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We study a random walk problem on the hierarchical network which is ascale-free network grown deterministically. The random walk problem is mappedonto a dynamical Ising spin chain system in one dimension with a nonlocal spinupdate rule, which allows an analytic approach. We show analytically that thecharacteristic relaxation time scale grows algebraically with the total numberof nodes $N$ as $T \sim N^z$. From a scaling argument, we also show thepower-law decay of the autocorrelation function $C_{\bfsigma}(t)\simt^{-\alpha}$, which is the probability to find the Ising spins in the initialstate ${\bfsigma}$ after $t$ time steps, with the state-dependent non-universalexponent $\alpha$. It turns out that the power-law scaling behavior has itsorigin in an quasi-ultrametric structure of the configuration space.Comment: 9 pages, 6 figure

Constrained spin-dynamics description of random walks on hierarchical scale-free networks