Betti-0 Curves: Topological Data Insights

• Betti-0 curves are topological descriptors that count connected components in a filtration, serving as key invariants in data analysis and geometric modeling.
• They are efficiently computed using methods like union-find and persistent homology, enabling robust analysis of cosmic structures and complex networks.
• Their application spans cosmology, network science, and algebraic geometry, where they aid in parameter estimation and distinguishing structural patterns.

A Betti-0 curve is a topological descriptor that records the evolution of the number of connected components (the zeroth Betti number) in a family of nested spaces or graphs parametrized by a continuous or discrete parameter, typically a scale, threshold, or filtration value. Appearing in applied topology, algebraic geometry, cosmological statistics, and network science, Betti-0 curves serve as fundamental invariants for characterizing connectivity patterns in data derived from point clouds, random fields, weighted matrices, and algebraic varieties. They are central to persistent homology and have significant applications in Topological Data Analysis (TDA), geometric modeling of networks, and the statistical analysis of large-scale structure in cosmology.

1. Mathematical Definition and Constructions

The Betti-0 curve $\beta_0(\lambda)$ arises in several settings, unified by the principle that it counts the number of connected components in a filtration. This can be formalized as follows:

• General Filtration Setting: For a filtration $\{X_\lambda\}_{\lambda}$—a nested sequence of topological spaces or graphs—define

$\beta_0(\lambda) = \mathrm{rank}~ H_0(X_\lambda)$

that is, the number of connected components present at parameter value $\lambda$ (Perez, 2022, Ouellette et al., 13 Feb 2025, Li et al., 8 Dec 2025, Caputi et al., 2024).

• Superlevel Set Filtration (Continuous Fields): For a function $f: X \rightarrow \mathbb{R}$ (continuous, e.g. on a manifold), the superlevel set at threshold $t$ is

$X^t := \{ x \in X : f(x) > t \}$

and the curve $t \mapsto \beta_0(t, f)$ records the number of connected superlevel regions as $t$ varies (Perez, 2022, Park et al., 2013).

• Point Clouds and Simplicial Complexes: For a finite set of points $X = \{x_i\}$ in $\{X_\lambda\}_{\lambda}$0, one builds a filtration of simplicial complexes (often α-complexes, Vietoris–Rips, or Čech complexes) parameterized by a scale $\{X_\lambda\}_{\lambda}$1; $\{X_\lambda\}_{\lambda}$2 is the number of components at that scale (Ouellette et al., 13 Feb 2025, Li et al., 8 Dec 2025, Tymchyshyn et al., 2023).
• Weighted Graph/Matrix Model: Given an $\{X_\lambda\}_{\lambda}$3 weighted symmetric matrix $\{X_\lambda\}_{\lambda}$4, one constructs a graph $\{X_\lambda\}_{\lambda}$5 by thresholding edges (distances or similarities). The Betti-0 curve is

$\{X_\lambda\}_{\lambda}$6

as $\{X_\lambda\}_{\lambda}$7 passes from the maximal to minimal matrix entry (Caputi et al., 2024, Curto et al., 2021).

• Algebraic Geometry (Projective Curves): The Betti-0 column in the minimal graded free resolution of a projective curve is the sequence $\{X_\lambda\}_{\lambda}$8 appearing as the first column (minimal generators) in the Betti table. For certain reducible curves, the term "Betti₀ curve" also refers to genus 0 line arrangements whose syzygy structure is completely determined and has a singular quadratic strand (Bruce et al., 2012).

2. Computation and Algorithmic Aspects

The computation of Betti-0 curves is highly efficient and can often be performed with near-linear complexity relative to the problem size:

• Union-Find Algorithm in Matrix Models: For a symmetric matrix $\{X_\lambda\}_{\lambda}$9, after sorting the off-diagonal entries, the Betti-0 curve can be computed by processing edge insertions via a disjoint-set union–find data structure (Caputi et al., 2024):
  1. List all distinct edge weights and sort.
  2. For each threshold, insert the corresponding edge and update the component count.
  3. Record $\beta_0(\lambda) = \mathrm{rank}~ H_0(X_\lambda)$0 after each insertion; overall complexity is $\beta_0(\lambda) = \mathrm{rank}~ H_0(X_\lambda)$1.

