  The concept of entropy rate for a dynamical process on a graph is introduced.
We study diffusion processes where the node degrees are used as a local
information by the random walkers. We describe analitically and numerically how
the degree heterogeneity and correlations affect the diffusion entropy rate. In
addition, the entropy rate is used to characterize complex networks from the
real world. Our results point out how to design optimal diffusion processes
that maximize the entropy for a given network structure, providing a new
theoretical tool with applications to social, technological and communication
networks.
