High density asymptotics of the Poisson random connection model

Abstract

Consider a sequence of independent Poisson point processes X₁,X₂,… with densities λ₁,λ₂,…, respectively, and connection functions g₁,g₂,… defined by g_n(r)=g(nr), for r>0 and for some integrable function g. The Poisson random connection model (X_n,λ_n,g_n) is a random graph with vertex set X_n and, for any two points x_i and x_j in X_n, the edge <x_i,x_j> is included in the random graph with a probability g_n(|x_i-x_j|) independent of the point process as well as other pairs of points. We show that if λ_n / n^d → λ, (0 < λ < ∞) as n→∞ then for the number I_(n)(K) of isolated vertices of X_n in a compact set K with non-empty interior, we have (Var(I_(n)^(K)))-½I_(n)(K)-E(I_(n)(K))) converges in distribution to a standard normal random variable. Similar results may be obtained for clusters of finite size. The importance of this result is in the statistical simulation of such random graphs.