  We study the bond percolation problem in random graphs of $N$ weighted
vertices, where each vertex $i$ has a prescribed weight $P_i$ and an edge can
connect vertices $i$ and $j$ with rate $P_iP_j$. The problem is solved by the
$q\to 1$ limit of the $q$-state Potts model with inhomogeneous interactions for
all pairs of spins. We apply this approach to the static model having
$P_i\propto i^{-\mu} (0<\mu<1)$ so that the resulting graph is scale-free with
the degree exponent $\lambda=1+1/\mu$. The number of loops as well as the giant
cluster size and the mean cluster size are obtained in the thermodynamic limit
as a function of the edge density, and their associated critical exponents are
also obtained. Finite-size scaling behaviors are derived using the largest
cluster size in the critical regime, which is calculated from the cluster size
distribution, and checked against numerical simulation results. We find that
the process of forming the giant cluster is qualitatively different between the
cases of $\lambda >3$ and $2 < \lambda <3$. While for the former, the giant
cluster forms abruptly at the percolation transition, for the latter, however,
the formation of the giant cluster is gradual and the mean cluster size for
finite $N$ shows double peaks.
