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

TL;DR

This paper analytically investigates the dynamics of random walks on hierarchical scale-free networks by mapping them onto a nonlocal Ising spin chain, revealing algebraic relaxation times and power-law autocorrelation decay.

Contribution

It introduces an analytic approach to study random walks on deterministic hierarchical scale-free networks via a spin chain model, uncovering scaling laws and the role of ultrametric structure.

Findings

Relaxation time scales algebraically with network size as T ~ N^z

Autocorrelation function decays as a power law C(t) ~ t^(-α)

Power-law behavior stems from ultrametric configuration space

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 $T \sim N^z$. 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.