Modeling traffic distribution and extracting optimal flows in multilayer networks is of the 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 the 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 a 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 for further analysis of optimal flows and navigability strategies in real, multilayer networks.
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