Classical results of Hausmann and Latschev show that Vietoris-Rips complexes can recover the homotopy type of a manifold, even from finite metric spaces that are nearby in Gromov-Hausdorff distance. We prove persistent homotopical versions of these theorems for metric spaces equipped with filtration functions. The central object of study is the so-called persistent homotopy type of the function-Rips complex, a filtered simplicial complex that combines a fixed Rips scale with the filtration data on the underlying space. Using techniques from CAT($κ$)-geometry and persistent simplicial homotopy theory, we generalize Latschev's and Hausmann's theorems to the setting of spaces with filtration functions and homotopical interleavings. A fundamental ingredient is a new homotopical stability theorem. The fixed-scale function-Rips construction is known not to be globally stable with respect to function Gromov-Hausdorff distance and homotopical interleaving distance. Here, we show that it is nevertheless stable for appropriate choices of the Rips parameter at such pairs $(M,f)$ for which $M$ is a complete metric space of curvature bounded above, and $f$ is a Lipschitz continuous multivariate
We introduce a flexible, categorical framework for large-scale geometry that clarifies basic behaviour of the metric Rips filtration and streamlines some constructions in geometric group theory. The paper has two main parts. First, we develop the theory of the metric Rips filtration and its colimit in natural coarse categories: informally, we characterise when the Rips colimit produces a canonical large-scale model of a metric space and use this to prove that the quasigeodesic subcategory is closed under colimits in the coarsely Lipschitz category. We also establish adjointness properties of the Rips colimit and use them to characterise extremal metrics and universal morphisms from quasigeodesic sources. Second, we apply this machinery to characterise universal quasigeodesic cones via an explicit Rips-Tuple recipe. In the HHS setting this yields a concrete, canonical model of the total space: an HHS is quasi-isometric to a Rips graph of the space of coarsely consistent tuples in the product of its factor spaces. Moreover, we give a local-to-global criterion that promotes uniformly controlled, factorwise retractions to a canonical global hierarchical retraction. Because the approach
We study the concepts of the $\ell_p$-Vietoris-Rips simplicial set and the $\ell_p$-Vietoris-Rips complex of a metric space, where $1\leq p \leq \infty.$ This theory unifies two established theories: for $p=\infty,$ this is the classical theory of Vietoris-Rips complexes, and for $p=1,$ this corresponds to the blurred magnitude homology theory. We prove several results that are known for the Vietoris-Rips complex in the general case: (1) we prove a stability theorem for the corresponding version of the persistent homology; (2) we show that, for a compact Riemannian manifold and a sufficiently small scale parameter, all the "$\ell_p$-Vietoris-Rips spaces" are homotopy equivalent to the manifold; (3) we demonstrate that the $\ell_p$-Vietoris-Rips spaces are invariant (up to homotopy) under taking the metric completion. Additionally, we show that the limit of the homology groups of the $\ell_p$-Vietoris-Rips spaces, as the scale parameter tends to zero, does not depend on $p$; and that the homology groups of the $\ell_p$-Vietoris-Rips spaces commute with filtered colimits of metric spaces.
The Delaunay-Rips filtration is a lighter and faster alternative to the well-known Rips filtration for low-dimensional Euclidean point clouds. Despite these advantages, it has seldom been studied. In this paper, we aim to bridge this gap by providing a thorough theoretical and empirical analysis of this construction. From a theoretical perspective, we show how the persistence diagrams associated with the Delaunay-Rips filtration approximate those obtained with the Rips filtration. Additionally, we describe the instabilities of the Delaunay-Rips persistence diagrams when the input point cloud is perturbed. Finally, we introduce an algorithm that computes persistence diagrams of Delaunay-Rips filtrations in any dimension. We show that our method is faster and has a lower memory footprint than traditional approaches in low dimensions. Our C++ implementation, which comes with Python bindings, is available at https://github.com/MClemot/GeoPH.
We develop a toric topological framework for studying the cohomology of Vietoris--Rips complexes $VR(Q_n;r)$ of hypercube graphs. Using total domination invariants and spectral methods, we establish general lower bounds on connectivity, which leads to infinite families of counterexamples to Shukla's conjecture, and derive first global upper bounds on coconnectivity. Our approach interprets Vietoris--Rips complexes via Stanley--Reisner rings, moment-angle complexes, and Tor algebras, allowing global topological information to be extracted from combinatorial data. In a second direction, we construct explicit cohomology classes using the Koszul resolution and show that they decomposable products of $1$-dimensional classes, and that their representatives can be combimbinatorially realised as the boundary of cross polytopes positively answering the question posed by Adams and Virk. We introduce ghost vertices as a new tool for detecting, extending, and proving linear independence of cohomology classes.
We introduce the monoidal Rips filtration, a filtered simplicial set for weighted directed graphs and other lattice-valued networks. Our construction generalizes the Vietoris-Rips filtration for metric spaces by replacing the maximum operator, determining the filtration values, with a more general monoidal product. We establish interleaving guarantees for the monoidal Rips persistent homology, capturing existing stability results for real-valued networks. When the lattice is a product of totally ordered sets, we are in the setting of multiparameter persistence. Here, the interleaving distance is bounded in terms of a generalized network distance. We use this to prove a novel stability result for the sublevel Rips bifiltration. Our experimental results show that our method performs better than Flagser in a graph regression task, and that combining different monoidal products in point cloud classification can improve performance.
