To deal with the problem of complex function projective synchronization in drive-response dynamical networks with $1+N$ nodes and multi-links in complex domain, an effective control scheme is proposed. Firstly, a more general drive-response complex network model is constructed by introducing $1+N$ nodes and multi-links. Then, the synchronization scheme is defined by making the scalar function a complex variable. Hybrid feedback controllers are designed and the corresponding positive definite function is constructed. Based on the Lyapunov stability theory and the linear matrix inequality, the sufficient condition of synchronization is given. Finally, the simulation results show the effectiveness of the synchronized methods, and the remarkable influences of multi-links on the synchronization speed are also revealed.