Contractible Vietoris–Rips complexes of ℤⁿ

We inspect Vietoris--Rips complexes ${\cal{VR}}_t(X)$ of certain metric spaces $X$ using a new generalization of Bestvina--Brady discrete Morse theory. Our main result is a pair of metric criteria on $X$, called the {\it Morse Criterion} and {\it Link Criterion}, that allow us to deduce information about the homotopy types of certain ${\cal{VR}}_t(X)$. One application is to topological data analysis, specifically persistence of homotopy type for certain Vietoris--Rips complexes. For example we recover some results of Adamaszek--Adams and Hausmann regarding homotopy types of ${\cal{VR}}_t(S^n)$. Another application is to geometric group theory; we prove that any group acting geometrically on a metric space satisfying a version of the Link Criterion admits a geometric action on a contractible simplicial complex, which has implications for the finiteness properties of the group. This applies for example to asymptotically ${\rm CAT}(0)$ groups. We also prove that any group with a word metric satisfying the Link Criterion in an appropriate range has a contractible Vietoris--Rips complex, and use combings to exhibit a family of groups with this property.

Parkinson's disease (PD) is a clinically heterogeneous disorder, which mainly affects patients' motor and non-motor function. Functional connectivity was preliminary explored and studied through resting state functional magnetic resonance imaging (rsfMRI). Through the topological analysis of 54 PD scans and 31 age-matched normal controls (NC) in the Neurocon dataset, leveraging on rsfMRI data, the brain functional connection and the Vietoris-Rips (VR) complex were constructed. The barcodes of the complex were calculated to reflect the changes of functional connectivity neural circuits (FCNC) in brain network. The 0-dimensional Betti number β0 means the number of connected branches in VR complex. The average number of connected branches in PD group was greater than that in NC group when the threshold δ ≤ 0.7. Two-sample Mann-Whitney U test and false discovery rate (FDR) correction were used for statistical analysis to investigate the FCNC changes between PD and NC groups. In PD group, under threshold of 0.7, the number of FCNC involved was significantly differences and these brain regions include the Cuneus_R, Lingual_R, Fusiform_R and Heschl_R. There are also significant differences in brain regions in the Frontal_Inf_Orb_R and Pallidum_R, when the threshold increased to 0.8 and 0.9 (p < 0.05). In addition, when the length of FCNC was medium, there was a significant statistical difference between the PD group and the NC group in the Neurocon dataset and the Parkinson's Progression Markers Initiative (PPMI) dataset. Topological analysis based on rsfMRI data may provide comprehensive information about the changes of FCNC and may provide an alternative for clinical differential diagnosis.

Vietoris–Rips complexes also provide topologically correct reconstructions of sampled shapes

Graph induced complex on point data

Fix a finite set of points in Euclidean n-space \(\mathbb{E}^{n}\) , thought of as a point-cloud sampling of a certain domain \(D\subset\mathbb{E}^{n}\) . The Vietoris–Rips complex is a combinatorial simplicial complex based on proximity of neighbors that serves as an easily-computed but high-dimensional approximation to the homotopy type of D. There is a natural “shadow” projection map from the Vietoris–Rips complex to \(\mathbb{E}^{n}\) that has as its image a more accurate n-dimensional approximation to the homotopy type of D.

For [Formula: see text] a metric space and [Formula: see text] a scale parameter, the Vietoris–Rips simplicial complex [Formula: see text] (resp. [Formula: see text]) has [Formula: see text] as its vertex set, and a finite subset [Formula: see text] as a simplex whenever the diameter of [Formula: see text] is less than [Formula: see text] (resp. at most [Formula: see text]). Though Vietoris–Rips complexes have been studied at small choices of scale by Hausmann and Latschev [13,16], they are not well-understood at larger scale parameters. In this paper we investigate the homotopy types of Vietoris–Rips complexes of ellipses [Formula: see text] of small eccentricity, meaning [Formula: see text]. Indeed, we show that there are constants [Formula: see text] such that for all [Formula: see text], we have [Formula: see text] and [Formula: see text], though only one of the two-spheres in [Formula: see text] is persistent. Furthermore, we show that for any scale parameter [Formula: see text], there are arbitrarily dense subsets of the ellipse such that the Vietoris–Rips complex of the subset is not homotopy equivalent to the Vietoris–Rips complex of the entire ellipse. As our main tool we link these homotopy types to the structure of infinite cyclic graphs.

