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Limit theory of sparse random geometric graphs in high dimensions

Gilles Bonnet, Christian Hirsch, Daniel Rosen, Daniel Willhalm

TL;DR

The paper develops a comprehensive limit theory for sparse, high-dimensional $l_ty$-random geometric graphs on a torus, establishing moment asymptotics, functional CLTs, and Poisson approximations for additive functionals such as subgraph counts and Betti numbers of the Rips complex, as well as multi-additive functionals including persistent Betti numbers. The authors introduce a unifying framework built on Mecke's formula, Stein's method, and cumulant techniques, showing that in the sparse regime the dominant contributions come from minimal-size components and that long-range dependencies vanish asymptotically. They provide concrete examples (subgraph counts, Betti numbers) and multi-additive extensions (dynamic counts, persistence) and prove both scalar and functional limit theorems, including a Poisson-approximation regime when the limiting intensity $ ho_d$ tends to a finite $K>0$. The results advance understanding of high-dimensional stochastic geometry and topological data analysis on large, sparse networks, with potential implications for high-dimensional clustering and persistent homology in data science contexts.

Abstract

We study topological and geometric functionals of $l_\infty$-random geometric graphs on the high-dimensional torus in a sparse regime, where the expected number of neighbors decays exponentially in the dimension. More precisely, we establish moment asymptotics, functional central limit theorems and Poisson approximation theorems for certain functionals that are additive under disjoint unions of graphs. For instance, this includes simplex counts and Betti numbers of the Rips complex, as well as general subgraph counts of the random geometric graph. We also present multi-additive extensions that cover the case of persistent Betti numbers of the Rips complex.

Limit theory of sparse random geometric graphs in high dimensions

TL;DR

The paper develops a comprehensive limit theory for sparse, high-dimensional -random geometric graphs on a torus, establishing moment asymptotics, functional CLTs, and Poisson approximations for additive functionals such as subgraph counts and Betti numbers of the Rips complex, as well as multi-additive functionals including persistent Betti numbers. The authors introduce a unifying framework built on Mecke's formula, Stein's method, and cumulant techniques, showing that in the sparse regime the dominant contributions come from minimal-size components and that long-range dependencies vanish asymptotically. They provide concrete examples (subgraph counts, Betti numbers) and multi-additive extensions (dynamic counts, persistence) and prove both scalar and functional limit theorems, including a Poisson-approximation regime when the limiting intensity tends to a finite . The results advance understanding of high-dimensional stochastic geometry and topological data analysis on large, sparse networks, with potential implications for high-dimensional clustering and persistent homology in data science contexts.

Abstract

We study topological and geometric functionals of -random geometric graphs on the high-dimensional torus in a sparse regime, where the expected number of neighbors decays exponentially in the dimension. More precisely, we establish moment asymptotics, functional central limit theorems and Poisson approximation theorems for certain functionals that are additive under disjoint unions of graphs. For instance, this includes simplex counts and Betti numbers of the Rips complex, as well as general subgraph counts of the random geometric graph. We also present multi-additive extensions that cover the case of persistent Betti numbers of the Rips complex.
Paper Structure (16 sections, 18 theorems, 157 equations)

This paper contains 16 sections, 18 theorems, 157 equations.

Table of Contents

  1. Introduction
  2. Model and main results
  3. Model definition
  4. Additive functionals
  5. Multi-additive functionals
  6. Examples
  7. Additive functionals
  8. Multi-additive functionals
  9. Proof of Theorem \ref{['fclt']}, parts 1 and 2
  10. Expectation and covariance asymptotics
  11. Multivariate CLT
  12. Proof of Theorem \ref{['pois_thm']}
  13. Proof of Theorem \ref{['pair_fclt']} -- multi-additive functionals
  14. Expectation and variance asymptotics
  15. CLT
  16. ...and 1 more sections

Key Result

Theorem 2.1

Let $\operatorname{\mathfrak{a}}$ be an additive nonnegative functional with $\operatorname{\mathfrak{a}}(G) \in e^{O(|G|)}$. Moreover, assume that $\lambda_d \to 0$ and $|W_d|^{1/d}\to \infty$.

Theorems & Definitions (52)

  • Theorem 2.1: CLT for additive functionals
  • Theorem 2.2: Poisson approximation
  • Theorem 2.3: CLT for multi-additive functionals
  • Remark 2.4
  • Example 3.1: Subgraph counts
  • Example 3.2: Betti numbers
  • Example 3.3: Linear combinations of univariate functionals
  • Example 3.4: Dynamic subgraph count
  • Example 3.5: Persistent Betti numbers
  • Lemma 4.1: Moment computations
  • ...and 42 more