Dynamical Crossover in Complex Networks near the Percolation Transition

Abstract

The return probability P 0 ( t ) of random walkers is investigated numerically for several scale-free fractal networks. Our results show that P 0 ( t ) is proportional to t - d s /2 with the non-integer spectral dimension d s as in the case of non-scale free fractal networks. We also study how the diffusion process is affected by the structural crossover from a fractal to a small-world architecture in a network near the percolation transition. It is elucidated that the corresponding dynamical crossover is scaled only by the unique characteristic time t ξ regardless of whether the network is scale free or not. In addition, the scaling relation d s = 2 D f / d w is found to be valid even for scale-free fractal networks, where D f and d w are the fractal and the walk dimensions. These results suggest that qualitative properties of P 0 ( t ) are irrelevant to the scale-free nature of networks.