Graph Neural Modeling of Network Flows

TL;DR

A novel graph learning architecture for network flow problems called Per-Edge Weights (PEW), which builds on a Graph Attention Network and uses distinctly parametrized message functions along each link to yield substantial gains over architectures whose global message function constrains the routing unnecessarily.

Abstract

Network flow problems, which involve distributing traffic such that the underlying infrastructure is used effectively, are ubiquitous in transportation and logistics. Among them, the general Multi-Commodity Network Flow (MCNF) problem 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 analyze 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.

Figures (26)

• Figure 1: Top. An illustration of the Multi-Commodity Network Flow family of problems. The requirements of the routing problem are defined using a matrix that specifies the total amount of traffic that has to be routed between each pair of nodes in a graph. We are also given a graph topology in which links are equipped with capacities. All flows have an entry and exit node and share the same underlying transportation infrastructure. Under a particular routing scheme, such as shortest path routing, the links are loaded by the total amount of traffic passing over them. Bottom. A model is trained using a dataset of the link utilizations for certain demand matrices and graph topologies, and is then used to predict the Maximum Link Utilization for an unseen demand matrix.
• Figure 2: Left. An illustration of the MPNN used in previous flow routing works, which uses the same message function $M^{(l)}$ for aggregating neighbor messages. Right. An illustration of our proposed Per-Edge Weights (PEW), which uses uniquely parametrized per-edge message functions.
• Figure 3: Normalized MSE obtained by the predictors on different topologies for the SSP (top) and ECMP (bottom) routing schemes. Lower values are better. PEW improves over vanilla GAT substantially and performs best out of all architectures. An MLP is competitive with the other GNNs.
• Figure 4: Difference in normalized MSE between the raw and sum demand input representations as a function of the number of training datapoints for PEW and GAT for the SSP (left) and ECMP routing schemes (right). As the dataset size increases, PEW is able to exploit the granular demand information, while GAT performs better with a lossy aggregation of the demand information.
• Figure 5: Impact of topological characteristics on the predictive performance of PEW. Performance degrades as the graph size increases (first 3 columns), but improves with higher levels of heterogeneity of the graph structure (last 3 columns).