  This paper presents a Wasserstein attraction approach for solving dynamic
mass transport problems over networks. In the transport problem over networks,
we start with a distribution over the set of nodes that needs to be
"transported" to a target distribution accounting for the network topology. We
exploit the specific structure of the problem, characterized by the computation
of implicit gradient steps, and formulate an approach based on discretized
flows. As a result, our proposed algorithm relies on the iterative computation
of constrained Wasserstein barycenters. We show how the proposed method finds
approximate solutions to the network transport problem, taking into account the
topology of the network, the capacity of the communication channels, and the
capacity of the individual nodes. Finally, we show the performance of this
approach applied to large-scale water transportation networks.
