High density asymptotics of the Poisson random connection model

Consider a sequence of independent Poisson point processes X1,X2,… with densities λ1,λ2,…, respectively, and connection functions g1,g2,… defined by gn(r)=g(nr), for r>0 and for some integrable function g. The Poisson random connection model (Xn,λn,gn) is a random graph with vertex set Xn and, for any two points xi and xj in Xn, the edge 〈xi,xj〉 is included in the random graph with a probability gn(|xi−xj|) independent of the point process as well as other pairs of points. We show that if λn/nd→λ,(0<λ<∞) as n→∞ then for the number I(n)(K) of isolated vertices of Xn in a compact set K with non-empty interior, we have (Var(I(n)(K)))−1/2(I(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.