Robust transport over networks

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.