  Most graph neural networks (GNNs) are prone to the phenomenon of
over-squashing in which node features become insensitive to information from
distant nodes in the graph. Recent works have shown that the topology of the
graph has the greatest impact on over-squashing, suggesting graph rewiring
approaches as a suitable solution. In this work, we explore whether
over-squashing can be mitigated through the embedding space of the GNN. In
particular, we consider the generalization of Hyperbolic GNNs (HGNNs) to
Riemannian manifolds of variable curvature in which the geometry of the
embedding space is faithful to the graph's topology. We derive bounds on the
sensitivity of the node features in these Riemannian GNNs as the number of
layers increases, which yield promising theoretical and empirical results for
alleviating over-squashing in graphs with negative curvature.
