Walks on Apollonian networks

We carry out comparative studies of random walks on deterministic Apollonian networks (DANs) and random Apollonian networks (RANs). We perform computer simulations for the mean first-passage time, the average return time, the mean-square displacement, and the network coverage for the unrestricted random walk. The diffusions both on DANs and RANs are proved to be sublinear. The effects of the network structure on the dynamics and the search efficiencies of walks with various strategies are also discussed. Contrary to intuition, it is shown that the self-avoiding random walk, which has been verified as an optimal local search strategy in networks, is not the best strategy for the DANs in the large size limit.