Abstract We revisit the classic problem of determining optimal routes in a graph for transporting two given distributions defined on its nodes, originally studied by Wardrop and Beckmann in the 1950s. The global congestion profile at any given time defines a dynamic metric on the graph, for which the routes must be geodesics. Our first contribution is the introduction of a dynamic version of the Beckmann problem, for which we derive the corresponding discrete partial differential equations governing the evolution of the system. These equations enable us to estimate the size of the support of the edge flow. Some of the main results include Theorems 3.14 and 4.9, which provide bounds on the support of solutions and prove the existence of free boundaries, as well as Theorem 4.10, which introduces a criterion to determine whether the support of the solution extends as the time horizon increases. Finally, we present a numerical model for mass transport through a junction, which reveals a connection with a system of obstacle problems.
Chang‐Lara et al. (2026) studied this question.