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) \u2013 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 \u201ctemperature\u201d parameter, tends to the solution of the \u201cmost economical\u201d 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
