Random Geometric Graph Diameter in the Unit Disk with ℓp Metric

  • Robert B. Ellis
  • Jeremy L. Martin
  • Catherine Yan
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3383)

Abstract

Let n be a positive integer, λ> 0 a real number, and 1≤ p≤ ∞. We study the unit disk random geometric graphGp(λ,n), defined to be the random graph on n vertices, independently distributed uniformly in the standard unit disk in \({\mathbb R}^2\), with two vertices adjacent if and only if their ℓp-distance is at most λ. Let \(\lambda=c\sqrt{\ln n/n}\), and let ap be the ratio of the (Lebesgue) areas of the ℓp- and ℓ2-unit disks. Almost always, Gp(λ,n) has no isolated vertices and is also connected if c>ap− − 1/2, and has \(n^{1-a_pc^2}(1+o(1))\) isolated vertices if c<ap− − 1/2. Furthermore, we find upper bounds (involving λ but independent of p) for the diameter of Gp(λ,n), building on a method originally due to M. Penrose.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Robert B. Ellis
    • 1
  • Jeremy L. Martin
    • 2
  • Catherine Yan
    • 1
  1. 1.Department of MathematicsTexas A&M UniversityCollege StationUSA
  2. 2.School of MathematicsUniversity of MinnesotaMinneapolisUSA

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