2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/22450We 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 $C_{\bfsigma}(t)\sim t^{-α}$, which is the probability to find the Ising spins in the initial state ${\bfsigma}$ after $t$ time steps, with the state-dependent non-universal exponent $α$. It turns out that the power-law scaling behavior has its origin in an quasi-ultrametric structure of the configuration space.9 pages, 6 figuresStatistical Mechanicstext