Lower Bounds on the Homology of Vietoris–Rips Complexes of Hypercube Graphs

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2024-03-04 DOI:10.1007/s40840-024-01663-x

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引用次数: 0

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 \(\textrm{VR}(Q_n;r)\) 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 \(\textrm{VR}(Q_n;r)\hookrightarrow \textrm{VR}(Q_n;r')\) is nullhomotopic, meaning no persistent homology bars have length longer than one, and we therefore focus attention on the individual spaces \(\textrm{VR}(Q_n;r)\) . We provide lower bounds on the ranks of homology groups of \(\textrm{VR}(Q_n;r)\) . For example, using cross-polytopal generators, we prove that the rank of \(H_{2^r-1}(\textrm{VR}(Q_n;r))\) is at least \(2^{n-(r+1)}\left( {\begin{array}{c}n\\ r+1\end{array}}\right) \) . We also prove a version of homology propagation: if \(q\ge 1\) and if p is the smallest integer for which \(\textrm{rank}H_q(\textrm{VR}(Q_p;r))\ne 0\) , then \(\textrm{rank}H_q(\textrm{VR}(Q_n;r)) \ge \sum _{i=p}^n 2^{i-p} \left( {\begin{array}{c}i-1\\ p-1\end{array}}\right) \cdot \textrm{rank}H_q(\textrm{VR}(Q_p;r))\) for all \(n \ge p\) . When \(r\le 3\) , this result and variants thereof provide tight lower bounds on the rank of \(H_q(\textrm{VR}(Q_n;r))\) 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 \(\textrm{VR}(Q_n;r)\) for \(n \ge 2r+1\) contain propagated homology not induced by the initial cross-polytopal generators.

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超立方图的 Vietoris-Rips 复合物同调的下限

摘要我们对所有维度和所有尺度的超立方图的 Vietoris-Rips 复数的贝蒂数提供了新的下界。更详细地说，设 \(Q_n\) 是 n 维超立方图中 \(2^n\) 个顶点的顶点集，并配有最短路径度量。让 \(\textrm{VR}(Q_n;r)\)成为它在尺度参数 \(r \ge 0\) 下的维托里斯-里普斯复数，它以\(Q_n\)为顶点集，以直径最大为 r 的所有子集为简集。对于整数 \(r<r'\)，包含 \(\textrm{VR}(Q_n;r)\hookrightarrow \textrm{VR}(Q_n;r')\) 是空同调的，这意味着没有持续同调条的长度长于 1，因此我们把注意力集中在单个空间 \(\textrm{VR}(Q_n;r)\) 上。我们提供了 \(\textrm{VR}(Q_n;r)\) 的同调群等级的下限。例如，使用交叉多聚生成器，我们证明了 \(H_{2^r-1}(\textrm{VR}(Q_n;r))\) 的秩至少是 \(2^{n-(r+1)}\left( {\begin{array}{c}n\ r+1\end{array}}\right) \)。我们还证明了同调传播的一个版本：如果 \(q\ge 1\) 并且如果 p 是 \(\textrm{rank}H_q(\textrm{VR}(Q_p;r))\ne 0\) 的最小整数，那么 \(textrm{rank}H_q(\textrm{VR}(Q_p;r))ne 0\)。那么（textrm{rank}H_q(\textrm{VR}(Q_n;r))\ge \sum _{i=p}^n 2^{i-p}\leave( {\begin{array}{c}i-1\ p-1\end{array}\right) \cdot textrm{rank}H_q(\textrm{VR}(Q_p;r))\) for all \(n \ge p\) .当 \(r\le 3\) 时，这个结果及其变体为所有 n 的 \(H_q(\textrm{VR}(Q_n;r))\的秩提供了严格的下界，并且对于每个 \(r\ge 4\) 我们都会产生新的同调群秩的下界。此外，我们还证明了对于每一个（r\ge 2\) ，对于（n\ge 2r+1\)的（\textrm{VR}(Q_n;r)）的同源性群包含不是由初始交叉多胞生成器诱导的传播同源性。

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.