  We consider discrete porous medium equations of the form \partial_t \rho_t =
\Delta \phi(\rho_t), where \Delta is the generator of a reversible continuous
time Markov chain on a finite set X, and \phi is an increasing function. We
show that these equations arise as gradient flows of certain entropy
functionals with respect to suitable non-local transportation metrics. This may
be seen as a discrete analogue of the Wasserstein gradient flow structure for
porous medium equations in R^n discovered by Otto. We present a one-dimensional
counterexample to geodesic convexity and discuss Gromov-Hausdorff convergence
to the Wasserstein metric.
