  Graph Neural Networks (GNNs) are limited in their propagation operators. In
many cases, these operators often contain non-negative elements only and are
shared across channels, limiting the expressiveness of GNNs. Moreover, some
GNNs suffer from over-smoothing, limiting their depth. On the other hand,
Convolutional Neural Networks (CNNs) can learn diverse propagation filters, and
phenomena like over-smoothing are typically not apparent in CNNs. In this
paper, we bridge these gaps by incorporating trainable channel-wise weighting
factors $\omega$ to learn and mix multiple smoothing and sharpening propagation
operators at each layer. Our generic method is called $\omega$GNN, and is easy
to 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- and
graph-classification tasks, where our $\omega$GCN and $\omega$GAT perform on
par with state-of-the-art methods.
