-W. Wei and Y. Tong, 3D hodge decompositions of edge-and face-based vector fields., ACM Transactions on Graphics (TOG), 38 (2019), 1-13. doi: 10.1145/3355089.3356546.
[48]	R. Zhao, M. Wang, J. Chen, Y. Tong and G.-W. Wei, The de Rham-Hodge analysis and modeling of biomolecules, Bull. Math. Biol., 82 (2020), 38pp. doi: 10.1007/s11538-020-00783-2.
[49]	A. Zomorodian and G. Carlsson, Computing persistent homology, Discrete Comput. Geom., 33 (2005), 249-274. doi: 10.1007/s00454-004-1146-y.

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

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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

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

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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} $

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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.

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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.

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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.

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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.

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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

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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

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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

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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

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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.

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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} $

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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} $

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