Modeling traffic distribution and extracting optimal flows in multilayer networks is of utmost importance to design efficient multi-modal network infrastructures. Recent results based on optimal transport theory provide powerful and computationally efficient methods to address this problem, but they are mainly focused on modeling single-layer networks. Here we adapt these results to study how optimal flows distribute on multilayer networks. We propose a model where optimal flows on different layers contribute differently to the total cost to be minimized. This is done by means of a parameter that varies with layers, which allows to flexibly tune the sensitivity to traffic congestion of the various layers. As an application, we consider transportation networks, where each layer is associated to a different transportation system and show how the traffic distribution varies as we tune this parameter across layers. We show an example of this result on the real 2-layer network of the city of Bordeaux with bus and tram, where we find that in certain regimes the presence of the tram network significantly unburdens the traffic on the road network. Our model paves the way to further analysis of optimal flows and navigability strategies in real multilayer networks.

中文翻译：

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多层网络中的最佳传输

在多层网络中建模流量分布和提取最佳流对于设计高效的多模式网络基础设施至关重要。最近基于最优传输理论的结果提供了强大且计算效率高的方法来解决这个问题，但它们主要集中在单层网络建模上。在这里，我们调整这些结果来研究最优流如何在多层网络上分布。我们提出了一个模型，其中不同层上的最佳流对要最小化的总成本有不同的贡献。这是通过一个随层变化的参数来完成的，它允许灵活地调整各个层对交通拥塞的敏感性。作为一个应用，我们考虑交通网络，其中每一层都与不同的交通系统相关联，并显示当我们跨层调整此参数时交通分布如何变化。我们在带有公共汽车和有轨电车的波尔多市真实 2 层网络上展示了这个结果的一个例子，我们发现在某些情况下，有轨电车网络的存在显着减轻了道路网络上的交通负担。我们的模型为进一步分析真实多层网络中的最佳流量和导航策略铺平了道路。