  We aim to deepen the theoretical understanding of Graph Neural Networks
(GNNs) on large graphs, with a focus on their expressive power. Existing
analyses relate this notion to the graph isomorphism problem, which is mostly
relevant for graphs of small sizes, or studied graph classification or
regression tasks, while prediction tasks on nodes are far more relevant on
large graphs. Recently, several works showed that, on very general random
graphs models, GNNs converge to certains functions as the number of nodes
grows. In this paper, we provide a more complete and intuitive description of
the function space generated by equivariant GNNs for node-tasks, through
general notions of convergence that encompass several previous examples. We
emphasize the role of input node features, and study the impact of node
Positional Encodings (PEs), a recent line of work that has been shown to yield
state-of-the-art results in practice. Through the study of several examples of
PEs on large random graphs, we extend previously known universality results to
significantly more general models. Our theoretical results hint at some
normalization tricks, which is shown numerically to have a positive impact on
GNN generalization on synthetic and real data. Our proofs contain new
concentration inequalities of independent interest.
