  A random graph process, $\Gorg[1](n)$, is a sequence of graphs on $n$
vertices which begins with the edgeless graph, and where at each step a single
edge is added according to a uniform distribution on the missing edges. It is
well known that in such a process a giant component (of linear size) typically
emerges after $(1+o(1))\frac{n}{2}$ edges (a phenomenon known as ``the double
jump''), i.e., at time $t=1$ when using a timescale of $n/2$ edges in each
step.
  We consider a generalization of this process, $\Gorg[K](n)$, which gives a
weight of size 1 to missing edges between pairs of isolated vertices, and a
weight of size $K \in [0,\infty)$ otherwise. This corresponds to a case where
links are added between $n$ initially isolated settlements, where the
probability of a new link in each step is biased according to whether or not
its two endpoint settlements are still isolated.
  Combining methods of \cite{SpencerWormald} with analytical techniques, we
describe the typical emerging time of a giant component in this process,
$t_c(K)$, as the singularity point of a solution to a set of differential
equations. We proceed to analyze these differential equations and obtain
properties of $\Gorg$, and in particular, we show that $t_c(K)$ strictly
decreases from 3/2 to 0 as $K$ increases from 0 to $\infty$, and that $t_c(K) =
\frac{4}{\sqrt{3K}}(1 + o(1))$. Numerical approximations of the differential
equations agree both with computer simulations of the process $\Gorg(n)$ and
with the analytical results.