• Persistent Homology for Point Clouds: The α-complex or Vietoris–Rips complex filtration is constructed using spatial data structures; $\beta_0(\lambda) = \mathrm{rank}~ H_0(X_\lambda)$2 can be extracted by standard persistent homology, using libraries such as GUDHI, across a range of scales (Li et al., 8 Dec 2025, Ouellette et al., 13 Feb 2025, Tymchyshyn et al., 2023).

• Minimal Free Resolutions in Algebraic Geometry: For projective curves, the Betti-0 column (minimal generators in each degree) is computed within the syzygy computation of the graded free resolution of the coordinate or ideal ring (Bruce et al., 2012).
• Stable Estimation from Data: For random fields or stochastic processes, the empirical Betti-0 curve is obtained by subsampling, kernel smoothing, or Monte Carlo averaging of the connected component count at each threshold or scale, with control over estimator stability via Wasserstein distance (Perez, 2022).

3. Analytical and Numerical Properties

• Persistent Homology Interpretation: The Betti-0 curve is equivalent to the 0th persistent barcode: for a persistence diagram $\beta_0(\lambda) = \mathrm{rank}~ H_0(X_\lambda)$3,

$\beta_0(\lambda) = \mathrm{rank}~ H_0(X_\lambda)$4

(Ouellette et al., 13 Feb 2025, Perez, 2022).

• Monotonicity: In all standard filtrations, $\beta_0(\lambda) = \mathrm{rank}~ H_0(X_\lambda)$5 is a nonincreasing, piecewise-constant function as the parameter increases (edges or simplices are added monotonically).
• Parameter Dependence in Random Fields: In Gaussian field models, the shape and amplitude of $\beta_0(\lambda) = \mathrm{rank}~ H_0(X_\lambda)$6 as a function of threshold $\beta_0(\lambda) = \mathrm{rank}~ H_0(X_\lambda)$7 depend on the power spectrum slope $\beta_0(\lambda) = \mathrm{rank}~ H_0(X_\lambda)$8. For $\beta_0(\lambda) = \mathrm{rank}~ H_0(X_\lambda)$9, $\lambda$0 peaks at $\lambda$1 and falls rapidly for larger $\lambda$2; as $\lambda$3 decreases, the curve broadens and the amplitude decreases less steeply, in contrast to the genus curve's universality (Park et al., 2013). In cosmological point clouds, similar scale dependencies are observed, with precise sensitivity to cosmological parameters (Li et al., 8 Dec 2025, Ouellette et al., 13 Feb 2025, Tymchyshyn et al., 2023).
• Stability Properties: Betti-0 curves are Lipschitz stable with respect to perturbations in the input within Sobolev spaces, and their distributions change smoothly under small Wasserstein deformations in distribution. Explicit $\lambda$4 bounds relate the difference in Betti-0 curves to $\lambda$5-Wasserstein distances of the underlying random fields (Perez, 2022).

4. Applications and Representative Contexts

• Cosmology and Large-Scale Structure: Betti-0 curves are central in quantifying the topology of excursion sets in cosmic density fields, providing direct measures for high-density cluster and void abundances as functions of threshold; they are more informative than genus statistics alone and exhibit non-Gaussian sensitivity. Normalization schemes (with mean inter-point separation and volume factors) permit robust parameter estimation and emulator-driven inference in cosmological surveys. Joint analysis with power spectra tightens constraints on cosmological parameters (Park et al., 2013, Li et al., 8 Dec 2025, Ouellette et al., 13 Feb 2025, Tymchyshyn et al., 2023).
• Complex Networks and Geometry Inference: Betti-0 curves, and especially their integrals (integral Betti-0 signature $\lambda$6), discriminate between random, Euclidean, spherical, and hyperbolic geometries in network or correlation matrices. Applied to empirical brain, climate, and financial networks, these signatures reveal underlying hyperbolic-like topological structure (Caputi et al., 2024).
• Data Analysis, Signal Processing, and Biological Systems: Betti-0 curves derived from similarity or correlation matrices detect nontrivial structure in biological data (e.g., neural activity, gene expression), exhibiting invariance under monotone transformations of the data, and can test for low-rank or modular structure efficiently (Curto et al., 2021).
• Algebraic Curves and Syzygy Theory: In projective algebraic geometry, the Betti-0 column of a reducible curve's Betti table captures the minimal number of generators. For reducible genus 0 line arrangements (termed "Betti₀ curves" in editor's term) the entire quadratic strand is given by a binomial formula:

