  Graph signals are signals with an irregular structure that can be described
by a graph. Graph neural networks (GNNs) are information processing
architectures tailored to these graph signals and made of stacked layers that
compose graph convolutional filters with nonlinear activation functions. Graph
convolutions endow GNNs with invariance to permutations of the graph nodes'
labels. In this paper, we consider the design of trainable nonlinear activation
functions that take into consideration the structure of the graph. This is
accomplished by using graph median filters and graph max filters, which mimic
linear graph convolutions and are shown to retain the permutation invariance of
GNNs. We also discuss modifications to the backpropagation algorithm necessary
to train local activation functions. The advantages of localized activation
function architectures are demonstrated in four numerical experiments: source
localization on synthetic graphs, authorship attribution of 19th century
novels, movie recommender systems and scientific article classification. In all
cases, localized activation functions are shown to improve model capacity.
