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  • Lower bounds on the homology of Vietoris–Rips complexes of hypercube graphs
    Adams, Henry, matematik ; Virk, Žiga, 1982-
    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.
    Source: Bulletin of the Malaysian Mathematical Sciences Society. - ISSN 0126-6705 (Vol. 47, iss. 3, [article no.] 72, May 2024, 32 str.)
    Type of material - article, component part ; adult, serious
    Publish date - 2024
    Language - english
    COBISS.SI-ID - 187729155

    Link(s):

    https://link.springer.com/article/10.1007/s40840-024-01663-x
    Repository of the University of Ljubljana – RUL
    DiRROS - Digital repository of Slovenian research organizations
    DOI

source: Bulletin of the Malaysian Mathematical Sciences Society. - ISSN 0126-6705 (Vol. 47, iss. 3, [article no.] 72, May 2024, 32 str.)
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