  This paper derives statistical guarantees for the performance of Graph Neural
Networks (GNNs) in link prediction tasks on graphs generated by a graphon. We
propose a linear GNN architecture (LG-GNN) that produces consistent estimators
for the underlying edge probabilities. We establish a bound on the mean squared
error and give guarantees on the ability of LG-GNN to detect high-probability
edges. Our guarantees hold for both sparse and dense graphs. Finally, we
demonstrate some of the shortcomings of the classical GCN architecture, as well
as verify our results on real and synthetic datasets.
