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Random geometric complexes
Matthew Kahle
TL;DR
The paper studies how homology in random Čech and Vietoris–Rips complexes changes as the radius varies, addressing higher-dimensional analogues of random-geometric-graph thresholds. It analyzes four radius regimes using asymptotic probability, Betti-number estimates, and discrete Morse theory, finding two thresholds for every higher homology group and contractibility or high connectivity in the connected regime.
Problem
The paper asks how the topological properties and expected homology ranks of random Čech and Vietoris–Rips complexes vary with radius, including their non-monotone higher homology.
Method
The paper analyzes subcritical, critical, supercritical, and connected regimes using geometric probability and discrete Morse theory.
Results
For every k ≥ 1, H_k is nonzero on an interval between two vanishing thresholds; Betti numbers have sparse-regime asymptotics, denser-regime bounds, and connected-regime contractibility or fixed-k connectivity.
Takeaways & Limitations
Higher homology in these random complexes is intrinsically non-monotone, while sufficiently large radii eliminate Čech homology and make Vietoris–Rips complexes highly connected.
Takeaways & Limitations
The connected-regime Vietoris–Rips result does not rule out nontrivial homology in dimensions k that grow with n.
Abstract
from arXiv · showhide
We study the expected topological properties of Cech and Vietoris-Rips complexes built on i.i.d. random points in R^d. We find higher dimensional analogues of known results for connectivity and component counts for random geometric graphs. However, higher homology H_k is not monotone when k > 0. In particular for every k > 0 we exhibit two thresholds, one where homology passes from vanishing to nonvanishing, and another where it passes back to vanishing. We give asymptotic formulas for the expectation of the Betti numbers in the sparser regimes, and bounds in the denser regimes. The main technical contribution of the article is in the application of discrete Morse theory in geometric probability.
1. Introduction
This paper studies the topology of random Čech and Vietoris–Rips complexes built from i.i.d. points in Euclidean space, with radius varying across scales. It identifies homology thresholds and expected Betti-number behavior, motivated partly by topological data analysis and probabilistic comparisons for point-cloud statistics.
- Scope and goals: The complexes are built on i.i.d. random points in R^d, and the paper studies vanishing, non-vanishing, and expected ranks of their homology groups.The results apply to both random Čech and Vietoris–Rips complexes, with homology coefficients over any field.
- Motivation: The study is motivated by topological data analysis, where a probabilistic null hypothesis is needed for comparison with topological statistics of point-cloud data.The paper also places random spaces in the broader context of probabilistic and geometric methods.
- Scope and goals: The radius r varies from 0 to ∞, allowing the paper to identify homology thresholds and derive asymptotic formulas and bounds for expected Betti numbers β_k.The analysis covers fairly general distributions on Euclidean space R^d.
- Relation to geometric probability: Higher-dimensional results extend threshold phenomena known for connectivity and component counts in geometric random graphs, while homology itself is non-monotone.For each k, homology is nonzero over an interval of radii and vanishes for sufficiently small or large radii.
- Definitions and comparison: Čech and Vietoris–Rips complexes behave similarly qualitatively but differ quantitatively, making those differences an explicit goal of the article.The Vietoris–Rips complex is equivalently the clique complex of the geometric random graph.
2. Summary of results
The paper divides random geometric complexes into four radius regimes and characterizes their homology across those scales. It gives general sparse-regime formulas, denser-regime bounds, and a discrete-Morse-theoretic treatment of Vietoris–Rips complexes.
- Overall regime structure: Four radius regimes have qualitatively different behavior: subcritical, critical, supercritical, and connected.The paper organizes its analysis around these regimes.
- Subcritical regime: In the subcritical regime r = o(n^-1/d), H_k has a vanishing-to-non-vanishing threshold, and E[β_k] has an asymptotic formula for k ≥ 1.These results hold for distributions on R^d with bounded measurable density functions.
- Critical regime: In the critical regime r = Θ(n^-1/d), E[β_k] = Θ(n) and Var[β_k] = Θ(n) for every k.This regime is where components begin connecting and the giant component emerges.
- Supercritical regime: In the supercritical regime r = ω(n^-1/d), the Vietoris–Rips complex has sub-linear expected Betti-number growth, established using a Morse-theoretic argument.Combining geometric probability with discrete Morse theory is the article’s main technical contribution.
- Connected regime: In the connected regime r = Ω((log n/n)^(1/d)), the Čech complex is contractible and the Vietoris–Rips complex is k-connected for every fixed k.The latter conclusion implies vanishing homology through dimension k.
- Non-monotonicity: Every H_k with k ≥ 1 is nonzero on an interval of radii and vanishes outside it, so each higher homology group passes through two thresholds.This non-monotonicity contrasts with simpler monotone connectivity phenomena.
3. Subcritical
In the subcritical regime, the paper identifies homology thresholds and derives expected Betti-number asymptotics for random Vietoris-Rips and Čech complexes, using feasible subgraph counts and minimal homology supports.
- The paper computes vanishing-to-nonvanishing thresholds and expected Betti-number asymptotics for H_k in the subcritical regime.
- Expectation: For random Vietoris-Rips complexes, E[β_k] has an asymptotic formula when r_n = O(n^-1/d-ε), for d ≥ 2 and k ≥ 1.
- Expectation: For random Čech complexes, the analogous asymptotic formula applies for d ≥ 2, 1 ≤ k ≤ d − 1, and r = O(n^-1/d−ε).
- Expectation: The Vietoris-Rips proof bounds nonminimal contributions through expected counts of connected feasible induced subgraphs and uses Penrose’s subgraph-count results.
- Expectation: In the sparse regime, almost all homology is contributed by vertex-minimal spheres: cross-polytope boundaries for Vietoris-Rips complexes and the empty simplex for Čech complexes.
