Random Geometric Graph Diameter in the Unit

Let n be a positive integer, and λ > 0 a real number. Let Vn be a set of n points randomly located within the unit disk, which are mutually independent. For 1 ≤ p ≤ ∞, define Gp(λ, n) to be the graph with the vertex set Vn, in which two vertices are adjacent if and only if their `p-distance is at most λ. We call this graph a unit disk random graph. Let λ = c √ ln n/n and let X be the number of isolated points in Gp(λ, n). Let ap be the (constant) ratio of the area of the `p-ball to the `2-ball of the same radius. Then, almost always, X = 0 when c > a −1/2 p , and X ∼ n1−apc2 when c < a−1/2 p . Penrose proved that with probability approaching 1, the graph Gp(λ, n) is connected when it has minimum degree 1. We extend Penrose’s method to prove that if Gp(λ, n) is connected, then there exists a constant K, independent of p, such that the diameter of Gp(λ, n) is bounded above by K/λ. We show in addition that when c exceeds a certain constant depending on p, the diameter of Gp(λ, n) is bounded above by (2 · 2 + o(1))/λ. More generally, there is a function cp(δ) such that the diameter is at most 2(1 + δ + o(1))/λ when c > cp(δ).