On the transitions between dynamical behaviors for small-world and scale-free networks

Abstract

"The effect of network topology on the transition between dynamical behaviors is investigated for two networks: the small-world and scale-free models. In our analysis, we consider a network of identical dynamical systems where each node, prior to being coupled into the network, has a stable equilibrium point. In a recent publication we established that, as the number of nodes in the network increases, the collective dynamics of a repellently coupled network transit from stable to bounded and ultimately to unbounded behavior. In this contribution we investigate how these transitions are affected by the choice of network model. We show that for small-world networks, the stable equilibrium point behavior transits to bounded complez behavior for an interval of networj sizes, before becoming unbounded for a sufficiently large number of nodes. While for scale-free networks, the transition from stable fixed point to unbounded behavior occurs directly as the number of nodes increases. To illustrate these results, we use numerical simulation of well known chaotic benchmark systems."