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High density asymptotics of the Poisson random connection model

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  • Roy, Rahul
  • Sarkar, Anish

Abstract

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.

Suggested Citation

  • Roy, Rahul & Sarkar, Anish, 2003. "High density asymptotics of the Poisson random connection model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 318(1), pages 230-242.
    • Handle: RePEc:eee:phsmap:v:318:y:2003:i:1:p:230-242
    DOI: 10.1016/S0378-4371(02)01420-6
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    Cited by:

    1. van de Brug, Tim & Meester, Ronald, 2004. "On central limit theorems in the random connection model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 332(C), pages 263-278.

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