title: Constrained spin-dynamics description of random walks on hierarchical scale-free networks.
abstract: We study a random walk problem on the hierarchical network which is a scale-free network grown deterministically. The random walk problem is mapped onto a dynamical Ising spin chain system in one dimension with a nonlocal spin update rule, which allows an analytic approach. We show analytically that the characteristic relaxation time scale grows algebraically with the total number of nodes N as T--N(z). From a scaling argument, we also show the power-law decay of the autocorrelation function C(sigma)(t)--t(-alpha), which is the probability to find the Ising spins in the initial state sigma after t time steps, with the state-dependent nonuniversal exponent alpha. It turns out that the power-law scaling behavior has its origin in a quasiultrametric structure of the configuration space.
authors.affiliation: Department of Physics, Chungnam National University, Daejeon 305-764, Korea.
authors.forename: Jae Dong
authors.initials: JD
authors.lastname: Noh
authors.forename: Heiko
authors.initials: H
authors.lastname: Rieger