  Many graph neural networks have been developed to learn graph representations
in either Euclidean or hyperbolic space, with all nodes' representations
embedded in a single space. However, a graph can have hyperbolic and Euclidean
geometries at different regions of the graph. Thus, it is sub-optimal to
indifferently embed an entire graph into a single space. In this paper, we
explore and analyze two notions of local hyperbolicity, describing the
underlying local geometry: geometric (Gromov) and model-based, to determine the
preferred space of embedding for each node. The two hyperbolicities'
distributions are aligned using the Wasserstein metric such that the calculated
geometric hyperbolicity guides the choice of the learned model hyperbolicity.
As such our model Joint Space Graph Neural Network (JSGNN) can leverage both
Euclidean and hyperbolic spaces during learning by allowing node-specific
geometry space selection. We evaluate our model on both node classification and
link prediction tasks and observe promising performance compared to baseline
models.
