Synchronization in Time-Varying and Evolving Complex Networks

Abstract

In this contribution, we present the synchronization in dynamical complex networks with varying couplings. We identify two kinds of variations—(i) Non autonomous (Time-varying) couplings: where the coupling strength depends exclusively on time, (ii) Autonomous or Varying couplings (evolution) where the coupling strength depends on the behavior of the interconnected systems. The coupling strength in (i) is exogenous whereas in (ii) the coupling strength is endogenous and is defined by the states of the systems in the nodes. The exponential stability of the synchronization is ensured for the non autonomous couplings, due to the imposition of the coupling strength. Whereas, in the case of evolutionary couplings the exponential stability of the synchronization is not guaranteed for all time, due to the couplings are not controlled or imposed. We present an overview of these features in complex networks and illustrated by means of numerical examples.

A Model for Evolutionary Structural Plasticity and Synchronization of a Network of Neurons

“…In order for a network with a fixed configuration to reach synchronization, it is enough to show that the error system converges to zero, as was reported in [12,34]. In this works, the synchronization is achieved in an isolated node _ μðtÞ = f ðμðtÞÞ.…”

“…The condition is satisfied if the second largest eigenvalue of the network matrix is less than a constant parameter d < 0 less than 0. These necessary conditions for synchronization were reported in 2002 by Chen in [12,16] and were used in [34], with timevarying coupling networks. The network topology is changing due to the couplings' evolutionary behavior between nodes in the functional network.…”

“…Although evolutionary behavior impedes to prove synchronization for all the time, the network's synchronization can be proved in the intervals where the topology is fixed. In these intervals, the synchronization of ( 5) can be proved if the network satisfies the necessary conditions reported in [12,34], where networks with time-varying couplings reach synchronization, as long as the following inequality holds:…”

“…Nevertheless, not for the moment when there is a change in a configuration from one network to another. However, it is possible to use the results reported in [34] for time intervals where the network matrix (4) is fixed.…”