  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.
