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 $T\sim N^z.$ From a scaling argument, we also show the power-law decay of the autocorrelation function $C_{\mathit{\sigma }}\mathrm{}\left(t\right)\mathrm{}\sim t^{-\alpha },$ which is the probability to find the Ising spins in the initial state σ after t time steps, with the state-dependent nonuniversal exponent α. It turns out that the power-law scaling behavior has its origin in a quasiultrametric structure of the configuration space.

• Received 15 October 2003

DOI:https://doi.org/10.1103/PhysRevE.69.036111