Random Vietoris–Rips Complexes

• Random Vietoris–Rips complexes are simplicial complexes built on random point clouds using a fixed distance threshold to capture key topological features.
• They exhibit sharp phase transitions where connectivity and higher-dimensional homological features emerge or vanish at critical scale thresholds.
• Efficient algorithms and sparsification techniques enable fast approximation of persistent homology, making these complexes valuable for topological data analysis.

Random Vietoris–Rips complexes are simplicial complexes constructed on point clouds arising from random processes or samples, most frequently with the aim of understanding the underlying topology or geometric features of the sampled space. These complexes serve as fundamental objects in stochastic topology and topological data analysis (TDA), and their combinatorial and topological properties exhibit rich and often subtle probabilistic behavior as a function of scale, sample size, and the geometry of the underlying space.

1. Definition, Model Setup, and Basic Properties

A Vietoris–Rips complex $\mathrm{VR}(X; r)$ is formed from a point set $X$ within a metric space (most commonly Euclidean), where a $k$-simplex is included for each subset of $k+1$ points whose pairwise distances are all less than (or at most) a fixed threshold $r$. In the random setting, $X$ is typically a realization of a random process; standard models include:

• $X$ given by $n$ independent random samples from a measure on a Riemannian manifold, convex body, or metric graph.
• $X$ as a homogeneous Poisson point process with intensity $n$ on a domain $K \subseteq \mathbb{R}^d$.
• The random geometric graph: the 1-skeleton of the Vietoris–Rips complex where edges are included for point pairs within distance $r$.

Key features:

• $\mathrm{VR}(X; r)$ is always a clique (flag) complex, determined by its 1-skeleton.
• The complex grows as $r$ increases, exhibiting topological phase transitions mirrored in its homology and persistent homology barcodes (Adamaszek et al., 2015, Adams et al., 2018, Akinwande et al., 2019).
• For $\ell_p$-metrics, the definition is generalized via an $l_p$-weight on distance matrices, recovering the classical $p = \infty$ case and the blurred magnitude homology at $p=1$ (Ivanov et al., 4 Nov 2024).

2. Phase Transitions and Homological Connectivity

Random Vietoris–Rips complexes exhibit sharp thresholds—critical values of $r_n$—where topological features (clusters, cycles, higher-dimensional voids) appear or vanish.

Sharp Vanishing Thresholds

For a random sample of $n$ points in a $d$-dimensional body,

$$r_n \sim \left( \frac{\log n + k \log \log n + \text{const}}{n \theta_d} \right)^{1/d}$$

is the radius at which isolated $k$-faces (i.e., $(k+1)$-simplices) vanish with high probability. Here, $\theta_d$ is the volume of the unit ball in $\mathbb{R}^d$ (Iyer et al., 2018). The full topology transitions from disconnected (many isolated points) to highly connected, and eventually to contractible, as $r_n$ increases.

Feature	Threshold Radius Scale	Topology
Connectivity	$r_n \sim (\log n / n)^{1/d}$	Giant connected component appears
$k$-face vanishing	$$r_n \sim \left( \frac{\log n + k\log\log n}{n} \right)^{1/d}$$	$k$th Betti number collapses
Contractibility	$$r_n \gtrsim c \left( \frac{\log n}{n} \right)^{1/d}$$	Complex becomes contractible

This sharpness includes log-log $n$ correction factors for higher Betti numbers and subtle distinctions between Čech and Vietoris–Rips models (Iyer et al., 2018).

Homotopy Recovery and Contractibility

• For a convex body $K \subset \mathbb{R}^d$, $\mathcal{R}(n, r_n)$ is a.a.s. contractible when $r_n \geq c (\log n / n)^{1/d}$ for suitable $c$ (Müller et al., 2021).
• For compact manifolds with boundary, there exist thresholds $c_1 (\log n / n)^{1/d} \leq r_n \leq c_2$ ensuring a.a.s. homotopy equivalence to $K$.

These thresholds reflect coverings, geometric properties, and nerve theorem arguments, with combinatorial deletions related to cop-win strategies in graph theory (Müller et al., 2021).

3. Topological Invariants, Phase Diagram, and Limiting Homology

Persistent Homology and Barcodes

The persistent homology of random Vietoris–Rips complexes captures the birth and death of homological features as $r$ varies:

• Barcodes (interval decompositions) summarize lifetimes of features; phase transitions are marked by birth or death of such intervals (Adamaszek et al., 2015, Adams et al., 2018).
• The expectation of invariants (e.g., Betti numbers, Euler characteristic) are Lipschitz functions of $r$ and converge to the values for the underlying manifold as $n \to \infty$ for scales $t$ below injectivity or convexity radii (Paik et al., 2022).

Explicit Phase Behavior

In the case of the circle $S^1$ (and regular polygons as approximations), the homotopy type of $\mathrm{VR}(S^1; r)$ transitions, as $r$ increases, through a series of odd-dimensional spheres $S^{2l+1}$, with phase boundaries at $r = l / (2l+1)$ (Adamaszek et al., 2015, Adams et al., 2018). Random samples mimic this with predictable "critical thresholds," but require dense sampling to match the continuum result.

