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Betti numbers in the Random Connection Model for higher-dimensional simplicial complexes and the Boolean model

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Pith's one-line read This paper proves a central limit theorem for Betti numbers in a broad class of random simplicial complexes, including the Boolean model with arbitrary convex grains, with an explicit asymptotic variance.

desk verdict Useful CLT for Betti numbers in marked RCM and Boolean models, but the variance normalization in Lemma 4.5 rests on an invalid application of Lemma 3.1 to cubes that are not unions of the coordinate unit cubes. read the letter →

arxiv 2506.13429 v1 pith:LKYA2MFZ submitted 2025-06-16 math.PR

Dominik Pabst This is my paper · ORCID
classification math.PR MSC 60F0560D0560B99
keywords RandomConnectionModelBettinumberscentrallimittheoremsBooleansimplicialcomplexesweaklystabilizingfunctionalsmarkedPoissonprocessesstochasticgeometry
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper establishes a central limit theorem for functionals of a marked, stationary random simplicial complex built from a Poisson process on $\mathbb{R}^d\times\mathbb{A}$, where each potential simplex appears with a probability given by a translation-invariant connection function. The main result says that for any weakly stabilizing functional satisfying a moment condition, the fluctuation around its mean, scaled by the square root of the window volume, converges to a normal distribution with an explicit variance. The paper proves that $p$-th Betti numbers are such functionals provided $\mathbb{E}[\pi(V)^{3(p+1)}]<\infty$, where $\pi(V)$ is a mixed-Poisson edge-degree parameter. It then transfers the theorem to Betti numbers of the Boolean model with arbitrary compact convex grains via the nerve theorem, removing a restriction to equal-radius grains that earlier results needed. What makes the statement useful is that it covers higher-dimensional cohomology, such as loops, voids, and higher holes, rather than only component counts.

What carries the argument

The load-bearing object is the difference operator from (5), $$\Lambda_{(x,a,t,u)}f(\Delta_W)=f(\$Delta_W^{{(x,a,t,u)}}$)-f(\$Delta_W^{{(x,a,t,u)\setminus\{x\}}$}),$$ which inserts a vertex and compares the functional on the enlarged and the unenlarged complex; unlike the classical add-one-cost operator, it keeps the rest of the configuration's coordinates unchanged when $t=1$, which makes conditional-expectation computations tractable. Lemma 3.1 gives the identity $$\operatorname{Var}(f(\Delta_W))=\gamma\$int_0^{1}$\int_W\int_{\mathbb{A}}\int_{\mathbb{M}}\mathbb{E}\big[\mathbb{E}[\Lambda_{(x,a,t,u)}f(\Delta_W)\mid\Psi_t]^2\big]\,\mathbb{Q}(\mathrm{d}u)\Theta(\mathrm{d}a)\,\mathrm{d}x\,\mathrm{d}t,$$ and Lemma 4.5 turns this into the variance asymptotics $\operatorname{Var}(f(\Delta_{W_n}))/|W_n|\to\sigma^2$ by showing the integrand stabilizes. The weak-stabilization condition in Definition 4.2 is what lets the difference operator converge to a limiting random variable $Z_t$, and the moment condition (7) upgrades this to $L^2$-convergence, which is what the two-sequence normal-approximation theorem in Theorem 4.1 needs.

What would settle it

For the unmarked random connection model on the line with an explicit connection function, take $f$ to be the number of isolated vertices, compute the limit $\sigma^2$ from Lemma 4.5 directly, and compare it with $\operatorname{Var}(f(\Delta_{W_n}))/|W_n|$ obtained by exact finite-window computation or high-precision simulation; agreement at growing $n$ supports the variance identity, while systematic mismatch would refute Lemma 4.5 and with it the central limit theorem's normalization.