We formalize an equivariant version of Bestvina-Brady discrete Morse theory, and apply it to Vietoris-Rips complexes in order to exhibit finite universal spaces for proper actions for all asymptotically CAT(0) groups.

We associate with each compact set X of Rn two real-valued functions cX and hX defined on R+ which provide two measures of how much the set X fails to be convex at a given scale. First, we show that, when P is a finite point set, an upper bound on cP(t) entails that the Rips complex of P at scale r collapses to the Cech complex of P at scale r for some suitable values of the parameters t and r. Second, we prove that, when P samples a compact set X, an upper bound on hX over some interval guarantees a topologically correct reconstruction of the shape X either with a Cech complex of P or with a Rips complex of P. Regarding the reconstruction with Cech complexes, our work compares well with previous approaches when X is a smooth set and surprisingly enough, even improves constants when X has a positive μ-reach. Most importantly, our work shows that Rips complexes can also be used to provide topologically correct reconstruction of shapes. This may be of some computational interest in high dimensions.

We study the algorithmic complexity of computing the persistence barcode of a randomly generated filtration. We provide a general technique to bound the expected complexity of reducing the boundary matrix in terms of the density of its reduced form. We apply this technique finding upper bounds for the average fill-in (number of non-zero entries) of the boundary matrix on Čech, Vietoris–Rips and Erdős–Rényi filtrations after matrix reduction, thus obtaining bounds on the expected complexity of the barcode computation. Our method is based on previous results on the expected Betti numbers of the corresponding complexes. Our fill-in bounds for Čech and Vietoris–Rips complexes are asymptotically tight up to a logarithmic factor. In particular, both our fill-in and computation bounds are better than the worst-case estimates. We also provide an Erdős–Rényi filtration realizing the worst-case fill-in and computation.

Given a metric space X and a distance threshold r&gt;0, the Vietoris-Rips\nsimplicial complex has as its simplices the finite subsets of X of diameter\nless than r. A theorem of Jean-Claude Hausmann states that if X is a Riemannian\nmanifold and r is sufficiently small, then the Vietoris-Rips complex is\nhomotopy equivalent to the original manifold. Little is known about the\nbehavior of Vietoris-Rips complexes for larger values of r, even though these\ncomplexes arise naturally in applications using persistent homology. We show\nthat as r increases, the Vietoris-Rips complex of the circle obtains the\nhomotopy types of the circle, the 3-sphere, the 5-sphere, the 7-sphere, ...,\nuntil finally it is contractible. As our main tool we introduce a directed\ngraph invariant, the winding fraction, which in some sense is dual to the\ncircular chromatic number. Using the winding fraction we classify the homotopy\ntypes of the Vietoris-Rips complex of an arbitrary (possibly infinite) subset\nof the circle, and we study the expected homotopy type of the Vietoris-Rips\ncomplex of a uniformly random sample from the circle. Moreover, we show that as\nthe distance parameter increases, the ambient Cech complex of the circle also\nobtains the homotopy types of the circle, the 3-sphere, the 5-sphere, the\n7-sphere, ..., until finally it is contractible.\n

We prove that for each positive integer n n , the Rips complexes of the n n -dimensional integer lattice in the d 1 d_1 metric (i.e., the Manhattan metric, also called the natural word metric in the Cayley graph) are contractible at scales above n 2 ( 2 n − 1 ) n^2(2n-1) , with the bounds arising from the Jung constants. We introduce a new concept of locally dominated vertices in a simplicial complex, upon which our proof strategy is based. This allows us to deduce the contractibility of the Rips complexes from a local geometric condition called local crushing. In the case of the integer lattices in dimension n n and a fixed scale r r , this condition entails the comparison of finitely many distances to conclude that the corresponding Rips complex is contractible. In particular, we are able to verify that for n = 1 , 2 , 3 n=1,2,3 , the Rips complex of the n n -dimensional integer lattice at scale greater or equal to n n is contractible. We conjecture that the same proof strategy can be used to extend this result to all dimensions n n .