Central Limit Theorems

For functionals such as the number of $k$-faces (or volume-weighted analogues), classical and multivariate central limit theorems (CLT) hold:

• For Poisson processes in $\mathbb{R}^d$, the numbers of $k$-faces, appropriately normalized, converge in law to Gaussian as $n \to \infty$, with explicit rates (Akinwande et al., 2019).
• This extends to multivariate CLT for the $f$-vector.

4. Efficient Approximation, Algorithms, and Sparsification

Vietoris–Rips complexes are computationally expensive due to exponential growth in the number of simplices. For random or large $n$:

• Linear-Size Approximation: Sparse filtrations of $O(n)$ size closely approximate the full persistence diagram by perturbing the metric with net-tree based deletion times and scale-dependent weights; they guarantee a multiplicative approximation in persistence and can be built in $O(n \log n)$ time, with constants depending on doubling dimension and approximation tightness (Sheehy, 2012).
• New Inductive Construction: Recent algorithms exploit explicit combinatorial structure, reducing simplex constructions to single edge checks, drastically improving empirically observed timings (e.g., $5$-$10\times$ faster than Incremental-VR for sparse random graphs), and allowing effective construction for high-dimensional or sparse random complexes (Rieser, 2023).
• Block and Split Decompositions: For structured (e.g., circular or split-decomposable) random metrics, direct-sum or wedge-sum decompositions of the homology allow modular computation and more tractable analysis, particularly relevant when the random complex nearly satisfies geometric or combinatorial constraints (Gómez, 23 Mar 2024).

5. Extended Models, Robustness, and Generalizations

• Lp and Metric Thickenings: The classical $\ell_\infty$ Vietoris–Rips complex is unified with other constructions such as the $\ell_1$ (blurred magnitude homology) via the theory of $\ell_p$-Vietoris–Rips complexes. These admit stability theorems (interleaving under Gromov–Hausdorff perturbations), invariance under metric completion, and the limit homology as $r \to 0$ is independent of $p$ (Ivanov et al., 4 Nov 2024).
• Metric Thickenings (Wasserstein): The Vietoris–Rips metric thickening (the space of probability measures supported on small subsets) remedies non-metrizability; for totally bounded spaces, it has persistence barcodes and homotopy types matching the classical complex (Adamaszek et al., 2017, Gillespie, 2023, Moy, 2022).
• Degree–Rips Complexes and Outlier Robustness: Degree–Rips bifiltrations filter points by local density and offer robustness to outliers, especially relevant for random samples with noise, and their limiting homotopy type can be computed using circle results (Rolle, 2022).

6. Special Cases and Covering Criteria

• Spheres and High-Dimensional Manifolds: For $S^n$ with geodesic metric and $r \to \pi^-$, the homotopy connectivity of $\mathrm{VR}(S^n; r)$ is controlled in terms of covering numbers of $S^n$ and $\mathbb{R}P^n$; as the scale increases, the homotopy type changes infinitely often and the connectivity diverges, with precise inequalities connecting first nontrivial homotopy group to sphere/projective space coverings (Adams et al., 22 Jul 2024).
• Structured Spaces: For Platonic solids, ellipses, polygons, and other combinatorially regular finite metric spaces, explicit computations reveal sequences of homotopy types, typically as wedges of spheres, across discrete ranges of $r$. In random models mimicking these structures, similar patterns with localized topological transitions and nontrivial persistent homology can be observed (Saleh et al., 2023, Adamaszek et al., 2017, Adams et al., 2018).

7. Applications, Statistical and Algorithmic Implications

• Topological Data Analysis: Random Vietoris–Rips complexes are a fundamental tool for statistical inference of the topology of sampled spaces via persistent homology (Adamaszek et al., 2015, Iyer et al., 2018, Paik et al., 2022).
• Phase Detection: Theoretical thresholds inform the choice of scale in algorithmic and statistical pipelines, enable confidence intervals for inferred Betti numbers, and ground the statistical meaning of persistence barcodes (Iyer et al., 2018, Paik et al., 2022).
• Efficient Computation in Practice: New algorithmic frameworks based on combinatorial structure, approximation via sparsification, and block decomposition dramatically enlarge the feasible scope for experimental and applied paper of large or high-dimensional random Vietoris–Rips complexes (Sheehy, 2012, Rieser, 2023, Gómez, 23 Mar 2024).
• Robustness and Stability: The convergence of Betti numbers in expectation and variance, as well as the stability under metric perturbation and metric completion, supports the use of random VR complexes for reliable topological inference in noisy or incompletely sampled data scenarios (Ivanov et al., 4 Nov 2024, Paik et al., 2022, Gillespie, 2023).

The interplay of probability, metric geometry, combinatorics, and computation makes the paper of random Vietoris–Rips complexes both rich and highly relevant across pure mathematics, statistical topology, manifold learning, sensor networks, and geometric data analysis.