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Extended reading notes

Core claim

The paper's central claim is Theorem 4.6: for the marked stationary random simplicial complex built from a Poisson process on $\mathbb{R}^d\times\mathbb{A}$ with translation-invariant connection functions, every weakly stabilizing functional $f$, meaning that the local effect at the origin measured by the difference operator $\Lambda_{(\mathbf{0},V,1,U)}f(\Delta_{W_n})$ converges in probability as the window $W_n$ grows, that also satisfies the moment bound $\sup_{W\ni \mathbf{0}}\mathbb{E}[|\Lambda_{(\mathbf{0},V,1,U)}f(\Delta_W)|^{2+\varepsilon}]<\infty$ obeys $$\frac{f(\Delta_{W_n})-\mathbb{E}[f(\Delta_{W_n})]}{\sqrt{|W_n|}}\xrightarrow{d}N_{\$sigma^{2}$},$$ with $\sigma^2=\gamma\int_0^1\mathbb{E}[\tilde{\mathbb{E}}[Z_t\mid\Psi_t]^2]\,\mathrm{d}t$ and with strict positivity of $\sigma^2$ whenever the stabilizing limit $Z=Z_1$ is nonzero with positive probability. The paper then shows that the $p$-th Betti number $\beta_p$ is weakly stabilizing whenever $\mathbb{E}[\pi(V)^{3(p+1)}]<\infty$, where $\pi(V)$ is the mean number of edges an added vertex attaches to; the proof uses the stability bound $|\beta_p(K)-\beta_p(L)|\le (f_p(K)-f_p(L))+(f_{p+1}(K)-f_{p+1}(L))$ to control the difference operator by simplex degrees, and mixed-Poisson moment calculus to control those degrees. In the Boolean model with arbitrary compact convex grains, the nerve theorem identifies the Betti numbers of the union set with Betti numbers of the model's $\alpha$-skeleton, so the central limit theorem carries over to the Boolean model without the equal-radius restriction of earlier results.

Load-bearing premise

The load-bearing premise is the variance representation of Lemma 3.1, carried over from the random-graph setting: if that identity does not survive the cube-coordinate construction of the simplicial complex, the variance normalization and the central limit theorem would not follow.

Editorial extensions

If this is right

  • A multivariate central limit theorem holds for any finite family of weakly stabilizing functionals satisfying the moment condition, with a covariance matrix given by the corresponding limits of the difference operators.
  • Under a boundedness condition on the connection function, Betti numbers satisfy a law of large numbers: $\beta_p(\Delta_{W_n})/|W_n|$ converges in probability to a constant.
  • The theorem applies to the Euler characteristic, counts of induced subcomplexes, counts of connected components isomorphic to a fixed complex, and vertex degree counts, under explicit moment assumptions.
  • For the Boolean model with arbitrary compact convex grains, the $p$-th Betti number of the union set satisfies a central limit theorem; choosing the dimension parameter $\alpha$ large enough removes any skeleton-dimension restriction.
  • A combinatorial positivity criterion is given: if there is a connected complex $K$ with $|f(K)-f(L)|>0$ for every induced subcomplex on one fewer vertex, and if such a $K$ appears with positive probability, then the asymptotic variance is strictly positive.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper does not state this, but the same stabilization machinery should prove central limit theorems for the Betti numbers of sublevel sets in a filtration, because the difference operator records only local homological changes; this would require the moment condition to hold for the filtration's local contribution.
  • A testable extension suggested by the variance formula is to use it for statistical inference in topological data analysis: asymptotic confidence intervals for estimated Betti numbers follow whenever the limit variance is positive and the stabilization condition is checked.
  • Because the mark space $\mathbb{A}$ is an arbitrary Borel space, the theorem is not tied to geometric marks; one could apply it to marked network models in which connection probabilities depend on both spatial distance and categorical attributes, with only the spatial component required to be translation invariant.
  • The Boolean-model section hints that the equal-radius condition is an artifact of the Cech-complex identification; for bounded grains the mixing parameter is bounded, and for unbounded grains the paper's integrability condition on the Minkowski difference moment appears to be the natural replacement.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. This paper studies a marked stationary generalization of the random connection model (RCM) for higher-dimensional simplicial complexes, introduced in a companion paper [21]. The model places a Poisson process on R^d × A, where A is a Borel mark space, and adds simplices of dimension up to α with probabilities determined by translation-invariant connection functions. The main results are: (i) a central limit theorem (Theorem 4.6) for weakly stabilizing functionals of the induced complexes on growing cubes, under a moment condition; (ii) a demonstration that Betti numbers and other functionals such as the Euler characteristic, subcomplex counts, and d_l^m are weakly stabilizing under appropriate integrability assumptions (Section 5); (iii) a corollary on positivity of the asymptotic variance (Section 6); and (iv) an application to Betti numbers of the Boolean model with compact convex grains (Section 7). The argument follows the stabilization approach of [4], using a variance representation (Lemma 3.1) and a cube decomposition.

