  Graph neural networks (GNNs) have been shown to replicate convolutional
neural networks' (CNNs) superior performance in many problems involving graphs.
By replacing regular convolutions with linear shift-invariant graph filters
(LSI-GFs), GNNs take into account the (irregular) structure of the graph and
provide meaningful representations of network data. However, LSI-GFs fail to
encode local nonlinear graph signal behavior, and so do regular activation
functions, which are nonlinear but pointwise. To address this issue, we propose
median activation functions with support on graph neighborhoods instead of
individual nodes. A GNN architecture with a trainable multirresolution version
of this activation function is then tested on synthetic and real-word datasets,
where we show that median activation functions can improve GNN capacity with
marginal increase in complexity.
