Theoretical Statistics and Mathematics Unit, ISI Delhi

High density asymptotics of the Poisson random connection model

by Rahul Roy and Anish Sarkar

Consider a sequence of independent Poisson point processes $X_1, X_2, \ldots,$ with densities $\lambda_1, \lambda_2, \ldots$ respectively and connection functions $g_1, g_2, \ldots$ defined by $g_n(r) = g(nr)$, for $r > 0$ and for some integrable functions $g$. The Poisson random connection model $(X_n, \lambda_n, g_n)$is a random graph with vertex set $X_n$ and , for any two points $x_i$ and $x_j$ in $X_n$, the edge $$ is included in the random graph with a prabability $g_n(|x_i-x_j|)$independent of the point process as well as other pairs of points. We show that if $\lambda_n/n^d \rightarrow \lambda$, $(0 < \lambda < \infty)$ as $n\rightarrow \infty$ the for the number $I_{(n)}(K)$ of isolated vertices of $X_n$ in a compact set $K$ with non-empty interior, we have $(Var(I_{(n)}(K)))^{-1/2} (I_{(n)}(K) - E(I_{(n)}(K))$ converges in distribution to a standard normal random variable. similar results may be obtained for clusters of finite size. The importance of this results is in the statistical simulation of such random graphs.

isid/ms/2002/16 [fulltext]

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