Chaotic synchronization on directed networks

The phenomenon of synchronization occurring in coupled chaotic maps on a directed random network is studied. The network is characterized by the average degree of its nodes and the fraction of directed links. It is found that the required coupling strength so that the chaotic synchronization emerges is smaller when the fraction of directed links is increased. In addition, the system undergoes a transition from an asynchronous phase to a synchronous one at some critical values of its parameters. The critical boundary separating the synchronous from the asynchronous regime is calculated on the parameter space of the system, given by the coupling strength and the fraction of directed links of the network. The phase transition between the two regimes is of second order for all values of the fraction of directed links, and the critical exponent depends of it.