The random connection model: Connectivity, edge lengths, and degree distributions. Issue 2 (2nd November 2017) - Iyer, Srikanth K.
Abstract: Consider the random graph G ( P n, r ) whose vertex set P n is a Poisson point process of intensity n on ( − 1 2, 1 2 ] d,   d   ≥   2 . Any two vertices X i, X j ∈ P n are connected by an edge with probability g ( d ( X i, X j ) r ), independently of all other edges, and independent of the other points of P n . d is the toroidal metric, r  > 0 and g : [ 0, ∞ ) → [ 0, 1 ] is non‐increasing and α = ∫ ℝ d g ( | x | ) d x < ∞ . Under suitable conditions on g, almost surely, the critical parameter Mn for which G ( P n, · ) does not have any isolated nodes satisfies lim ⁡ n → ∞ α n M n d log ⁡ n = 1 . Let β = inf ⁡ { x > 0 : x g ( α x θ ) > 1 }, and θ be the volume of the unit ball in ℝ d . Then for all γ > β, G ( P n, ( γ log ⁡ n α n ) 1 d ) is connected with probability approaching one as n → ∞ . The bound can be seen to be tight for the usual random geometric graph obtained by setting g = 1 [ 0, 1 ] . We also prove some useful results on the asymptotic behavior of the length of the edges and the degree distribution in the connectivity regime . The results in this paper work for connection functions g that are not necessarily compactly supported but satisfy g ( r ) = o ( r − c ) .