  Consider a graph on $n$ uniform random points in the unit square, each pair
being connected by an edge with probability $p$ if the inter-point distance is
at most $r$. We show that as $n\to\infty$ the probability of full connectivity
is governed by that of having no isolated vertices, itself governed by a
Poisson approximation for the number of isolated vertices, uniformly over all
choices of $p,r$. We determine the asymptotic probability of connectivity for
all $(p_n,r_n)$ subject to $r_n=O(n^{-\varepsilon})$, some $\varepsilon >0$. We
generalize the first result to higher dimensions and to a larger class of
connection probability functions.
