Designing optimal networks for multi-commodity transport problem

TL;DR

This paper develops a new optimal transport-based model for designing multi-commodity networks, revealing how loops can naturally emerge and offering a faster convergence method than traditional gradient-based optimization.

Contribution

It introduces a novel dynamics-based model for multi-commodity network optimization that generalizes single-commodity results and provides insights into network topology formation.

Findings

Loops can arise due to flow differentiation.

The model converges faster than standard gradient methods.

Provides a unified framework for multi-commodity network design.

Abstract

Designing and optimizing different flows in networks is a relevant problem in many contexts. While a number of methods have been proposed in the physics and optimal transport literature for the one-commodity case, we lack similar results for the multi-commodity scenario. In this paper we present a model based on optimal transport theory for finding optimal multi-commodity flow configurations on networks. This model introduces a dynamics that regulates the edge conductivities to achieve, at infinite times, a minimum of a Lyapunov functional given by the sum of a convex transport cost and a concave infrastructure cost. We show that the long time asymptotics of this dynamics are the solutions of a standard constrained optimization problem that generalizes the one-commodity framework. Our results provide new insights into the nature and properties of optimal network topologies. In particular, they show that loops can arise as a consequence of distinguishing different flow types, complementing previous results where loops, in the one-commodity case, were obtained as a consequence of imposing dynamical rules to the sources and sinks or when enforcing robustness to damage. Finally, we provide an efficient implementation of our model which convergences faster than standard optimization methods based on gradient descent.