Traditional feature engineering approaches for molecular sequence classification suffer from sparsity issues and computational complexity, while deep learning models often underperform on tabular biological data. This paper introduces a novel topological approach that transforms molecular sequences into images by combining Chaos Game Representation (CGR) with Rips complex construction from algebraic topology. Our method maps sequence elements to 2D coordinates via CGR, computes pairwise distances, and constructs Rips complexes to capture both local structural and global topological features. We provide formal guarantees on representation uniqueness, topological stability, and information preservation. Extensive experiments on anticancer peptide datasets demonstrate superior performance over vector-based, sequence language models, and existing image-based methods, achieving 86.8\% and 94.5\% accuracy on breast and lung cancer datasets, respectively. The topological representation preserves critical sequence information while enabling effective utilization of vision-based deep learning architectures for molecular sequence analysis.
The long computational time and large memory requirements for computing Vietoris Rips persistent homology from point clouds remains a significant deterrent to its application to big data. This paper aims to reduce the memory footprint of these computations. It presents a new construction, the distilled Vietoris Rips filtration, and proves that its persistent homology is isomorphic to that of standard Vietoris Rips. The distilled complex is constructed using a discrete Morse vector field defined on the reduced Vietoris Rips complex. The algorithm for building and reducing the distilled filtration boundary matrix is highly parallelisable and memory efficient. It can be implemented for point clouds in any metric space given the pairwise distance matrix.
We prove that for each positive integer $n$, the Rips complexes of the $n$-dimensional integer lattice in the $d_1$ metric (i.e., the Manhattan metric, also called the natural word metric in the Cayley graph) are contractible at scales above $n^2(2n-1)$, with the bounds arising from the Jung's 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 $n$ and a fixed scale $r$, 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 $n=1,2,3$, the Rips complex of the $n$-dimensional integer lattice at scale greater or equal to $n$ is contractible. We conjecture that the same proof strategy can be used to extend this result to all dimensions $n$
Vietoris-Rips metric thickenings have previously been proposed as an alternate approach to understanding Vietoris-Rips simplicial complexes and their persistent homology. Recent work has shown that for totally bounded metric spaces, Vietoris-Rips metric thickenings have persistent homology barcodes that agree with those of Vietoris-Rips simplicial complexes, ignoring whether endpoints of bars are open or closed. Combining this result with the known homotopy types and barcodes of the Vietoris-Rips simplicial complexes of the circle, the barcodes of the Vietoris-Rips metric thickenings of the circle can be deduced up to endpoints, and conjectures have been made about their homotopy types. We confirm these conjectures are correct, proving that the Vietoris-Rips metric thickenings of the circle are homotopy equivalent to odd-dimensional spheres at the expected scale parameters. Our approach is to find quotients of the metric thickenings that preserve homotopy type and show that the quotient spaces can be described as CW complexes. The quotient maps are also natural with respect to the scale parameter and thus provide a direct proof of the persistent homology of the metric thickenings.
We present a unified dark energy framework capable of generating a continuous spectrum of cosmological ``rip'' scenarios -- including the Big Rip, Grand Rip, Mild Rip, Little Rip, Little Sibling of the Big Rip, and the newly found Dollhouse Rip -- while ensuring a physically consistent evolution across cosmic history. Building on earlier phenomenological proposals, we introduce a barotropic equation-of-state parameter with a sigmoid-like correction to guarantee a strictly positive dark energy density and to avoid early-time pathologies commonly present in previous models. Using this formulation, closed-form analytic expressions for the energy density can be obtained. This, in turn, enables a systematic classification of future singularities based on the signs and magnitudes of two key parameters of the model. We test these scenarios with state-of-the-art cosmological probes, including DESI DR2 BAO, cosmic chronometers, CMB compressed likelihoods, and the Pantheon+ supernovae sample. According to our Bayesian analysis, all rip scenarios yield best-fit parameters compatible with $Λ$CDM at the $1σ$ level, with Bayes factors weakly favoring $Λ$CDM. The mild, logarithmic evolution of th
We present a new, inductive construction of the Vietoris-Rips complex, in which we take advantage of a small amount of unexploited combinatorial structure in the $k$-skeleton of the complex in order to avoid unnecessary comparisons when identifying its $(k+1)$-simplices. In doing so, we achieve a significant reduction in the number of comparisons required to construct the Vietoris-Rips compared to state-of-the-art algorithms, which is seen here by examining the computational complexity of the critical step in the algorithms. In experiments comparing a C/C++ implementation of our algorithm to the GUDHI v3.9.0 software package, this results in an observed $5$-$10$-fold improvement in speed of on sufficiently sparse Erdős-Rényi graphs with the best advantages as the graphs become sparser, as well as for higher dimensional Vietoris-Rips complexes. We further clarify that the algorithm described in Boissonnat and Maria (https://doi.org/10.1007/978-3-642-33090-2_63) for the construction of the Vietoris-Rips complex is exactly the Incremental Algorithm from Zomorodian (https://doi.org/10.1016/j.cag.2010.03.007), albeit with the additional requirement that the result be stored in a tree st
We consider persistent homology obtained by applying homology to the open Rips filtration of a compact metric space $(X,d)$. We show that each decrease in zero-dimensional persistence and each increase in one-dimensional persistence is induced by local minima of the distance function $d$. When $d$ attains local minimum at only finitely many pairs of points, we prove that each above mentioned change in persistence is induced by a specific critical edge in Rips complexes, which represents a local minimum of $d$. We use this fact to develop a theory (including interpretation) of critical edges of persistence. The obtained results include upper bounds for the rank of one-dimensional persistence and a corresponding reconstruction result. Of potential computational interest is a simple geometric criterion recognizing local minima of $d$ that induce a change in persistence. We conclude with a proof that each locally isolated minimum of $d$ can be detected through persistent homology with selective Rips complexes. The results of this paper offer the first interpretation of critical scales of persistent homology (obtained via Rips complexes) for general compact metric spaces.
