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Existence of a plane without edge crossings in projections of the random geometric graph

Lianne de Jonge, Kinga Nagy

Abstract

Consider a random geometric graph $G$ with a vertex set defined by a Poisson point process with intensity $t>0$ in a convex body. We can generate a drawing of the graph by projecting the construction onto some plane $L$. Choosing different planes leads to different drawings, and in particular, potentially more or fewer edge crossings. In this paper, we prove that if the connection radius is smaller than a given threshold, the probability that there exists a plane with zero crossings tends to one as $t\to \infty$. We also state the asymptotic probability that such a plane is found after considering a given number of randomly chosen planes.

Existence of a plane without edge crossings in projections of the random geometric graph

Abstract

Consider a random geometric graph with a vertex set defined by a Poisson point process with intensity in a convex body. We can generate a drawing of the graph by projecting the construction onto some plane . Choosing different planes leads to different drawings, and in particular, potentially more or fewer edge crossings. In this paper, we prove that if the connection radius is smaller than a given threshold, the probability that there exists a plane with zero crossings tends to one as . We also state the asymptotic probability that such a plane is found after considering a given number of randomly chosen planes.

Paper Structure

This paper contains 7 sections, 4 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Definitions and results
  3. Preliminaries
  4. Poisson approximation for crossings on multiple planes
  5. Proof of theorems
  6. Acknowledgements
  7. Funding information

Key Result

Theorem 1

Let $X_t$ be the measure of planes that contain zero crossings: Then, there exists $c^*=c^*(d,W)>0$ such that if $c'<c^*$ and for large enough $t$, then

Theorems & Definitions (11)

  • Theorem 1: Existence of a plane without crossings
  • Theorem 2: Finding a plane without crossings
  • Remark
  • Proposition 1
  • Lemma 1
  • proof : Proof of \ref{['prop:Poisson']}
  • proof : Proof of \ref{['lemma:covbound']}.
  • proof : Proof of \ref{['thm:existence']}
  • Claim 1
  • proof
  • ...and 1 more