  Network flow problems, which involve distributing traffic over a network such
that the underlying infrastructure is used effectively, are ubiquitous in
transportation and logistics. Among them, the Multi-Commodity Network Flow
(MCNF) problem is of general interest, as it concerns the distribution of
multiple flows of different sizes between several sources and sinks, while
achieving effective utilization of the links. Due to the appeal of data-driven
optimization, these problems have increasingly been approached using graph
learning methods. In this paper, we propose a novel graph learning architecture
for network flow problems called Per-Edge Weights (PEW). This method builds on
a Graph Attention Network and uses distinctly parametrized message functions
along each link. We extensively evaluate the proposed solution through an
Internet flow routing case study using $17$ Service Provider topologies and $2$
routing schemes. We show that PEW yields substantial gains over architectures
whose global message function constrains the routing unnecessarily. We also
find that an MLP is competitive with other standard architectures. Furthermore,
we shed some light on the relationship between graph structure and predictive
performance for data-driven routing of flows, an aspect that has not been
considered by existing work in the area.

Machine Learning;Artificial Intelligence;Networking and Internet Architecture