The effect of asymmetric disorder on the diffusion in arbitrary networks

TL;DR

This paper establishes a precise relationship between weak asymmetric disorder and electric resistance in networks, revealing stability conditions and a new dynamical exponent for diffusion processes.

Contribution

It introduces an exact relationship linking disorder strength and resistance, and identifies a new dynamical exponent governing diffusion in fractal networks.

Findings

Weak disorder stability when resistance exponent $$ is negative

Logarithmic scaling of mean first-passage time with distance for $$

Introduction of a new dynamical exponent $$ for fractal lattices

Abstract

Considering diffusion in the presence of asymmetric disorder, an exact relationship between the strength of weak disorder and the electric resistance of the corresponding resistor network is revealed, which is valid in arbitrary networks. This implies that the dynamics are stable against weak asymmetric disorder if the resistance exponent $\zeta$ of the network is negative. In the case of $\zeta>0$, numerical analyses of the mean first-passage time $\tau$ on various fractal lattices show that the logarithmic scaling of $\tau$ with the distance $l$, $\ln\tau\sim l^{\psi}$, is a general rule, characterized by a new dynamical exponent $\psi$ of the underlying lattice.