In this document, we propose a bridge between the graphs and the geometric realizations of their Vietoris Rips complexes, i.e. Graphs, with their canonical Čech closure structure, have the same homotopy type that the realization of their Vietoris Rips complex.
Selective Rips complexes corresponding to a sequence of parameters are a generalization of Vietoris-Rips complexes utilizing the idea of thin simplices. We prove that if a metric space $Y$ is close (in Gromov-Hausdorff distance) to a closed Riemannian manifold $X$, then selective Rips complexes of $Y$ for certain parameters attain the homotopy type of $X$. This result is a generalization of Latchev's reconstruction result from Vietoris-Rips complexes to selective Rips complexes. In particular, we present a novel proof for the Latschev's theorem as a special case. We also present a functorial setting, which is new even in the case of Vietoris-Rips complexes.
Interest on the possible future scenarios the universe could have has grew substantially with breakthroughs on late-time acceleration. Holographic dark energy (HDE) presents a very interesting approach towards addressing late-time acceleration, presenting an intriguing interface of ideas from quantum gravity and cosmology. In this work we present an extensive discussion of possible late-time scenarios, focusing on rips and similar events, in a universe with holographic dark energy. We discuss these events in the realm of the generalized Nojiri-Odintsov cutoff and also for the more primitive holographic cutoffs like Hubble, particle and event horizon cutoffs. We also discuss the validity of the generalized second law of thermodynamics and various energy conditions in these regimes. Our work points towards the idea that it is not possible to have alternatives of the big rip consistently in the simpler HDE cutoffs, and shows the flexibility of the generalized HDE cutoff as well.
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 homotopy types of the circle, the 3-sphere, the 5-sphere, the 7-sphere, ..., until finally it is contractible. As our main tool we introduce a directed graph invariant, the winding fraction, which in some sense is dual to the circular chromatic number. Using the winding fraction we classify the homotopy types of the Vietoris-Rips complex of an arbitrary (possibly infinite) subset of the circle, and we study the expected homotopy type of the Vietoris-Rips complex of a uniformly random sample from the circle. Moreover, we show that as the distance parameter increases, the ambient Cech complex
Understanding the late-time acceleration of the universe and its subtleties is one of the biggest mysteries in cosmology. A lot of different approaches have been put forward to deal with this, ranging from the conventional cosmological constant to various models of dark energy and beyond. Recently one very interesting approach to explaining the late time acceleration has been put forward, where the the expansion of the universe is driven by mergers with other "baby" universes and has been shown to be quite viable as well from the point of view of recent observational data. So in this work we examine the possibility of various rip scenarios and other future cosmological singularities in such "multiversal" scenario, probing such singularities for the first time in a multi universe scenario. We examine two models of such a baby universe merging cosmology, and show that remarkably no rip scenario or future cosmological singularity is possible in such models.
Fix a finite set of points in Euclidean $n$-space $\euc^n$, thought of as a point-cloud sampling of a certain domain $D\subset\euc^n$. The 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 Rips complex to $\euc^n$ that has as its image a more accurate $n$-dimensional approximation to the homotopy type of $D$. We demonstrate that this projection map is 1-connected for the planar case $n=2$. That is, for planar domains, the Rips complex accurately captures connectivity and fundamental group data. This implies that the fundamental group of a Rips complex for a planar point set is a free group. We show that, in contrast, introducing even a small amount of uncertainty in proximity detection leads to `quasi'-Rips complexes with nearly arbitrary fundamental groups. This topological noise can be mitigated by examining a pair of quasi-Rips complexes and using ideas from persistent topology. Finally, we show that the projection map does not preserve higher-order topological data for planar sets, nor does it pr
We determine the homotopy type of the Vietoris-Rips complexes of the (vertex sets of the) platonic solids. The most interesting case is that the Vietoris-Rips complex of the dodecahedron is a wedge of nine 3-spheres when the parameter is between combinatorial distance 3 and 4.