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By recursively summing node features over entire neighborhoods, spatial graphconvolution operators have been heralded as key to the success of Graph NeuralNetworks (GNNs). Yet, despite the multiplication of GNN methods across tasksand applications, the impact of this aggregation operation on their performancestill has yet to be extensively analysed. In fact, while efforts have mostlyfocused on optimizing the architecture of the neural network, fewer works haveattempted to characterize (a) the different classes of spatial convolutionoperators, (b) how the choice of a particular class relates to properties ofthe data , and (c) its impact on the geometry of the embedding space. In thispaper, we propose to answer all three questions by dividing existing operatorsinto two main classes ( symmetrized vs. row-normalized spatial convolutions),and show how these translate into different implicit biases on the nature ofthe data. Finally, we show that this aggregation operator is in fact tunable,and explicit regimes in which certain choices of operators -- and therefore,embedding geometries -- might be more appropriate.

Tuning the Geometry of Graph Neural Networks