Random-walk and pair-annihilation processes on scale-free networks

We investigate the dynamic scaling properties of stochastic particle systems on a non-deterministic scale-free network. It is known that the dynamic scaling behavior depends on the degree distribution exponent of the underlying scale-free network. Our study shows that it also depends on the global structure of the underlying network. In random walks on the tree-structure scale-free network, we find that the relaxation time follows a power-law scaling T ∼ N with the network size N and the random-walker return probability decays algebraically with the decay exponent, which varies from node to node. On the other hand, in random walks on the looped scale-free network, they do not show the power-law scaling. We also study a pair-annihilation process on the scale-free network with the tree and the looped structure, respectively. We find that the particle density decays algebraically in time in both cases, but with a different exponent.