  We consider random networks whose dynamics is described by a rate equation,
with transition rates $w_{nm}$ that form a symmetric matrix. The long time
evolution of the system is characterized by a diffusion coefficient $D$. In one
dimension it is well known that $D$ can display an abrupt percolation-like
transition from diffusion ($D>0$) to sub-diffusion (D=0). A question arises
whether such a transition happens in higher dimensions. Numerically $D$ can be
evaluated using a resistor network calculation, or optionally it can be deduced
from the spectral properties of the system. Contrary to a recent expectation
that is based on a renormalization-group analysis, we deduce that $D$ is
finite; suggest an "effective-range-hopping" procedure to evaluate it; and
contrast the results with the linear estimate. The same approach is useful for
the analysis of networks that are described by quasi-one-dimensional sparse
banded matrices.
