No abstract

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.

Network routing approaches are widely used to study the evolution in time of self-adapting systems. However, few advances have been made for problems where adaptation is governed by time-dependent inputs. In this work, we study a dynamical systems where the edge conductivities-capacities-of a network are regulated by time-varying mass loads injected on nodes. Motivated by empirical observations, we assume that conductivities adapt slowly with respect to the characteristic time of the loads. Furthermore, assuming the loads to be periodic, we derive a new dynamics where the evolution of the system is governed by a matrix obtained with the Fourier coefficients of the input loads. Remarkably, we find a sufficient condition on these coefficients that determines when the resulting network topologies are trees. We show an example of this on the Bordeaux bus network where we tune the input loads to interpolate between loopy and tree topologies. We validate our model on several synthetic networks and provide an expression for long-time solutions of the original conductivities, supported by numerical observations and analytical arguments.

Optimizing passengers routes is crucial to design efficient transportation networks. Recent results show that optimal transport provides an efficient alternative to standard optimization methods. However, it is not yet clear if this formalism has empirical validity on engineering networks. We address this issue by considering different response functions—quantities determining the interaction between passengers—in the dynamics implementing the optimal transport formulation. Particularly, we couple passengers’ fluxes by taking their sum or the sum of their squares. The first choice naturally reflects edges occupancy in transportation networks, however the second guarantees convergence to an optimal configuration of flows. Both modeling choices are applied to the Paris metro. We measure the extent of traffic bottlenecks and infrastructure resilience to node removal, showing that the two settings are equivalent in the congested transport regime, but different in the branched one. In the latter, the two formulations differ on how fluxes are distributed, with one function favoring routes consolidation, thus potentially being prone to generate traffic overload. Additionally, we compare our method to Dijkstra’s algorithm to show its capacity to efficiently recover shortest-path-like graphs. Finally, we observe that optimal transport networks lie in the Pareto front drawn by the energy dissipated by passengers, and the cost to build the infrastructure.

In classification tasks, it is crucial to meaningfully exploit the information contained in the data. While much of the work in addressing these tasks is focused on building complex algorithmic infrastructures to process inputs in a black-box fashion, little is known about how to exploit the various facets of the data before inputting this into an algorithm. Here, we focus on this latter perspective by proposing a physics-inspired dynamical system that adapts optimal transport principles to effectively leverage color distributions of images. Our dynamics regulates immiscible fluxes of colors traveling on a network built from images. Instead of aggregating colors together, it treats them as different commodities that interact with a shared capacity on the edges. The resulting optimal flows can then be fed into standard classifiers to distinguish images in different classes. We show how our method can outperform competing approaches on image classification tasks in datasets where color information matters.