  Activation functions are crucial in graph neural networks (GNNs) as they
allow defining a nonlinear family of functions to capture the relationship
between the input graph data and their representations. This paper proposes
activation functions for GNNs that not only adapt to the graph into the
nonlinearity, but are also distributable. To incorporate the feature-topology
coupling into all GNN components, nodal features are nonlinearized and combined
with a set of trainable parameters in a form akin to graph convolutions. The
latter leads to a graph-adaptive trainable nonlinear component of the GNN that
can be implemented directly or via kernel transformations, therefore, enriching
the class of functions to represent the network data. Whether in the direct or
kernel form, we show permutation equivariance is always preserved. We also
prove the subclass of graph-adaptive max activation functions are Lipschitz
stable to input perturbations. Numerical experiments with distributed source
localization, finite-time consensus, distributed regression, and recommender
systems corroborate our findings and show improved performance compared with
pointwise as well as state-of-the-art localized nonlinearities.
