Mean first-passage time for random walks on random growth tree networks.

Abstract

As opposed to most previous works focusing on random walks on deterministic tree networks, in this paper, we pay more attention on random growth tree networks. Specifically, we propose two distinct types of random growth tree networks T(1;t,p) and T(2;t,p) where t represents time step and p is a probability parameter (0≤p≤1). Tree T(1;t,p) is iteratively built in a fractal manner; however, T(2;t,p) is generated using a nonfractal operation. Then we study random walks on tree network T(i;t,p) (i=1,2) and derive the analytical solution to mean first-passage time 〈F_{T(i;t,p)}〉. The results suggest that two growth ways have remarkably different influence on parameters 〈F_{T(i;t,p)}〉. More precisely, the fractal-growth manner makes topological structure of tree network more loose than the nonfractal one and thus increases drastically mean first-passage time in the large graph limit. Finally, we extensively conduct experimental simulations, and the results demonstrate that computer simulations are in strong agreement with the theoretical analysis.