Significance. If the proof were completed, the results would be a significant advance in stochastic topology: they would provide CLTs for Betti numbers in a general marked RCM, including marked graphs, and extend known Boolean model results to grains of arbitrary compact convex shape (earlier work [27] handled only equal-radius balls). The paper is methodical in identifying the class of weakly stabilizing functionals and provides a general framework that includes several existing models. The companion paper [21] supplies the model definition and simplex-count asymptotics, and the present paper contributes the CLT machinery for Betti numbers. The proof of the stabilization class is detailed, and the moment conditions are explicit, yielding an explicit asymptotic variance formula.

major comments (2)
  1. [Section 3, Lemma 3.1] The variance representation is asserted for every bounded measurable W, but the construction of Δ_W via the coordinate system (k,i) means that Δ_W is determined by Ψ_W only when W is a union of the unit cubes Q_i, as the paper itself notes in Section 3. For a general cube, adding a point x ∉ W that shares a unit cube with points of W can change the coordinates (and hence the simplices) of points in W, so Λ_{x,a,t,u} f(Δ_W) does not vanish for x ∉ W. The proof's statement that Λ vanishes outside W is therefore false in the generic case, and Lemma 3.1 is not established for the cubes W_n = [-n^{1/d}/2, n^{1/d}/2)^d used in Lemma 4.5. Since Lemma 4.5 and Lemma 3.2 depend on Lemma 3.1, the variance normalization and the Poincaré inequality used in Theorem 4.6 are not justified.
  2. [Theorem 4.6 proof] The proof reduces at the outset to the special sequence W_n = [-n^{1/d}/2, n^{1/d}/2)^d, claiming that this is permissible by translation invariance. Translation invariance only allows shifting cubes in space, not changing their side lengths; an arbitrary cube sequence has side lengths a_n → ∞ and cannot be mapped to the special sequence. No approximation or subsequence argument is provided to pass from the special sequence to all cube sequences, so the theorem's statement for any sequence of cubes is not proven. This gap is separate from, and compounds, the Lemma 3.1 issue.
minor comments (4)
  1. [Lemma 4.5 proof] In the proof of Lemma 4.5, 'Theorem 3.1' should be 'Lemma 3.1', and equation (11) has the same expression on both sides; the right-hand side should presumably contain Λ_{(0,V,1,U)} f(Δ_W).
  2. [Remark 4.3] Remark 4.3 is deferred to the author's thesis [20] but is used in Theorem 4.6 to ensure finite variance of f(Δ_{G_1}); please include a proof or at least a precise statement of the result from [20] for self-containedness.
  3. [Theorem 4.6] Theorem 4.6 contains a typo in the definition of W_n ('d√n' should be n^{1/d}), and the displayed cube should be half-open consistently.
  4. [Section 7, equation (25)] Equation (25) uses the symbol ∫_{K_d} ambiguously; the integral over the mark space should be written with a variable name (e.g., Θ(dK)) to avoid confusion.

Circularity Check

0 steps flagged · score 2.0 of 10

No material circularity; the central CLT is built on external stabilization, variance, and normal-approximation frameworks, with only peripheral same-author citations.

full rationale

The paper's central derivation is not circular. Lemma 3.1's variance representation is imported from Theorem B.1 of [18], an external source, and is then used to prove the Poincare-type inequality and Lemma 4.5; no fitted parameter is later renamed as a prediction. Lemma 4.5 derives the asymptotic variance sigma^2 as a limit of variance integrals, and Theorem 4.6 follows the external stabilization and normal-approximation framework of [4] and [25] rather than assuming the conclusion. The weak stabilization of Betti numbers in Proposition 5.2 is verified by bounding the Betti difference operator by simplex degrees via Lemma 2.1(iv), cited to [27], and then checking the moment condition from the mixed-Poisson degree estimate; the target CLT is not used as an input. Same-author citations [20] and [21] supply the model construction, a proof of Remark 4.3, and simplex-count asymptotics used for the law of large numbers in Corollary 5.3, but these are peripheral to the Betti CLT and do not force it. The noted concern about Lemma 3.1's validity for cubes that are not unions of the lattice unit cubes is a potential technical gap in the variance normalization, not an equivalence between inputs and conclusions, so it does not constitute circularity. The score of 2 reflects only the presence of minor, non-load-bearing self-citations; no circular step can be exhibited by reducing an equation or theorem to its own input.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

No free parameters are fitted. The central claim rests on a product-form Poisson process, translation-invariant connection functions, external variance formulas, Lipschitz properties of Betti numbers, mixed Poisson moment transfer, the nerve theorem, and companion simplex-count results. None of the target theorem's conclusions are assumed as inputs.

