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Asymptotic critical transmission radii in random geometry graphs over three-dimensional regions

Jie Ding, Shuai Ma, Xiang Wei, Xiaohua Xu, Xinshan Zhu

TL;DR

The work determines the precise asymptotic distributions of critical transmission radii for random geometric graphs on 3D regions, addressing both k-connectivity and minimum-degree criteria. It introduces an explicit radius form $r_n=\left(\dfrac{\log n+(\tfrac{3k}{2}-1)\log\log n+\xi}{\pi n}\right)^{1/3}$ with $\xi$ tied to the boundary area via $\mathrm{Area}(\partial\Omega)\frac{e^{-\frac{2\xi}{3}}}{\pi^{1/3}}(\frac{2}{3})^k\frac{1}{k!}=e^{-c}$, and proves that $n\int_{\Omega} \psi^k_{n,r}(x)\,dx\to e^{-c}$ leading to $\Pr\{\rho(\chi_n;\delta\ge k+1)\le r_n\}\to e^{-e^{-c}}$ and $\Pr\{\rho(\chi_n;\kappa\ge k+1)\le r_n\}\to e^{-e^{-c}}$. The analysis combines Poissonization/de-Poissonization with refined near-boundary geometry, extending two-dimensional techniques to 3D and revealing explicit boundary-driven corrections absent in 2D.

Abstract

This article presents the precise asymptotical distribution of two types of critical transmission radii, defined in terms of k-connectivity and the minimum vertex degree, for random geometry graphs distributed over three-dimensional regions.

Asymptotic critical transmission radii in random geometry graphs over three-dimensional regions

TL;DR

The work determines the precise asymptotic distributions of critical transmission radii for random geometric graphs on 3D regions, addressing both k-connectivity and minimum-degree criteria. It introduces an explicit radius form with tied to the boundary area via , and proves that leading to and . The analysis combines Poissonization/de-Poissonization with refined near-boundary geometry, extending two-dimensional techniques to 3D and revealing explicit boundary-driven corrections absent in 2D.

Abstract

This article presents the precise asymptotical distribution of two types of critical transmission radii, defined in terms of k-connectivity and the minimum vertex degree, for random geometry graphs distributed over three-dimensional regions.

Paper Structure

This paper contains 15 sections, 17 theorems, 173 equations, 10 figures, 2 tables.

Table of Contents

  1. Introduction and main results
  2. Preliminary results about regions in $\mathbb{R}^3$
  3. Spheres in $\mathbb{R}^3$
  4. Compact region $\Omega$ in $\mathbb{R}^3$
  5. Bounds of $|B(x,r)\cap\Omega|$ under condition $C_{\mathrm{I}}$
  6. Bounds of $|B(x,r)\cap\Omega|$ under condition $C_{\mathrm{II}}$
  7. Proof of Proposition \ref{['pro:Explicit-Form']} under condition $C_{\mathrm{I}}$
  8. Proof of Proposition \ref{['pro:Explicit-Form']} under condition $C_{\mathrm{II}}$
  9. Proof of Proposition \ref{['pro:conclusion-1']} under condition $C_{\mathrm{I}}$
  10. Poissonized version of Proposition \ref{['pro:conclusion-1']}
  11. A conclusion on Poisson process
  12. Proof of Proposition \ref{['pro:Poisson-Version']} under condition $C_{\mathrm{I}}$
  13. Proof of Proposition \ref{['pro:conclusion-1']} under condition $C_\mathrm{II}$
  14. Proof of Proposition \ref{['pro:conclusion-2']}
  15. Conclusions and Discussions

Key Result

Theorem A

(Penrose-RGG-bookPenrose-k-connectivity) Let $k>0,\lambda\in \mathbb{R}^+$, $\Omega\subset\mathbb{R}^d\;(d\geq 2)$ be a unit-volume hypercube. If sequence $\left(r_n\right)_{n\geq 1}$ satisfies then as $n\rightarrow\infty$, the probabilities of the two events $\rho(\chi_n;\delta\geq k+1)\leq r_n$ and $\rho(\chi_n;\kappa\geq k+1)\leq r_n$ both converge to $e^{-\lambda}$.

Figures (10)

  • Figure 1: Illustration of Lemma \ref{['lem:shadow-low-bound']}
  • Figure 2: $\mathcal{T}_O(\partial\Omega)$ is tangent to $\partial \Omega$ at point $O$
  • Figure 3: Illustration of Lemma \ref{['lemma:C2-Boundary-volume']}
  • Figure 4: $t(x)$ is nonnegative and major spherical segment contained in $\Omega$
  • Figure 5: Lower and Upper bounds of $|B(x,r)\cap \Omega|$: a 2D illustration (1)
  • ...and 5 more figures

Theorems & Definitions (26)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem 1
  • Remark 1
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Lemma 1
  • Lemma 2
  • ...and 16 more