  We study the synchronizability and the synchronization dynamics of networks
of nonlinear oscillators. We investigate how the synchronization of the network
is influenced by some of its topological features such as variations of the
power law exponent $\gamma$ and the degree correlation coefficient $r$. Using
an appropriate construction algorithm based on clustering the network vertices
in $p$ classes according to their degrees, we construct networks with an
assigned power law distribution but changing degree correlation properties. We
find that the network synchronizability improves when the network becomes
disassortative, i.e. when nodes with low degree are more likely to be connected
to nodes with higher degree. We consider the case of both weighed and unweighed
networks. The analytical results reported in the paper are then confirmed by a
set of numerical observations obtained on weighed and unweighed networks of
nonlinear R\"ossler oscillators. Using a nonlinear optimization strategy we
also show that negative degree correlation is an emerging property of networks
when synchronizability is to be optimized. This suggests that negative degree
correlation observed experimentally in a number of physical and biological
networks might be motivated by their need to synchronize better.
