Contents I. Introduction
Over the last years, complex networks have been intensively investigated across many fields science and engineering [1]–[5]. The synchronization in complex network is one of the interesting and significant phenomena in complex networks, which has been a favorite topic for research in complex networks. Since the pioneering work of Pecora and Carroll [6], there has been a large amount of work on synchronization in complex networks. Pinning synchronization control method can reduce the number of controllers, improve control efficiency. For complex networks with non-delay coupling, many results [7]–[12] have been obtained about the pinning synchronization in complex networks. Complex networks with delay coupling are more common because of the network traffic congestions as well as the finite speed of signal transmission. The synchronization for various types of networks with delay coupling has been extensively studied. Li et al. [13] introduced complex dynamical network models with coupling delays for both continuous and discrete-time cases and then investigated their synchronization conditions and criteria. Gao et al. [14] presented several new delay-dependent conditions for a general complex network model with coupling delays. Wu et al. [15] generalized some previous results for delay-independent and delay-dependent synchronization. Li et al. [16] investigated the synchronization problem of some general complex dynamical networks with time-varying delays. Zhang et al. [17] divided the delay interval into some variable subintervals, and derived new synchronization criteria for complex networks with time-varying delays. However, studies on pinning synchronization in complex networks with delay coupling are fewer. Guo et al. [18] investigated pinning synchronization in complex networks with non-delay and delay coupling, and obtained some sufficient conditions for pinning synchronization. In the paper, pinning synchronization criteria are obtained for complex networks with non-delay and delay couplings. Sufficient conditions are derived for the synchronization to need the minimum number of pinning nodes. A numerical simulation gives effectiveness of the scheme.
