Combined with RBF neural network and sliding mode control, the synchronization between drive system and response system was achieved in module space and phase space, respectively (module-phase synchronization). The RBF neural network is used to estimate the unknown nonlinear function in the system. The module-phase synchronization of two fractional-order complex chaotic systems is implemented by the Lyapunov stability theory of fractional-order systems. Numerical simulations are provided to show the effectiveness of the analytical results.

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The phase transition in information spreading was investigated. Two phase transitions were studied, respectively, from the point of view of network information and the node information, and the corresponding simulations were implemented on high-order networks. In order to explore the phase transition in information dissemination, this paper analyzes the characteristics of social networks and constructs a high-order network model with explicit, subexplicit, and implicit relationships. The factors affecting interpersonal communication were analyzed, the social cumulative effect was introduced, and two transmission models with characteristics of permanent forgetting and temporary forgetting were proposed. The comparison between simulated experiment and real case shows that the network mode accord with the characteristics of the actual network, and the two models of phase transitions are closer to the actual.

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The iterative propagation of information between nodes will strengthen the connection strength between nodes, and the network can evolve into different groups according to difference edge strength. Based on this observation, we present the user engagement to quantify the influences of users different propagation modes to network propagation, and construct weight network to simulate real social network, and proposed the community detection method in social networks based on information propagation and user engagement. Our method can produce different scale communities and overlapping community. We also applied our method to real-world social networks. The experiment proved that the network spread and the community division interact with each other. The community structure is significantly different in the network propagation of different scales.