  Graph neural networks (GNNs) achieve remarkable performance in graph machine
learning tasks but can be hard to train on large-graph data, where their
learning dynamics are not well understood. We investigate the training dynamics
of large-graph GNNs using graph neural tangent kernels (GNTKs) and graphons. In
the limit of large width, optimization of an overparametrized NN is equivalent
to kernel regression on the NTK. Here, we investigate how the GNTK evolves as
another independent dimension is varied: the graph size. We use graphons to
define limit objects -- graphon NNs for GNNs, and graphon NTKs for GNTKs -- ,
and prove that, on a sequence of graphs, the GNTKs converge to the graphon NTK.
We further prove that the spectrum of the GNTK, which is related to the
directions of fastest learning which becomes relevant during early stopping,
converges to the spectrum of the graphon NTK. This implies that in the
large-graph limit, the GNTK fitted on a graph of moderate size can be used to
solve the same task on the large graph, and to infer the learning dynamics of
the large-graph GNN. These results are verified empirically on node regression
and classification tasks.
