# Optimal Transport in Multilayer Networks for Traffic Flow Optimization
## 1 Introduction  
Investigating how a network operates and assessing an optimal network design in interconnected networks is a critical problem in several areas [1]. Examples of these include economics [2], climate systems [3], epidemic spreading [4; 5; 6] and transportation networks [7]. The main challenge of these problems is to account for the various types of connections that nodes can use to travel through the network efficiently. For example, in transportation networks, the main application considered here, passengers can travel using various means of transport within the same journey. The different transportation modes can operate in significantly different ways [8; 9]. For instance, traveling along a rail network (e.g., by tram or subway) is usually faster than along a road network (e.g., by car or bus). The rail network is less sensitive to traffic congestion but the road network has wider coverage and thus allows to reach more destinations. The question is how to combine all these different features to design optimal networks and predict the optimal trajectories of passengers.  
Multilayer networks [1; 10; 11; 12] are a powerful tool to study multi-modal transportation networks [13; 14; 15]. Transport in a multilayer network, where layers correspond to transport modes, is often studied using diffusion or spreading processes [1; 16; 17; 18]. Many of these works use shortest-path minimization [14; 19; 20; 21] as the main method to extract the passengers' trajectories. However, this can be a restrictive choice: on one side, this assumes that different layers share the same cost function to be minimized; on the other side, shortest-path minimization is not sensitive to traffic congestion and thus, may not be realistic in certain scenarios. Empirical studies [22] have also indicated that passengers may not necessarily choose the shortest paths.  
Here, instead, we propose a model that considers more general transport cost minimization, based on a regularized version of the Monge-Kantorovich optimal transport problem [23]. The regularization is obtained via a parameter \(\beta\) that allows to flexibly tune the cost between settings where traffic is penalized or consolidated. Optimal transport is a proven, powerful tool to model traffic in networks and optimal network design [24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34; 35; 36; 37; 38; 39]. Recent works [30; 40] extended this formalism to a multi-commodity case that properly accounts for passengers with different origins and destinations. All these studies consider the case of a single-layer network, i.e., one transportation mode. The existence of multiple connections on different layers invites a generalization of these recent results of optimal transport to cope with multilayer networks.  
Here, we make this effort and propose a model that uses optimal transport theory to design optimal multilayer networks and finds optimal path trajectories on them. We show how such networks operate under various transport costs tuned by \(\beta\) on both synthetic and real data. We see how the traffic evolves from being more homogeneous to a more unbalanced traffic distribution when a second layer is present and the cost to travel through it changes.  
In summary, the goal of this work is to propose an efficient optimal transport-based method for modeling optimal network flows in multilayer networks. Our model finds optimal flows by naturally incorporating the different nature of transportation modes and is computationally efficient. While here, we focus on transportation networks, our method is applicable to a broader set of practical applications involving flows on multilayer networks.