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

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

2003-10-15

arxiv.org/abs/cond-mat/0310344v2

Phys. Rev. E 69, 036111 (2004)

Physics

Condensed Matter

Statistical Mechanics

9 pages, 6 figures

Scientific paper

10.1103/PhysRevE.69.036111

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 $C_{\bfsigma}(t)\sim t^{-\alpha}$, 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 $\alpha$. It turns out that the power-law scaling behavior has its origin in an quasi-ultrametric structure of the configuration space.

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