  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.
