Lorenzo Buffa — Università di Roma Tor Vergata, CREF #
From dense to sparse - Optimal Transport in a random graph framework #
Optimal transport (OT) is a mathematical framework that deals with the efficient transportation of mass or resources from one configuration to another while minimizing an associated cost.  We propose a null model for bipartite networks that can measure the distance of a real system from its optimal configuration. This distance is measured by a $\beta$ parameter, that controls the relative importance of the cost function. Notably, the $\beta\to 0$ limit of this model can be mapped into the Bipartite Weighted Configuration Model (BiWCM), the weighted version of the Bipartite Configuration Model (BiCM). These are Maximum Entropy null models, widely used in Network Theory for the reconstruction and validation of networks. On the other end, OT is reached at the limit for $\beta\to\infty$. We study this change as a phase transition, in which the order parameter is linked to the connectivity of the graph. The two known limits are the energy-driven state (OT solution) and the entropy-driven state (BIWCM) of the system. Finally, we propose a full study of the critical properties of this model, highlighting the link between the Optimal Transport problem and graph theory.