  Several recent works use positional encodings to extend the receptive fields
of graph neural network (GNN) layers equipped with attention mechanisms. These
techniques, however, extend receptive fields to the complete graph, at
substantial computational cost and risking a change in the inductive biases of
conventional GNNs, or require complex architecture adjustments. As a
conservative alternative, we use positional encodings to expand receptive
fields to $r$-hop neighborhoods. More specifically, our method augments the
input graph with additional nodes/edges and uses positional encodings as node
and/or edge features. We thus modify graphs before inputting them to a
downstream GNN model, instead of modifying the model itself. This makes our
method model-agnostic, i.e., compatible with any of the existing GNN
architectures. We also provide examples of positional encodings that are
lossless with a one-to-one map between the original and the modified graphs. We
demonstrate that extending receptive fields via positional encodings and a
virtual fully-connected node significantly improves GNN performance and
alleviates over-squashing using small $r$. We obtain improvements on a variety
of models and datasets and reach competitive performance using traditional GNNs
or graph Transformers.