Random cyclic dynamical systems

We give an [Formula: see text] algorithm for computing the [Formula: see text]-dimensional persistent homology of a filtration of clique complexes of cyclic graphs on [Formula: see text] vertices. This is nearly quadratic in the number of vertices [Formula: see text], and therefore a large improvement upon the traditional persistent homology algorithm, which is cubic in the number of simplices of dimension at most [Formula: see text], and hence of running time [Formula: see text] in the number of vertices [Formula: see text]. Our algorithm applies, for example, to Vietoris–Rips complexes of points sampled from a curve in [Formula: see text] when the scale is bounded depending on the geometry of the curve, but still large enough so that the Vietoris–Rips complex may have non-trivial homology in arbitrarily high dimensions [Formula: see text]. In the case of the plane [Formula: see text], we prove that our algorithm applies for all scale parameters if the [Formula: see text] vertices are sampled from a convex closed differentiable curve whose convex hull contains its evolute. We ask if there are other geometric settings in which computing persistent homology is (say) quadratic or cubic in the number of vertices, instead of in the number of simplices.

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 Qn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Q_n$$\\end{document} be the vertex set of 2n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2^n$$\\end{document} vertices in the n-dimensional hypercube graph, equipped with the shortest path metric. Let VR(Qn;r)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{VR}(Q_n;r)$$\\end{document} be its Vietoris–Rips complex at scale parameter r≥0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r \\ge 0$$\\end{document}, which has Qn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Q_n$$\\end{document} as its vertex set, and all subsets of diameter at most r as its simplices. For integers r<r′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r<r'$$\\end{document} the inclusion VR(Qn;r)↪VR(Qn;r′)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{VR}(Q_n;r)\\hookrightarrow \ extrm{VR}(Q_n;r')$$\\end{document} is nullhomotopic, meaning no persistent homology bars have length longer than one, and we therefore focus attention on the individual spaces VR(Qn;r)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{VR}(Q_n;r)$$\\end{document}. We provide lower bounds on the ranks of homology groups of VR(Qn;r)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{VR}(Q_n;r)$$\\end{document}. For example, using cross-polytopal generators, we prove that the rank of H2r-1(VR(Qn;r))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_{2^r-1}(\ extrm{VR}(Q_n;r))$$\\end{document} is at least 2n-(r+1)nr+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2^{n-(r+1)}\\left( {\\begin{array}{c}n\\\\ r+1\\end{array}}\\right) $$\\end{document}. We also prove a version of homology propagation: if q≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q\\ge 1$$\\end{document} and if p is the smallest integer for which rankHq(VR(Qp;r))≠0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{rank}H_q(\ extrm{VR}(Q_p;r))\ e 0$$\\end{document}, then rankHq(VR(Qn;r))≥∑i=pn2i-pi-1p-1·rankHq(VR(Qp;r))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{rank}H_q(\ extrm{VR}(Q_n;r)) \\ge \\sum _{i=p}^n 2^{i-p} \\left( {\\begin{array}{c}i-1\\\\ p-1\\end{array}}\\right) \\cdot \ extrm{rank}H_q(\ extrm{VR}(Q_p;r))$$\\end{document} for all n≥p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n \\ge p$$\\end{document}. When r≤3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r\\le 3$$\\end{document}, this result and variants thereof provide tight lower bounds on the rank of Hq(VR(Qn;r))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_q(\ extrm{VR}(Q_n;r))$$\\end{document} for all n, and for each r≥4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r \\ge 4$$\\end{document} we produce novel lower bounds on the ranks of homology groups. Furthermore, we show that for each r≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r\\ge 2$$\\end{document}, the homology groups of VR(Qn;r)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{VR}(Q_n;r)$$\\end{document} for n≥2r+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n \\ge 2r+1$$\\end{document} contain propagated homology not induced by the initial cross-polytopal generators.

We show that the Vietoris–Rips complex $${\mathcal R}(n,r)$$ built over n points sampled at random from a uniformly positive probability measure on a convex body $$K\subseteq \mathbb R^d$$ is a.a.s. contractible when $$r\ge c({\ln n}/{n})^{1/d}$$ for a certain constant that depends on K and the probability measure used. This answers a question of Kahle (Discrete Comput. Geom. 45(3), 553–573 (2011)). We also extend the proof to show that if K is a compact, smooth d-manifold with boundary—but not necessarily convex—then $${\mathcal R}(n,r)$$ is a.a.s. homotopy equivalent to K when $$c_1(\ln n/{n})^{1/d} \le r\le c_2$$ for constants $$c_1=c_1(K)$$ , $$c_2=c_2(K)$$ . Our proofs expose a connection with the game of cops and robbers.