$\lambda$7

and higher strands vanish; these serve as canonical examples for minimal free resolutions in syzygy theory (Bruce et al., 2012).

5. Integral and Statistical Summaries

• Integral Betti-0 Signature: The area under the normalized Betti-0 curve, $\lambda$8, acts as a robust summary statistic, efficiently distinguishing model geometries and their influence on network topology. Smaller $\lambda$9 signals faster coalescence of components (e.g., hyperbolic networks), while larger $f: X \rightarrow \mathbb{R}$0 indicates more random or spherically organized networks (Caputi et al., 2024).
• Fisher Information and Parameter Sensitivity: Betti-0 curves supply strong Fisher information for cosmological parameter estimation, outperforming two-point statistics on non-linear scales. Combined Betti-0 and Betti-1 (loops) curves explain nearly all topological information content, with minimal additional gain from higher Betti numbers (Ouellette et al., 13 Feb 2025).

6. Special Cases and Explicit Examples

• Rank-One Matrices: For symmetric matrices $f: X \rightarrow \mathbb{R}$1 (rank-one), the Betti-0 curve has a staircase structure: each edge added connects a new vertex to a "cone-point," with each threshold corresponding to an off-diagonal entry. For mixed-sign $f: X \rightarrow \mathbb{R}$2, the Betti-0 curve displays two regimes, reflecting clique formation in blocks and final merging at the threshold $f: X \rightarrow \mathbb{R}$3 (Curto et al., 2021).
• Genus Zero Reducible Curves: For a line arrangement in $f: X \rightarrow \mathbb{R}$4 of genus 0, the Betti table has quadratic strand entries determined completely by

$f: X \rightarrow \mathbb{R}$5

for $f: X \rightarrow \mathbb{R}$6, resulting in an explicit, parameter-free description of their minimal graded free resolution (Bruce et al., 2012).

• Cosmic Web Invariance: In cosmic structure point clouds, the scale at which $f: X \rightarrow \mathbb{R}$7 (for fixed $f: X \rightarrow \mathbb{R}$8) is nearly independent of redshift within fixed halo mass bins, suggesting a new, robust topological invariant reflecting the persistent length scale of halo connectivity structures across the evolution of the Universe (Tymchyshyn et al., 2023).

7. Comparative and Interpretive Remarks

Context	Definition of $f: X \rightarrow \mathbb{R}$9	Key Computational Method
Random fields	Number of connected superlevel components	Voxel labeling, connected region counting (per $t$0 threshold)
Point clouds (cosmology)	Number of components in $t$1-complex	Persistent homology via α-complexes or Delaunay triangulations
Weighted matrices	Number of components in thresholded graph	Union-find (disjoint-set) on sorted edge list
Projective curves	Minimal generators in graded resolution	Syzygy computation, minimal free resolutions

The Betti-0 curve serves as a universal and highly interpretable summary of connectivity in a wide variety of topological filtrations. Its computational simplicity, stability properties, and robust parameter sensitivity underpin its widespread adoption in applied topology, cosmology, network science, and algebraic geometry (Park et al., 2013, Li et al., 8 Dec 2025, Perez, 2022, Ouellette et al., 13 Feb 2025, Curto et al., 2021, Tymchyshyn et al., 2023, Caputi et al., 2024, Bruce et al., 2012).