Quantum and classical diffusion on small-world networks

We study numerically quantum diffusion of a particle on small-world networks by integrating the time-dependent Schrödinger equation with a localized initial state. The participation ratio, which corresponds to the number of visited sites in the case of classical diffusion, as a function of time is measured and the corresponding diffusion time, $\tau$ is computed. In a local regular network, i.e., in the network with the rewiring probability $p=0,$ the diffusion time depends on the network size N as $\tau \sim N,$ while the behavior $\tau \sim \log N$ is observed as p becomes finite. Such fast diffusion of a particle on a complex network suggests that the small-world transition is also the fast-world transition from a dynamic point of view. The classical diffusion behavior is also studied and compared with the quantum behavior.