  We consider transport over a strongly connected, directed graph. The
scheduling amounts to selecting transition probabilities for a discrete-time
Markov evolution which is designed to be consistent with certain initial and
final marginals. The random evolution is selected to be closest to a prior
measure on paths in the relative entropy sense, i.e., a Schroedinger bridge
between the two marginals. This is an atypical stochastic control problem where
the control consists in suitably modifying the transition mechanism. The prior
can incorporate cost of traversing edges or allocate equal probability to all
paths of equal length connecting any two given nodes, i.e., a uniform measure
on paths. This latter choice relies on the so-called Ruelle-Bowen random walk
and gives rise to a scheduling that tends to utilize all paths as uniformly as
the topology allows. Thus, when the Ruelle-Bowen law is taken as prior, the
transportation plan tends to lessen congestion and ensure a level of
robustness. We show that the Ruelle-Bowen law is itself a Schroedinger bridge
albeit with a prior that is not a probability measure. The paradigm of
Schroedinger bridges as a mechanism for scheduling transport on networks can be
adapted to graphs that are not strongly connected as well as to weighted
graphs. The latter leads to transportation plans that effect a compromise
between robustness and transportation cost.
