Lower bounds on the homology of Vietoris–Rips complexes of hypercube graphs

Adams, Henry (avtor), Virk, Žiga (avtor)

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.

• Jezik: Angleški jezik
• Ključne besede: Vietoris–Rips complexes, clique complexes, hypercubes, Betti numbers
• Vrsta gradiva: Članek v reviji
• Tipologija: 1.01 - Izvirni znanstveni članek
• Organizacija: FRI - Fakulteta za računalništvo in informatiko
• Status publikacije: Objavljeno
• Različica publikacije: Objavljena publikacija
• Leto izida: 2024
• Št. strani: 32 str.
• Številčenje: Vol. 47, iss. 3, art. 72
• PID: 20.500.12556/RUL-155778
• UDK: 515.1:519.1
• ISSN pri članku: 0126-6705
• DOI: 10.1007/s40840-024-01663-x
• COBISS.SI-ID: 187729155
• Datum objave v RUL: 17.04.2024

Gradivo je del revije

• Naslov: Bulletin of the Malaysian Mathematical Sciences Society
• Skrajšan naslov: Bull. Malays. Math. Sci. Soc.
• Založnik: Springer
• ISSN: 0126-6705
• COBISS.SI-ID: 515781657

Licence

• Licenca: CC BY 4.0, Creative Commons Priznanje avtorstva 4.0 Mednarodna
• Povezava: http://creativecommons.org/licenses/by/4.0/deed.sl
• Opis: To je standardna licenca Creative Commons, ki daje uporabnikom največ možnosti za nadaljnjo uporabo dela, pri čemer morajo navesti avtorja.