  We study the evolution of random graphs where edges are added one by one
between pairs of weighted vertices so that resulting graphs are scale-free with
the degree exponent $\gamma$. We use the branching process approach to obtain
scaling forms for the cluster size distribution and the largest cluster size as
functions of the number of edges $L$ and vertices $N$. We find that the process
of forming a spanning cluster is qualitatively different between the cases of
$\gamma>3$ and $2<\gamma<3$. While for the former, a spanning cluster forms
abruptly at a critical number of edges $L_c$, generating a single peak in the
mean cluster size $<s>$ as a function of $L$, for the latter, however, the
formation of a spanning cluster occurs in a broad range of $L$, generating
double peaks in $<s>$.
