Graph convolutional neural networks (GCNNs) learn compositional representations from network data by nesting linear graph convolutions into nonlinearities. In this work, we approach GCNNs from a state-space perspective revealing that the graph convolutional module is a minimalistic linear state-space model, in which the state update matrix is the graph shift operator. We show that this state update may be problematic because it is nonparametric, and depending on the graph spectrum it may explode or vanish. Therefore, the GCNN has to trade its degrees of freedom between extracting features from data and handling these instabilities. To improve such trade-off, we propose a novel family of nodal aggregation rules that aggregate node features within a layer in a nonlinear state-space parametric fashion allowing for a better trade-off. We develop two architectures within this family inspired by the recurrence with and without nodal gating mechanisms. The proposed solutions generalize the GCNN and provide an additional handle to control the state update and learn from the data. Numerical results on source localization and authorship attribution show the superiority of the nonlinear state-space generalization models over the baseline GCNN.@en

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.@en