# Optimal Transport in Multilayer Networks for Traffic Flow Optimization
## 5 Conclusions  
In this work, we proposed a model that uses optimal transport theory to find optimal path trajectories on multilayer networks. By means of the regularization parameter \(\beta_{\alpha}\), we were able to take into account different contributions from the different layers for the total transportation cost. We illustrated the model on both synthetic and real data and  
Figure 5: Example of optimal paths in the city of Bordeaux for a bus and tram network. The paths are obtained with (**left**) and without (**right**) the tram layer. Here, \(\beta_{1}=0.5\) in both cases, while \(\beta_{2}=1.5\) in the second case. The width of the edges is proportional to the optimal \(||F_{e}||_{2}\). The reported \(Gini_{1}\) coefficient for the bus network (layer 1) is calculated using \(||F_{e}||_{2}\). The total percentage flux \(f_{2}=\sum_{e\in\mathcal{E}_{2}}||F_{e}||_{2}/(\sum_{e\in\mathcal{E}_{2}}||F_{ e}||_{1}+\sum_{e\in\mathcal{E}_{2}}||F_{e}||_{2})=0.17\), distributed over \(E_{2}=112\) tram edges, compared to \(E_{1}=2347\) bus edges.  
showed how the optimal distribution of passenger flows on network edges is influenced by different parameters used for the construction of the model (i.e., \(w\), \(p\), \(\beta_{\alpha}\)).  
In the absence of real data, we simulated the entry and exit destination of passengers. However, if travel demands are known, for instance, using mobile data [50], it would be interesting to investigate the distribution of traffic obtained with our model and compare it with real usage data as done in [51]. We considered a cost assigned on edges where \(\beta_{\alpha}\) tunes the impact of traffic on them, but one can generalize this to include penalties on nodes based on their degrees, as considered in [52]. Our model can be used to extract the main features of multilayer transportation networks [53] or to study the existence of several congestion regimes in both synthetic and real data [21] and investigate how this changes, varying \(\beta_{\alpha}\). Finally, in our experiments, we fixed the weight of inter-super nodes to be small. Potentially, one could suitably increase this to account for the cost of changing transportation modes within a journey and use our model to see how optimal trajectories change. This would be relevant in scenarios where the passengers' comfort contributes to the total transport cost. To facilitate future analysis, we provide an open source implementation of our code at [[https://github.com/cdebacco/MultiOT](https://github.com/cdebacco/MultiOT)]([https://github.com/cdebacco/MultiOT](https://github.com/cdebacco/MultiOT)) (accessed on 28 May 2021).  
All authors contributed to developing the models, conceived the experiments, analyzing the results and reviewing the manuscript. A.A.I. conducted the experiments. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
**Data Availability Statement**: Publicly available data sets were analyzed in this study. These data can be found here: [[http://transportetworks.cs.aalto.fi](http://transportetworks.cs.aalto.fi)]([http://transportetworks.cs.aalto.fi](http://transportetworks.cs.aalto.fi)) (accessed on 28 May 2021).  
**Acknowledgments**: The authors thank the International Max Planck Research School for Intelligent Systems (IMPRS-IS) for supporting [PERSON] and [PERSON].  
**Conflicts of Interest**: The authors declare no conflict of interest.