- Expectation: For k = 2, nonminimal seven-vertex contributions are negligible compared with octahedral components, yielding E[β_2] = Θ(n^6r^5d).
- Expectation: Regular 2k-gons show that cross-polytope 1-skeletons are geometrically feasible in the plane for every k.
4. Critical
In the critical regime r = Θ(n^-1/d), expected kth Betti numbers grow linearly for both random Vietoris-Rips and Čech complexes, while the regime coincides with geometric-graph percolation.
- E[β_k] = Θ(n) for every fixed k ≥ 1 in both random Vietoris-Rips and Čech complexes when r = Θ(n^-1/d).
- The proof extends the sparse-regime component-count argument to the thermodynamic limit, using E[e_o_k] = Θ(n) and bounds on larger components.
- This regime is especially relevant because percolation occurs in the underlying random geometric graph.
- The paper poses whether the face-incidence graph has a unique k-dimensional giant component above a threshold and only small components below it.
5. Supercritical
In the supercritical regime r = ω(n^-1/d), discrete Morse theory bounds Betti numbers for random Vietoris-Rips complexes on points uniformly sampled from a smoothly bounded convex body. The resulting upper bound is sub-linear, showing that Betti numbers grow fastest in the thermodynamic limit.
- The expected Betti numbers have a sub-linear upper bound in the supercritical regime, so they grow fastest in the thermodynamic limit.The theorem’s constant c depends on the convex body K but not on k, and in fact depends only on K’s volume.
- Discrete Morse theory is the main tool for bounding Betti numbers through critical faces of the complex.The construction uses a discrete vector field whose decreasing indices prevent closed V-paths, making it a discrete gradient vector field.
- r = ω(n^-1/d) defines the supercritical regime analyzed for random Vietoris-Rips complexes.The points are sampled uniformly from a smoothly bounded convex body K.
- A geometric lemma supplies a constant ǫd > 0 and a region I with measure at least ǫdr^d, uniformly supporting the critical-face estimate.The scaled lemma applies when one pair of points is farther than r while all other pairwise distances are at most r.
- Vertices falling in region I force a face F to be paired, so F can be critical only when no sampled vertex falls in that region.This yields an upper bound on the probability that F is critical, using independence of the random points.
6. Connected
In the connected regime, above a radius on the order of (log n/n)^(1/d), random Čech complexes become contractible while Vietoris–Rips complexes become highly connected. The section also establishes a second homology threshold for Vietoris–Rips complexes, using geometric random-graph components and discrete Morse theory.
- Čech complexes: r = Ω((log n/n)^(1/d)) suffices for random Čech complexes to be a.a.s. contractible.The proof uses coverage by radius-r/2 balls and the nerve theorem.
- Čech complexes: A lower constant multiple of (log n/n)^(1/d) leaves the random Čech complex a.a.s. disconnected, making the contractibility threshold sharp up to constants.
- Vietoris–Rips complexes: r = Ω((log n/n)^(1/d)) yields a Vietoris–Rips complex that is a.a.s. k-connected for every fixed k, with the constant depending on the body’s volume and k.Discrete Morse theory shows that, asymptotically, only the lowest-dimensional critical cell remains.
- Vietoris–Rips complexes: The Vietoris–Rips result does not establish contractibility, because nontrivial homology may remain in dimensions k that grow with n.
- Homology thresholds: A second threshold marks the return from nonvanishing to vanishing kth homology, with an intermediate regime containing a component homeomorphic to the sphere S^k.The component result relies on a feasible-subgraph lemma for geometric random graphs.
- Scope: The connected-regime arguments assume a uniform distribution on a smoothly bounded convex body, although similar methods may apply more generally.
7. Further directions
The paper connects its results to statistical persistent homology, while identifying open questions about long-lived classes, manifold settings, torsion, and the radius dependence of expected Betti numbers.
- Persistent homology: The results bound the number of nontrivial homology classes and suggest that most classes, arising from vertex minimal spheres, do not persist for long.Ruling out long-persisting classes remains an important step toward quantifying the statistical significance of persistent homology.
- Persistent homology: A theorem excluding homology classes that persist for a long time altogether would help quantify the statistical significance of persistent homology.
- Manifold extensions: The results are stated for Euclidean space, although analogous homology results are expected for d-dimensional compact Riemannian manifolds.In the supercritical regime, manifold homology would coexist with noise, while persistent homology is expected to detect the manifold’s homology.
- Unresolved topology: The analysis bounds Betti numbers but does not address coefficients or the torsion in Z-homology of random complexes.The paper states that more refined tools appear necessary for detecting this torsion.
- Open structural questions: The studied topological properties are non-monotone and appear roughly unimodal, but whether E[β_k] is eventually monotone in r remains open.
Appendix: discrete Morse theory
The appendix introduces discrete Morse theory through discrete vector fields, gradient paths, and critical simplices, then uses the fundamental theorem to relate critical cells to homotopy and Betti-number bounds.
- Definitions: A discrete vector field is a collection of face pairs in which each face belongs to at most one pair.
- Definitions: A discrete gradient vector field is a discrete vector field with no closed V-paths.A closed V-path alternates paired faces and unpaired codimension-one incidences before returning to its starting face.
- Definitions: A simplex not contained in any vector-field pair is called critical.
- Main theorem: The fundamental theorem states that a simplicial complex with a discrete gradient vector field is homotopy equivalent to a CW complex with one k-cell for each critical k-dimensional simplex.
- Application: Counting cells gives the bound β_k ≤ f_k, which the paper uses to bound the expected dimension of homology.The method exploits this cellular-homology inequality in Sections 5 and 6.