Hierarchical synchronization in complex networks with heterogeneous degrees
We study synchronization behavior in networks of coupled chaotic oscillators with heterogeneous connection degrees. Our focus is on regimes away from the complete synchronization state, when the coupling is not strong enough, when the oscillators are under the influence of noise or when the oscillators are nonidentical. We have found a hierarchical organization of the synchronization behavior with respect to the collective dynamics of the network. Oscillators with more connections ͑hubs͒ are
... chronized more closely by the collective dynamics and constitute the dynamical core of the network. The numerical observation of this hierarchical synchronization is supported with an analysis based on a mean field approximation and the master stability function. Complex networks are playing an increasing role in the understanding of complex systems. The analysis of various real-world complex systems using the approach of complex networks has uncovered general and important principles in the structure organization of realistic systems. In particular, many complex networks are scalefree, characterized by a heterogeneous power-law distribution of the degrees. A problem of fundamental importance is the impact of the network topology on the dynamics of the complex systems, which has been recently studied intensively in the context of synchronization of coupled oscillators. Many previous works have focused on the global synchronizability, i.e., the ability of the network to synchronize completely for fully identical oscillators. In this paper we consider more natural situations where the networks are not in the complete synchronization state, for example, when the coupling is not strong enough, when the oscillators are in the presence of noise or when the oscillators are nonidentical. We have shown that complex networks of chaotic oscillators display significant collective oscillations in such regimes. More interestingly, we have found that in networks with heterogeneous degrees, the individual oscillators have different levels of synchronization with respect to the collective oscillations and they exhibit a hierarchical dependence on the connection degrees. The behavior can be understood analytically based on a mean field approximation and the linear stability analysis. Our results demonstrate that, in the context of synchronization, hubs having large degrees play the leading role in the formation of the dynamical core, which is the main contributor to the collective dynamics of the network. In the future, it is interesting to study hierarchical synchronization in more realistic networks whose connection topology and connection strengths are time varying and the results could have meaningful applications in the dynamics of real-world complex systems, such as the human brain.