Synchronization transition in Sakaguchi-Kuramoto model on complex networks with partial degree-frequency correlation

We investigate transition to synchronization in Sakaguchi-Kuramoto (SK) model on complex networks analytically as well as numerically. Natural frequencies of a percentage ($f$) of higher degree nodes of the network are assumed to be correlated with their degrees and that of the remaining nodes are drawn from some standard distribution namely Lorenz distribution. The effects of variation of $f$ and phase frustration parameter $\alpha$ on transition to synchronization are investigated in detail. Self-consistent equations involving critical coupling strength ($\lambda_c$) and group angular velocity ($\Omega_c$) at the onset of synchronization have been derived analytically in the thermodynamic limit. For the detailed investigation we considered SK model on scale-free as well as Erd\H{o}s-R\'{e}nyi (ER) networks. Interestingly explosive synchronization (ES) has been observed in both the networks for different ranges of values of $\alpha$ and $f$. For scale-free networks, as the value of $f$ is set within $10\% \leq f \leq 70\%$, the range of the values of $\alpha$ for existence of the ES is greatly enhanced compared to the fully degree-frequency correlated case. On the other hand, for random networks, ES observed in a narrow window of $\alpha$ when the value of $f$ is taken within $30\% \leq f \leq 50\%$. In all the cases critical coupling strengths for transition to synchronization computed from the analytically derived self-consistent equations show a very good agreement with the numerical results.