Return probability of random walks on hierarchical networks

초록

Random walks can be a model for various dynamical processes on complex networks including disease spreading and data packet transport. Applying the renormalization group approach to the Gaussian model, we derive the return-to-origin probability of random walks and the spectral dimension on a hierarchical scale-free network, which may be fractal or small-world depending on the weight of shortcut links. We find that the shortcuts make the spectral dimension larger and the return-to-origin probability smaller. While the spectral dimension varies between $1$ and $2$ for the fractal case, it is fixed at $2$ without regard to the variation of the network structure when the network is non-fractal. The distribution of local spectral dimensions is also discussed.