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Synchronization and Anti-synchroniztion of Dynamically Coupled Networks

We consider the coupling between two networks, each having N nodes whose individual dynamics is modeled by a two-state master equation. The intra-network interactions are all to all, whereas the inter-network interactions involve only a small percentage of the total number of nodes. We demonstrate that the dynamics of the mean field for a single network has an equivalent description in terms of a Langevin equation for a particle in a double-well potential. The coupling of two networks or equivalent coupling of two Langevin equations demonstrates synchronization or antisynchronization between two systems, depending on the sign of the interaction. The anti-synchronized behavior is explained in terms of the potential function and the inter-network interaction. The relative entropy is used to establish that the conditions for maximum information transfer between the networks are consistent with the Principle of Complexity Management and occurs when one system is near the critical state. The limitations of the Langevin modeling of the network coupling are also discussed.