Reconstruction of example point clouds from the McGill dataset [SZM*08]. First row: Input Point cloud. Second row: Our reconstruction of the shape in the first row. Third row: Poisson surface reconstruction [KBH06] of the shape in the first row. Shapes showcased (left to right) are two chairs, cup, octopus, snake, dolphin, teddy and table respectively.

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 any r-ring. Our method augments the input graph with additional nodes/edges and uses positional encodings as node and/or edge features. Thus, it is compatible with many existing GNN architectures. We also provide examples of positional encodings that are non-invasive, i.e., there is a oneto-one map between the original and the modified graphs. Our experiments 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 across models, showing state-ofthe-art performance even using older architectures than recent Transformer models adapted to graphs.

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In this work we produce a framework for constructing universal function approximators on graph isomorphism classes. Additionally, we prove how this framework comes with a collection of theoretically desirable properties and enables novel analysis. We show how this allows us to outperform state of the art on four different well known datasets in graph classification and how our method can separate classes of graphs that other graph-learning methods cannot. Our approach is inspired by persistence homology, dependency parsing for Natural Language Processing, and multivalued functions. The complexity of the underlying algorithm is O(mn) and code is publicly available 1 .