Eigenvector-based analysis of cluster synchronization in general complex networks of coupled chaotic oscillators

Whereas topological symmetries have been recognized as crucially important to the exploration of synchronization patterns in complex networks of coupled dynamical oscillators, the identification of the symmetries in large-size complex networks remains as a challenge. Additionally, even though the topological symmetries of a complex network are known, it is still not clear how the system dynamics is transited among different synchronization patterns with respect to the coupling strength of the oscillators. We propose here the framework of eigenvector-based analysis to identify the synchronization patterns in the general complex networks and, incorporating the conventional method of eigenvalue-based analysis, investigate the emergence and transition of the cluster synchronization states. We are able to argue and demonstrate that, without a prior knowledge of the network symmetries, the method is able to predict not only all the cluster synchronization states observable in the network, but also the critical couplings where the states become stable and the sequence of these states in the process of synchronization transition. The efficacy and generality of the proposed method are verified by different network models of coupled chaotic oscillators, including artificial networks of perfect symmetries and empirical networks of non-perfect symmetries. The new framework paves a way to the investigation of synchronization patterns in large-size, general complex networks.