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 $C_{\bfsigma}(t)\sim t^{-\alpha}$, 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.