Karhunen–Loève decomposition approach to analyzing complex network synchronization

In this paper, the approach of the Karhunen–Loève decomposition, known also as the proper orthogonal modes (POMs), is taken to analyze phase synchronization of various complex networks with different topologies, namely the classic Kuramoto model, coupled chaotic maps with Gaussian delays and a chain of diffusively coupled bistable oscillators. In the case of the Kuramoto model, the POMs reveal the tendency and the level of synchronization with the increase of the coupling strength for globally coupled networks and scale-free networks, while periodic POMs are found in nearest-neighbor coupled networks. Furthermore, for cluster networks on the Kuramoto model, the first leading POMs based on different time intervals reveal that different sub-groups of nodes synchronize gradually to different levels, eventually leading to the complete phase synchronization. In the case of coupled chaotic maps, some properties of phase synchronization change with the coupling strength value. In the case of the chain of diffusively coupled bistable oscillators, several main POMs not only determine the network phase synchronization but also provide good reconstruction of the network responses.