Persistent Mayer homology and persistent Mayer Laplacian

Li Shen; Jian Liu; Guo-Wei Wei

doi:10.3934/fods.2024032

Article Contents

2024, Volume 6, Issue 4: 584-612. Doi: 10.3934/fods.2024032

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Persistent Mayer homology and persistent Mayer Laplacian

1.
Department of Mathematics, Michigan State University, MI 48824, USA
2.
Mathematical Science Research Center, Chongqing University of Technology, Chongqing 400054, China
3.
Department of Electrical and Computer Engineering, Michigan State University, MI 48824, USA
4.
Department of Biochemistry and Molecular Biology, Michigan State University, MI 48824, USA

^*Corresponding author: Guo-Wei Wei
^*Corresponding author: Guo-Wei Wei

Received: April 2024

Revised: June 2024

Early access: July 4, 2024

Published online: July 4, 2024

Abstract / Introduction Full Text(HTML) Figure(14) / Table(5) Related Papers Cited by

Abstract

In algebraic topology, the differential (i.e., boundary operator) typically satisfies $ d^{2} = 0 $. However, the generalized differential $ d^{N} = 0 $ for an integer $ N\geq 2 $ has been studied in terms of Mayer homology on $ N $-chain complexes for more than eighty years. We introduce Mayer Laplacians on $ N $-chain complexes. We show that both Mayer homology and Mayer Laplacians offer considerable application potential, providing topological and geometric insights to spaces. We also introduce persistent Mayer homology and persistent Mayer Laplacians at various $ N $. The bottleneck distance and stability of persistence diagrams associated with Mayer homology are investigated. Our computational experiments indicate that the topological features offered by persistent Mayer homology and spectrum given by persistent Mayer Laplacians hold substantial promise for large, complex, and diverse data. We envision that the present work serves as an inaugural step towards integrating Mayer homology and Mayer Laplacians into the realm of topological data analysis.

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Mathematics Subject Classification: Primary: 55N31.

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Figure 1. Illustration of the boundary operators and chain, cycle, and boundary groups of the $ N $-chain complex for $ N = 3 $

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Figure 2. The simplicial triangulations of the Möbius strip, hexagon, torus, and octahedron

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Figure 3. Illustration of the Vietoris-Rips complexes at different filtration radius for pointset $ X_1 $. Note that for the point set $ X_1 $ in this example, we can obtain a maximum of 12 Vietoris-Rips complexes with different filtration radius. For simplicity, we have omitted 5 complexes between $ r_5 $ and $ r_6 $

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Figure 4. Comparison of persistent Betti numbers between the cases $ N = 2 $, $ N = 3 $

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Figure 5. Illustration of persistent Betti numbers between the cases $ N = 5 $, $ N = 7 $. The Mayer degree, denoted by $ q $, refers to the stage of Mayer homology

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Figure 6. Illustration of the Vietoris-Rips complexes at different filtration radius for pointset $ X_2 $

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Figure 7. Comparison of persistent Betti numbers and the smallest positive eigenvalues of persistent Laplacians for the case that $ N = 2 $ (classical). The blue curves denote the Betti curves, while the red curves represent changes of the smallest eigenvalues. The notion $ \beta^{r}_{n, q} $ denotes the $ n $-dimensional Betti number at stage $ q $ of the Vietoris-Rips complex at distance $ r $. The notion $ \lambda_{n, q}^{r}(1) $ represents the smallest eigenvalue of the non-harmonic component of the Laplacian $ \Delta_{n, q}^{r} $ at distance parameter $ r $

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Figure 8. Comparison of persistent Betti numbers and the smallest positive eigenvalues of persistent Laplacians for the case that $ N = 3 $. The blue curves denote the Betti curves, while the red curves represent changes of the smallest eigenvalues. The notion $ \beta^{r}_{n, q} $ denotes the $ n $-dimensional Betti number at stage $ q $ of the Vietoris-Rips complex at distance $ r $. The notion $ \lambda_{n, q}^{r}(1) $ represents the smallest eigenvalue of the non-harmonic component of the Laplacian $ \Delta_{n, q}^{r} $ at filtration parameter $ r $

