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

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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 Nᶻ. From a scaling argument, we also show the power-law decay of the autocorrelation function C(t) t⁻α, which is the probability to find the Ising spins in the initial state 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.