On the local stochastic stability of nonlinear complex networks

The local stochastic stability of nonlinear complex networks is studied, subject to stochastic perturbations to the coupling strengths and stochastic parametric excitations to the nodes. The complex network is first linearized at its trivial solution and then the linearized network is reduced to N independent subsystems by using a suitable linear transformation, where N is the size of the network. The largest Lyapunov exponent for each subsystem is then calculated and all the approximate analytical solutions are evaluated for some specific cases. It is found that the largest Lyapunov exponent among all subsystems is the one associated with the subsystem that has the largest or the smallest eigenvalue of the configuration matrix of the network. Finally, an example is given to demonstrate the validity and accuracy of the theoretical analysis. © 2010 World Scientific Publishing Company.