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Figure 9. Comparison of persistent Betti numbers and the smallest positive eigenvalues of persistent Laplacians for the case that $ N = 5 $. The blue curves denote the Betti curves, while the red curves represent changes of the smallest eigenvalues. The notion $ \beta^{r}_{n, q} $ denotes the $ n $-dimensional Betti number at stage $ q $ of the Vietoris-Rips complex at distance $ r $. The notion $ \lambda_{n, q}^{r}(1) $ represents the smallest eigenvalue of the non-harmonic component of the Laplacian $ \Delta_{n, q}^{r} $ at filtration parameter $ r $

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Figure 10. Structures of the fullerene $ \mathrm{C}_{60} $ (Left) and the cucurbit[7]uril $ \mathrm{CB}7 $ (Right)

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Figure 11. Comparison of persistent Betti numbers and the smallest positive eigenvalues of persistent Laplacians for fullerene $ \mathrm{C}_{60} $ in cases where $ N = 2 $, $ N = 3 $, and $ N = 5 $. Here, $ \beta_{n, q} $ denotes the $ n $-dimensional Betti number at stage $ q $ for a given distance parameter. Similarly, $ \lambda_{n, q} $ represents the smallest eigenvalue of the non-harmonic component of the Laplacian $ \Delta_{n, q} $ at a given distance parameter

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Figure 12. Comparison of persistent Betti numbers and the smallest positive eigenvalues of persistent Laplacians for fullerene $ \mathrm{C}_{60} $ in cases where $ N = 2 $, $ N = 3 $, and $ N = 5 $. Here, $ \beta_{n, q} $ denotes the $ n $-dimensional Betti number at stage $ q $ for a given distance parameter. Similarly, $ \lambda_{n, q} $ represents the smallest eigenvalue of the non-harmonic component of the Laplacian $ \Delta_{n, q} $ at a given distance parameter

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Figure 13. Comparison of persistent Betti numbers and the smallest positive eigenvalues of persistent Laplacians for cucurbit[7]uril $ \mathrm{CB}7 $ in cases where $ N = 2 $, $ N = 3 $, and $ N = 5 $. Here, $ \beta_{n, q} $ denotes the $ n $-dimensional Betti number at stage $ q $ for a given distance parameter. Similarly, $ \lambda_{n, q} $ represents the smallest eigenvalue of the non-harmonic component of the Laplacian $ \Delta_{n, q} $ at a given distance parameter

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Figure 14. Comparison of persistent Betti numbers and the smallest positive eigenvalues of persistent Laplacians for cucurbit[7]uril $ \mathrm{CB}7 $ in cases where $ N = 2 $, $ N = 3 $, and $ N = 5 $. Here, $ \beta_{n, q} $ denotes the $ n $-dimensional Betti number at stage $ q $ for a given distance parameter. Similarly, $ \lambda_{n, q} $ represents the smallest eigenvalue of the non-harmonic component of the Laplacian $ \Delta_{n, q} $ at a given distance parameter

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Table 1. The Mayer Betti numbers for the simplicial complexes $ \Delta[3] $, $ \partial\Delta[3] $, a hexagon, and the simplicial triangulations of the Möbius strip, torus, and octahedron

simplicial complexes	$ \beta_{0, 1} $	$ \beta_{1, 1} $	$ \beta_{2, 1} $	$ \beta_{0, 2} $	$ \beta_{1, 2} $	$ \beta_{2, 2} $
$ \Delta[3] $	1	1	0	0	2	0
$ \partial\Delta[3] $	1	2	0	0	2	1
Hexagon	6	0	0	0	6	0
Möbius trip	1	6	0	0	6	1
Torus	1	18	0	0	9	10
Octahedron	1	3	1	0	2	3

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Table 2. Illustration of Mayer Laplacians for $ N = 3 $

n, q	n=0, q=1	n=0, q=2
$ L_{n, q} $	$ {\mathbf{O}}_{6\times 6} $	$ \left( \begin{array}{cccccc} 2&\xi^{2}&0&0&0&1\\ \xi&2&\xi^{2}&0&0&0\\ 0&\xi&2&\xi^{2}&0&0\\ 0&0&\xi&2&\xi^{2}&0\\ 0&0&0&\xi&2&1\\ 1&0&0&0&1&2\\ \end{array} \right) $
$ \beta_{n, q} $	6	0
$ {\mathbf{Spec}}(L_{n, q}) $	{0, 0, 0, 0, 0, 0}	{0.12, 0.47, 1.65, 2.35, 3.53, 3.88}

