  We consider discrete stochastic processes, modeled by classical master
equations, on networks. The temporal growth of the lack of information about
the system is captured by its non-equilibrium entropy, defined via the
transition probabilities between different nodes of the network. We derive a
relation between the entropy and the spectrum of the master equation's transfer
matrix. Our findings indicate that the temporal growth of the entropy is
proportional to the logarithm of time if the spectral density shows scaling. In
analogy to chaos theory, the proportionality factor is called (stochastic)
information dimension and gives a global characterization of the dynamics on
the network. These general results are corroborated by examples of regular and
of fractal networks.
