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Dynamics of networks with time-varying connections: on the interplay of node dynamics, coupling delays and network fluctuations

Abstract: Coupling delays are ubiquitous in many real-life networks: it takes time for information to travel in communication networks, or between coupled optical elements. In the brain a coupling delay between neurons arises from the conduction time of an electric signal along the axon.Here, we study the effect of a topology that changes over time in such delay-coupled networks. Network fluctuations are essential features of, for instance, interacting neurons, where synaptic plasticity continuously changes the topology or networks modeling social interactions. We concentrate on the synchronization properties of chaotic maps coupled with an interaction delay Td. The coupling topology fluctuates between an ensemble of directed small-world networks, while keeping the mean degree constant. The dynamics is characterized by three timescales: the internal time scale of the node dynamics Tin, the connection delay along the links Td, and the timescale of the network fluctuations Tn.

When the network fluctuations are faster than the coupling delay and the internal time scale Tn<< Tin}, T_d the synchronized state can be stabilized by the fluctuations: synchronization canbe stable even if most or all temporary network typologies are unstable, as predicted by the fast switching approximation. As the network time scale Tn increases, the synchronized state becomes unstable when both time scales collide T_n ~T_d. Synchronization is more probable as the network time scale increases further. However, in the slow network regime

(Tn>>Td>>Tin) we find that the long-term dynamics is desynchronized whenever the probability of reaching anon-synchronizing network is finite.

We complement these results with an analytical theory in the linear limit. Two limit cases allow an interpretation in terms of an``effective network'': When the network fluctuations are much faster than the internal time scale and the coupling delay (Tn<<Tin, T_d), the effective network topology is the average over the different typologies.

When coupling delay and network fluctuation time scales collide (Tin<<Tn = Td), the effective topology is the geometric mean over the different typologies.