We describe the homotopy types of Vietoris-Rips complexes of hypercube graphs at small scale parameters. In more detail, let $Q_n$ be the vertex set of the hypercube graph with $2^n$ vertices, equipped with the shortest path metric. Equivalently, $Q_n$ is the set of all binary strings of length $n$, equipped with the Hamming distance. The Vietoris-Rips complex of $Q_n$ at scale parameter zero is $2^n$ points, and the Vietoris-Rips complex of $Q_n$ at scale parameter one is the hypercube graph, which is homotopy equivalent to a wedge sum of circles. We show that the Vietoris-Rips complex of $Q_n$ at scale parameter two is homotopy equivalent to a wedge sum of 3-spheres, and furthermore we provide a formula for the number of 3-spheres. Many questions about the Vietoris-Rips complexes of $Q_n$ at larger scale parameters remain open.
We prove a Kunneth theorem for the Vietoris-Rips homology and cohomology of a semi-uniform space. We then interpret this result for graphs, where we show that the Kunneth theorem holds for graphs with respect to the strong graph product. We finish by computing the Vietoris-Rips cohomology of the torus endowed with diferent semi-uniform structures.
Given a metric space X and a distance threshold r>0, the Vietoris-Rips simplicial complex has as its simplices the finite subsets of X of diameter less than r. A theorem of Jean-Claude Hausmann states that if X is a Riemannian manifold and r is sufficiently small, then the Vietoris-Rips complex is homotopy equivalent to the original manifold. Little is known about the behavior of Vietoris-Rips complexes for larger values of r, even though these complexes arise naturally in applications using persistent homology. We show that as r increases, the Vietoris-Rips complex of the circle obtains the h…
Upper bounds on the topological Betti numbers of Vietoris-Rips complexes are established, and examples of such complexes with high Betti numbers are given.
We study combinatorial connectivity for two models of random geometric complexes. These two models - Čech and Vietoris-Rips complexes - are built on a homogeneous Poisson point process of intensity $n$ on a $d$-dimensional torus using balls of radius $r_n$. In the former, the $k$-simplices/faces are formed by subsets of $(k+1)$ Poisson points such that the balls of radius $r_n$ centred at these points have a mutual interesection and in the latter, we require only a pairwise intersection of the balls. Given a (simplicial) complex (i.e., a collection of $k$-simplices for all $k \geq 1$), we can …
Efficient Decomposition of Forman-Ricci Curvature on Vietoris-Rips Complexes and Data Applications
Basque Center for Applied Mathematics（巴斯克应用数学中心）；Universidade Federal de Pernambuco（伯南布哥联邦大学）；Dutch Institute for Emergent Phenomenal（荷兰涌现现象研究所）；School of Electronic Information and Electrical Engineering, Shanghai Jiao Tong University（上海交通大学电子信息与电气工程学院）；Inria, Montpellier（法国国家信息与自动化研究所（蒙彼利埃））；Max Planck Institute for Mathematics in the Sciences（马克斯·普朗克科学数学研究所）；Center for Scalable Data Analytics and Artificial Intelligence, Leipzig University（莱比锡大学可扩展数据分析与人工智能中心）；Santa Fe Institute（圣塔菲研究所）
Component graphs $Γ_{0}(F)$ are defined for arrays of sets $F$, and in particular for arrays of path components for Vietoris-Rips complexes and Lesnick complexes. The path components of $Γ_{0}(F)$ are the {\it stable components} of the array $F$. The stable components for the system of Lesnick complexes $\{ L_{s,k}(X) \}$ for a finite data set $X$ decompose into layers, which are themselves path components of a graph. Combinatorial scoring functions are defined for layers and stable components.
For X a finite subset of the circle and for 0 < r <= 1 fixed, consider the function f_r : X -> X which maps each point to the clockwise furthest element of X within angular distance less than 2 pi r. We study the discrete dynamical system on X generated by f_r, and especially its expected behavior when X is a large random set. We show that, as |X| -> infinity, the expected fraction of periodic points of f_r tends to 0 if r is irrational and to 1/q if r = p/q is rational with p and q coprime. These results are obtained via more refined statistics of f_r which we compute explicitly in terms of (…
Random abstract simplicial complex representation provides a mathematical description of wireless networks and their topology. In order to reduce the energy consumption in this type of network, we intend to reduce the number of network nodes without modifying neither the connectivity nor the coverage of the network. In this paper, we present a reduction algorithm that lower the number of points of an abstract simplicial complex in an optimal order while maintaining its topology. Then, we study the complexity of such an algorithm for a network simulated by a binomial point process and represent…
The family of $\mathfrak{I}$-contractible graphs and contractible transformations was defined by A. Ivashchenko in the mid-90's. In this paper we study the collapsibility and homological properties of the clique complex associated to $\mathfrak{I}$-contractible graphs. We show that for any graph in a special subfamily of the $\mathfrak{I}$-contractible graphs (the strong $\mathfrak{I}$-contractible ones) its clique complex is collapsible. Moreover, we present an algorithm that allows us to verify if any graph is strong $\mathfrak{I}$-contractible, as well as an algorithm to delete those vertic…
