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 $g_n(r) = g(nr),
\mbox{ for } r>0$ and for some integrable function $g$. 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 probability
$g_n(|x_i - x_j|)$ 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 $n \rightarrow \infty$ then for the number
$I_{(n)} (K)$ of isolated vertices of $X_n$ in a compact set $K$ with
non-empty interior, we have
$(\text{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.