assumptions (8)
  • domain assumption Connection functions phi_j are measurable, symmetric, and translation-invariant in the sense of equation (4).
    This defines the model and is needed for stationarity; it is not derived in the paper.
  • domain assumption The vertex process is a marked Poisson process on R^d times A with intensity gamma times Lebesgue times Theta.
    Existence and uniqueness follow from [19]; the product structure and independence of marks are assumed.
  • standard math Lemma 3.1 gives a variance representation for the special difference operator (5), transferred from Theorem B.1 of [18].
    This representation underlies the variance normalization in Lemma 4.5 and the final CLT.
  • standard math Lemma 2.1(iv), the Lipschitz-type bound for Betti numbers in terms of simplex counts, and Lemma 2.1(iii), weak additivity.
    Used in Proposition 5.2 to prove weak stabilization and the moment condition for Betti numbers.
  • standard math Mixed Poisson moment transfer from Proposition 1 of [17]: finiteness of moments of the mixing parameter implies finiteness of moments of the edge degree.
    Used to verify the moment condition (7) for example functionals.
  • standard math The nerve theorem for finite unions of non-empty compact convex sets.
    Used in Section 7 to transfer Betti CLTs from the nerve complex to the Boolean model.
  • domain assumption Integrability conditions E[pi(V)^{3(p+1)}] < infinity for Betti CLTs and the boundedness condition (22) for the law of large numbers.
    These conditions control the difference operator moments; in the Boolean model they follow when grains are bounded.
  • ad hoc to paper Companion results from [21] for expected simplex counts, used in Corollary 5.3 and Proposition 5.5.
    The current preprint relies on the same-author companion preprint [21] for these asymptotic counts; they are not independently verified here.

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Pith. "Pith review of Betti numbers in the Random Connection Model for higher-dimensional simplicial complexes and the Boolean model." pith.science (2026). https://pith.science/paper/LKYA2MFZ

@misc{pith2026250613429,
  author       = {Pith},
  title        = {Pith review of: Betti numbers in the Random Connection Model for higher-dimensional simplicial complexes and the Boolean model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LKYA2MFZ}},
  note         = {Machine review of arXiv:2506.13429}
}
abstract

Random simplicial complexes, as generalizations of random graphs, have become increasingly popular in the literature in recent years. In this paper, we consider a new model for a random simplicial complex that was introduced in arXiv:2506.11918, which generalizes the Random Connection Model in a natural way and includes several models used in the literature as special cases. We focus on the marked stationary case with vertices in $R^d\times A$, where the mark space $A$ is an arbitrary Borel space. We will derive a central limit theorem for an abstract class of functionals and show that many of the typical functionals considered in the study of simplicial complexes, such as Betti numbers, fall into this class. As an important special case, we obtain a central limit theorem for Betti numbers in the Boolean model.

Figures

Figures reproduced from arXiv: 2506.13429 by the authors.

Figure 1
Figure 1. (a) A realisation of the model with the connection functions from (1) and (b) a realisation of the model with the connection functions from (2) based on the same realisation of a Poisson process we choose the connection functions 𝜑1 (𝑥, 𝑦) = 1 { ‖𝑥 − 𝑦‖ ≤ 𝑟 } , 𝜑2 (𝑥, 𝑦, 𝑧) ≡ 𝑝 (1) for 𝑟 > 0 and 𝑝 ∈ [0, 1]. The model is built in two steps: first, a geometric graph with parameter 𝑟 is formed; second, each potential t… view at source ↗
Figure 2
Figure 2. (a) A union of convex sets and (b) the corresponding nerve end, we choose as the marking space the set 𝔸 = 𝑑 of all non-empty, compact and convex subsets of ℝ𝑑 , equipped with the 𝜎-field induced by the Hausdorff metric. According to the Theorems A.19 and A.26 in [19], this is a Borel space. As before, let Θ be a probability measure on 𝑑 . Additionally, we choose the connection functions 𝜑𝑗 ( (𝑥0 , 𝐾0 ), …, (𝑥𝑗 , … view at source ↗

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Percolation in the marked stationary Random Connection Model for higher-dimensional simplicial complexes

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    A sharp phase transition for q-percolation is proven for the marked stationary random connection model on higher-dimensional simplicial complexes, unifying the Boolean, Vietoris-Rips, and Cech cases.

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