Your browser does not allow JavaScript!
JavaScript is necessary for the proper functioning of this website. Please enable JavaScript or use a modern browser.
Open Science Slovenia
Open Science
DiKUL
slv
|
eng
Search
Browse
New in RUL
About RUL
In numbers
Help
Sign in
Lower bounds on the homology of Vietoris–Rips complexes of hypercube graphs
ID
Adams, Henry
(
Author
),
ID
Virk, Žiga
(
Author
)
PDF - Presentation file,
Download
(853,79 KB)
MD5: C0198322F18108E85074DAD543CA6E08
URL - Source URL, Visit
https://link.springer.com/article/10.1007/s40840-024-01663-x
Image galllery
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.
Language:
English
Keywords:
Vietoris–Rips complexes
,
clique complexes
,
hypercubes
,
Betti numbers
Work type:
Article
Typology:
1.01 - Original Scientific Article
Organization:
FRI - Faculty of Computer and Information Science
Publication status:
Published
Publication version:
Version of Record
Year:
2024
Number of pages:
32 str.
Numbering:
Vol. 47, iss. 3, art. 72
PID:
20.500.12556/RUL-155778
UDC:
515.1:519.1
ISSN on article:
0126-6705
DOI:
10.1007/s40840-024-01663-x
COBISS.SI-ID:
187729155
Publication date in RUL:
17.04.2024
Views:
129
Downloads:
5
Metadata:
Cite this work
Plain text
BibTeX
EndNote XML
EndNote/Refer
RIS
ABNT
ACM Ref
AMA
APA
Chicago 17th Author-Date
Harvard
IEEE
ISO 690
MLA
Vancouver
:
Copy citation
Share:
Record is a part of a journal
Title:
Bulletin of the Malaysian Mathematical Sciences Society
Shortened title:
Bull. Malays. Math. Sci. Soc.
Publisher:
Springer
ISSN:
0126-6705
COBISS.SI-ID:
515781657
Licences
License:
CC BY 4.0, Creative Commons Attribution 4.0 International
Link:
http://creativecommons.org/licenses/by/4.0/
Description:
This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.
Projects
Funder:
ARIS - Slovenian Research and Innovation Agency
Project number:
N1-0114-2019
Name:
Algebrajski odtisi geometrijskih značilnosti v homologiji
Funder:
ARIS - Slovenian Research and Innovation Agency
Project number:
J1-4001-2022
Name:
Izbrani problemi iz uporabne in računske topologije
Funder:
ARIS - Slovenian Research and Innovation Agency
Project number:
P1-0292-2022
Name:
Topologija in njena uporaba
Similar documents
Similar works from RUL:
Similar works from other Slovenian collections:
Back