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Robust Transport Over Networks

机译：可靠的网络传输

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We consider transportation 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 initial and final marginal constraints on mass transport. We address the situation where initially the mass is concentrated on certain nodes and needs to be transported over a certain time period to another set of nodes, possibly disjoint from the first. The evolution is selected to be closest to a prior measure on paths in the relative entropy sense-such a construction is known as a Schrodinger bridge between the two given marginals. It may be viewed as an atypical stochastic control problem where the control consists in suitably modifying the prior transition mechanism. The prior can be chosen to incorporate constraints and costs for traversing specific edges of the graph, but it can also be selected to allocate equal probability to all paths of equal length connecting any two nodes (i.e., a uniform distribution on paths). This latter choice for prior transitions relies on the so-called Ruelle-Bowen random walker and gives rise to scheduling that tends to utilize all paths as uniformly as the topology allows. Thus, this RuelleBowen law (9)IRB ) taken as prior, leads to a transportation plan that tends to lessen congestion and ensures a level of robustness. We also show that the distribution 9)IRB on paths, which attains the maximum entropy rate for the random walker given by the topological entropy, can itself be obtained as the time-homogeneous solution of a maximum entropy problem for measures on paths (also a Schrodinger bridge problem, albeit with prior that is not a probability measure). Finally we show that the paradigm of Schrodinger 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. In the latter case, our approach may be used to design a transportation plan which effectively compromises between robustness and other criteria such as cost. Indeed, we explicitly provide a robust transportation plan which assigns maximum probability to minimum cost paths and therefore compares favorably with Optimal Mass Transportation strategies.
机译：我们考虑通过紧密联系的有向图进行运输。调度等于为离散时间马尔可夫演化选择过渡概率，该概率被设计为与大规模运输的初始和最终边际约束一致。我们解决的情况是，质量最初集中在某些节点上，需要在一定时间段内传输到另一组节点，可能与第一组节点不相交。在相对熵的意义上，将演化选择为最接近路径上的先验度量-这种构造称为两个给定边际之间的薛定rod桥。可以将其视为非典型随机控制问题，其中控制在于适当修改现有的过渡机制。可以选择先验以合并约束和遍历图的特定边缘的成本，但是也可以选择先验以将相等的概率分配给连接任意两个节点的等长的所有路径（即，路径上的均匀分布）。先前转换的后一种选择依赖于所谓的Ruelle-Bowen随机沃克，并导致调度，该调度倾向于在拓扑结构允许的范围内统一利用所有路径。因此，该RuelleBowen定律（9）IRB）被当作是先例，从而导致了一种交通计划，该计划倾向于减轻拥堵并确保鲁棒性。我们还表明，路径上的分布9）IRB可以通过拓扑熵给出的随机沃克获得最大熵率，它本身可以作为路径上测度的最大熵问题的时间均匀解（也可以是薛定inger桥问题，尽管先验不是概率度量。最终，我们证明了Schrodinger桥的范式作为一种用于调度网络上的传输的机制，可以适应于没有强连接的图以及加权图。在后一种情况下，我们的方法可用于设计运输计划，该计划可在健壮性和其他标准（例如成本）之间做出有效折衷。实际上，我们明确提供了一个强大的运输计划，该计划将最大的可能性分配给最小的成本路径，因此可以与“最佳大众运输”策略进行比较。

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