  In this paper we propose a pooling approach for convolutional information
processing on graphs relying on the theory of graphons and limits of dense
graph sequences. We present three methods that exploit the induced graphon
representation of graphs and graph signals on partitions of [0, 1]2 in the
graphon space. As a result we derive low dimensional representations of the
convolutional operators, while a dimensionality reduction of the signals is
achieved by simple local interpolation of functions in L2([0, 1]). We prove
that those low dimensional representations constitute a convergent sequence of
graphs and graph signals, respectively. The methods proposed and the
theoretical guarantees that we provide show that the reduced graphs and signals
inherit spectral-structural properties of the original quantities. We evaluate
our approach with a set of numerical experiments performed on graph neural
networks (GNNs) that rely on graphon pooling. We observe that graphon pooling
performs significantly better than other approaches proposed in the literature
when dimensionality reduction ratios between layers are large. We also observe
that when graphon pooling is used we have, in general, less overfitting and
lower computational cost.
