Random geometric graph diameter in the unit disk with ℓp metric

Abstract

Let n be a positive integer, λ > 0 a real number, and 1 ≤ p ≤ ∞. We study the unit disk random geometric graph G p(λ,n), defined to be the random graph on n vertices, independently distributed uniformly in the standard unit disk in ℝ2, with two vertices adjacent if and only if their ℓp-distance is at most λ. Let λ = c-√ln n/n, and let ap be the ratio of the (Lebesgue) areas of the ℓp- and ℓ2-unit disks. Almost always, G p(λ, n) has no isolated vertices and is also connected if c > ap-1/2, and has n1-apc2 (1 + o(1)) isolated vertices if c