  We generate correlated scale-free networks in the configuration model through
a new rewiring algorithm which allows to tune the Newman assortativity
coefficient $r$ and the average degree of the nearest neighbors $K$ (in the
range $-1\le r \le 1$, $K\ge \langle k \rangle$). At each attempted rewiring
step, local variations $\Delta r$ and $\Delta K$ are computed and then the step
is accepted according to a standard Metropolis probability $ \exp(\pm\Delta
r/T)$, where $T$ is a variable temperature. We prove a general relation between
$\Delta r$ and $\Delta K$, thus finding a connection between two variables
which have very different definitions and topological meaning. We describe
rewiring trajectories in the $r$-$K$ plane and explore the limits of maximally
assortative and disassortative networks, including the case of small minimum
degree ($k_{min} \ge 1$) which has previously not been considered. The size of
the giant component and the entropy of the network are monitored in the
rewiring. The average number of second neighbours in the branching
approximation $\bar{z}_{2,B}$ is proven to be constant in the rewiring, and
independent from the correlations for Markovian networks. As a function of the
degree, however, the number of second neighbors gives useful information on the
network connectivity and is also monitored.
