The effect of asymmetric disorder on the diffusion in arbitrary networks

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.