  We study spectral graph convolutional neural networks (GCNNs), where filters
are defined as continuous functions of the graph shift operator (GSO) through
functional calculus. A spectral GCNN is not tailored to one specific graph and
can be transferred between different graphs. It is hence important to study the
GCNN transferability: the capacity of the network to have approximately the
same repercussion on different graphs that represent the same phenomenon.
Transferability ensures that GCNNs trained on certain graphs generalize if the
graphs in the test set represent the same phenomena as the graphs in the
training set. In this paper, we consider a model of transferability based on
graphon analysis. Graphons are limit objects of graphs, and, in the graph
paradigm, two graphs represent the same phenomenon if both approximate the same
graphon. Our main contributions can be summarized as follows: 1) we prove that
any fixed GCNN with continuous filters is transferable under graphs that
approximate the same graphon, 2) we prove transferability for graphs that
approximate unbounded graphon shift operators, which are defined in this paper,
and, 3) we obtain non-asymptotic approximation results, proving linear
stability of GCNNs. This extends current state-of-the-art results which show
asymptotic transferability for polynomial filters under graphs that approximate
bounded graphons.