n, q	n=1, q=1		n=1, q=2
$L_{n, q}$	$\left( \begin{array}{cccccc} 2&\xi^{2}&0&0&0&1\\ \xi&2&\xi^{2}&0&0&0\\ 0&\xi&2&\xi^{2}&0&0\\ 0&0&\xi&2&\xi^{2}&0\\ 0&0&0&\xi&2&1\\ 1&0&0&0&1&2\\ \end{array} \right)$		${\mathbf{O}}_{6\times 6}$
$\beta_{n, q}$	0		6
${\mathbf{Spec}}(L_{n, q})$	{0.12, 0.47, 1.65, 2.35, 3.53, 3.88}		{0, 0, 0, 0, 0, 0}

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Table 3. Illustration of Mayer Laplacians for $ N = 5 $

$ n, q $	$ n=0 $, $ q=1 $	$ n=0 $, $ q=2 $	$ n=0 $, $ q=3 $		$ n=0 $, $ q=4 $
$L_{n, q}$	${\mathbf{O}}_{6\times 6}$	${\mathbf{O}}_{6\times 6}$	${\mathbf{O}}_{6\times 6}$		$\left( \begin{array}{cccccc} 2& \xi^{4}_{5}& 0& 0& 0& 1\\ \xi_{5}& 2& \xi^{4}_{5}& 0& 0& 0\\ 0& \xi_{5}& 2& \xi^{4}_{5}& 0& 0\\ 0& 0& \xi_{5}& 2& \xi^{4}_{5}& 0\\ 0& 0& 0& \xi_{5}& 2& 1\\ 1& 0& 0& 0& 1& 2\\ \end{array} \right)$
$\beta_{n, q}$	6	6	6		0
${\mathbf{Spec}}(L_{n, q})$	{0, 0, 0, 0, 0, 0}	{0, 0, 0, 0, 0, 0}	{0, 0, 0, 0, 0, 0}		{0.04, 0.66, 1.38, 2.62, 3.34, 3.96}

$n, q$	$n=1$, $q=1$			$n=1$, $q=2$		$n=1$, $q=3$	$n=1$, $q=4$
$L_{n, q}$	$\left( \begin{array}{cccccc} 2& \xi^{4}_{5}& 0& 0& 0& 1\\ \xi_{5}& 2& \xi^{4}_{5}& 0& 0& 0\\ 0& \xi_{5}& 2& \xi^{4}_{5}& 0& 0\\ 0& 0& \xi_{5}& 2& \xi^{4}_{5}& 0\\ 0& 0& 0& \xi_{5}& 2& 1\\ 1& 0& 0& 0& 1& 2\\ \end{array} \right)$			${\mathbf{O}}_{6\times 6}$		${\mathbf{O}}_{6\times 6}$	${\mathbf{O}}_{6\times 6}$
$\beta_{n, q}$	0			6		6	6
${\mathbf{Spec}}(L_{n, q})$	{0.04, 0.66, 1.38, 2.62, 3.34, 3.96}			{0, 0, 0, 0, 0, 0}		{0, 0, 0, 0, 0, 0}	{0, 0, 0, 0, 0, 0}

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Table 4. A statistics of the Mayer Betti curves variation for different $ N $ value

N value	variations
N value	Betti 0	Avg. Betti 0	Betti 1	Avg. Betti 1
2	3	3	2	2
3	7	3.5	12	6
5	15	3.75	33	8.25
7	17	2.83	54	9

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Table 5. A comparison of variation detection of the Mayer Betti numbers with the Mayer Laplacian's first non-zero eigenvalues for $ N = 2, 3 $, and $ 5 $

Mayer features	$ N=2 $	$ N=3 $	$ N=3 $	$ N=5 $	$ N=5 $	$ N=5 $	$ N=5 $
Mayer features	$ q=1 $	$ q=1 $	$ q=2 $	$ q=1 $	$ q=2 $	$ q=3 $	$ q=4 $
$ \beta_{0, q} $	2	3	2	2	1	3	2
$ \lambda_{0, q}(1) $	3	4	4	2	2	4	4
$ \beta_{1, q} $	0	4	3	3	3	4	3
$ \lambda_{1, q}(1) $	4	4	4	4	2	2	4

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