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

Effect of assortative mixing in the second-order Kuramoto model