  Consider a network where all nodes are distributed on a unit square following
a Poisson distribution with known density $\rho$ and a pair of nodes separated
by an Euclidean distance $x$ are directly connected with probability
$g(\frac{x}{r_{\rho}})$, where $g:[0,\infty)\rightarrow[0,1]$ satisfies three
conditions: rotational invariance, non-increasing monotonicity and integral
boundedness, $r_{\rho}=\sqrt{\frac{\log\rho+b}{C\rho}}$,
$C=\int_{\Re^{2}}g(\Vert \boldsymbol{x}\Vert)d\boldsymbol{x}$ and $b$ is a
constant, independent of the event that another pair of nodes are directly
connected. In this paper, we analyze the asymptotic distribution of the number
of isolated nodes in the above network using the Chen-Stein technique and the
impact of the boundary effect on the number of isolated nodes as
$\rho\rightarrow\infty$. On that basis we derive a necessary condition for the
above network to be asymptotically almost surely connected. These results form
an important link in expanding recent results on the connectivity of the random
geometric graphs from the commonly used unit disk model to the more generic and
more practical random connection model.
