  We study single commodity network flows with suitable robustness and
efficiency specs. An original use of a maximum entropy problem for
distributions on the paths of the graph turns this problem into a steering
problem for Markov chains with prescribed initial and final marginals. From a
computational standpoint, viewing scheduling this way is especially attractive
in light of the existence of an iterative algorithm to compute the solution.
The present paper builds on [13] by introducing an index of efficiency of a
transportation plan and points, accordingly, to efficient-robust transport
policies. In developing the theory, we establish two new invariance properties
of the solution (called bridge) - an iterated bridge invariance property and
the invariance of the most probable paths. These properties, which were
tangentially mentioned in our previous work, are fully developed here. We also
show that the distribution on paths of the optimal transport policy, which
depends on a "temperature" parameter, tends to the solution of the "most
economical" but possibly less robust optimal mass transport problem as the
temperature goes to zero. The relevance of all of these properties for
transport over networks is illustrated in an example.
