| Title: | Lower bounds on the homology of Vietoris–Rips complexes of hypercube graphs |
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| Authors: | ID Adams, Henry (Author)
ID Virk, Žiga (Author) |
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| Files: |  URL - Source URL, visit https://link.springer.com/article/10.1007/s40840-024-01663-x
 PDF - Presentation file, download (867,00 KB)
MD5: 6764E993953E72C897F06E591CEA6AB9
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| Language: | English |
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| Typology: | 1.01 - Original Scientific Article |
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| Organization: |  IMFM - Institute of Mathematics, Physics, and Mechanics
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| Abstract: | We provide novel lower bounds on the Betti numbers of Vietoris-Rips complexes of hypercube graphs of all dimensions, and at all scales. In more detail, let $Q_n$ be the vertex set of $2^n$ vertices in the $n$-dimensional hypercube graph, equipped with the shortest path metric. Let ${\rm VR}(Q_n;r)$ be its Vietoris-Rips complex at scale parameter $r \ge 0$, which has $Q_n$ as its vertex set, and all subsets of diameter at most $r$ as its simplices. For integers $r < r'$ the inclusion ${\rm VR}(Q_n;r) \hookrightarrow {\rm VR}(Q_n;r')$ is nullhomotopic, meaning no persistent homology bars have length longer than one, and we therefore focus attention on the individual spaces ${\rm VR}(Q_n;r)$. We provide lower bounds on the ranks of homology groups of ${\rm VR}(Q_n;r)$. For example, using cross-polytopal generators, we prove that the rank of $H_{2^r-1}({\rm VR}(Q_n;r))$ is at least $2^{n-(r+1)}\binom{n}{r+1}$. We also prove a version of homology propagation: if $q\ge 1$ and if $p$ is the smallest integer for which ${\rm rank} H_q({\rm VR}(Q_p;r)) \neq 0$, then ${\rm rank} H_q({\rm VR}(Q_n;r)) \ge \sum_{i=p}^n 2^{i-p} \binom{i-1}{p-1} \cdot {\rm rank} H_q({\rm VR}(Q_p;r))$ for all $n \ge p$. When $r \le 3$, this result and variants thereof provide tight lower bounds on the rank of $H_q({\rm VR}(Q_n;r))$ for all $n$, and for each $r \ge 4$ we produce novel lower bounds on the ranks of homology groups. Furthermore, we show that for each $r\ge 2$, the homology groups of ${\rm VR}(Q_n;r)$ for $n \ge 2r+1$ contain propagated homology not induced by the initial cross-polytopal generators. |
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| Keywords: | Vietoris–Rips complexes, clique complexes, hypercubes, Betti numbers |
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| Publication status: | Published |
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| Publication version: | Version of Record |
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| Publication date: | 01.05.2024 |
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| Year of publishing: | 2024 |
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| Number of pages: | 32 str. |
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| Numbering: | Vol. 47, iss. 3, [article no.] 72 |
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| PID: | 20.500.12556/DiRROS-18318  |
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| UDC: | 515.1:519.1 |
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| ISSN on article: | 0126-6705 |
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| DOI: | 10.1007/s40840-024-01663-x |
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| COBISS.SI-ID: | 187729155 |
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| Note: |
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| Publication date in DiRROS: | 05.03.2024 |
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| Views: | 137 |
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| Downloads: | 44 |
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