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Graph Neural Networks (GNNs) are limited in their propagation operators. Inmany cases, these operators often contain non-negative elements only and areshared across channels, limiting the expressiveness of GNNs. Moreover, someGNNs suffer from over-smoothing, limiting their depth. On the other hand,Convolutional Neural Networks (CNNs) can learn diverse propagation filters, andphenomena like over-smoothing are typically not apparent in CNNs. In thispaper, we bridge these gaps by incorporating trainable channel-wise weightingfactors $\omega$ to learn and mix multiple smoothing and sharpening propagationoperators at each layer. Our generic method is called $\omega$GNN, and is easyto implement. We study two variants: $\omega$GCN and $\omega$GAT. For$\omega$GCN, we theoretically analyse its behaviour and the impact of $\omega$on the obtained node features. Our experiments confirm these findings,demonstrating and explaining how both variants do not over-smooth.Additionally, we experiment with 15 real-world datasets on node- andgraph-classification tasks, where our $\omega$GCN and $\omega$GAT perform onpar with state-of-the-art methods.

Improving Graph Neural Networks with Learnable Propagation Operators