  Graph neural networks (GNNs) are composed of layers consisting of graph
convolutions and pointwise nonlinearities. Due to their invariance and
stability properties, GNNs are provably successful at learning representations
from data supported on moderate-scale graphs. However, they are difficult to
learn on large-scale graphs. In this paper, we study the problem of training
GNNs on graphs of moderate size and transferring them to large-scale graphs. We
use graph limits called graphons to define limit objects for graph filters and
GNNs -- graphon filters and graphon neural networks (WNNs) -- which we
interpret as generative models for graph filters and GNNs. We then show that
graphon filters and WNNs can be approximated by graph filters and GNNs sampled
from them on weighted and stochastic graphs. Because the error of these
approximations can be upper bounded, by a triangle inequality argument we can
further bound the error of transferring a graph filter or a GNN across graphs.
Our results show that (i) the transference error decreases with the graph size,
and (ii) that graph filters have a transferability-discriminability tradeoff
that in GNNs is alleviated by the scattering behavior of the nonlinearity.
These findings are demonstrated empirically in a movie recommendation problem
and in a decentralized control task.
