Collective Chaos Induced by Structures of Complex Networks

Mapping a complex network of $N$coupled identical oscillators to a quantum system, the nearest neighbor level spacing (NNLS) distribution is used to identify collective chaos in the corresponding classical dynamics on the complex network. The classical dynamics on an Erdos-Renyi network with the wiring probability $p_{ER} \le \frac{1}{N}$ is in the state of collective order, while that on an Erdos-Renyi network with $p_{ER} > \frac{1}{N}$ in the state of collective chaos. The dynamics on a WS Small-world complex network evolves from collective order to collective chaos rapidly in the region of the rewiring probability $p_r \in [0.0,0.1]$, and then keeps chaotic up to $p_r = 1.0$. The dynamics on a Growing Random Network (GRN) is in a special state deviates from order significantly in a way opposite to that on WS small-world networks. Each network can be measured by a couple values of two parameters $(\beta ,\eta)$.