  In this paper we analyze the second-order Kuramoto model presenting a
positive correlation between the heterogeneity of the connections and the
natural frequencies in scale-free networks. We numerically show that
discontinuous transitions emerge not just in disassortative but also in
assortative networks, in contrast with the first-order model. We also find that
the effect of assortativity on network synchronization can be compensated by
adjusting the phase damping. Our results show that it is possible to control
collective behavior of damped Kuramoto oscillators by tuning the network
structure or by adjusting the dissipation related to the phases movement.
