20.500.12556/RUL-155778 Lower bounds on the homology of Vietoris–Rips complexes of hypercube graphs 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. Vietoris–Rips complexes clique complexes hypercubes Betti numbers true false true Angleški jezik Ni določen Članek v reviji 2024-04-17 14:00:51 2024-04-17 14:00:54 2024-05-14 03:49:31 0000-00-00 00:00:00 2024 0 0 32 str. iss. 3, art. 72 Vol. 47 May 2024 0000-00-00 Zaloznikova Objavljeno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 515.1:519.1 0126-6705 10.1007/s40840-024-01663-x 187729155 s40840-024-01663-x.pdf s40840-024-01663-x.pdf 1 E93F4324DBCC8A2680D5BF6E212E0A50 3bbe72dc6d585e6b5542a7b564221642d663d7e19bca3df34474c91dc11a4cb2 82c791fd-1108-11ef-adc7-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=183496 https://link.springer.com/article/10.1007/s40840-024-01663-x 1 257625cc-fcb2-11ee-85a5-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=182869 Fakulteta za računalništvo in informatiko 0 0 0