  Interest has been rising lately towards methods representing data in
non-Euclidean spaces, e.g. hyperbolic or spherical, that provide specific
inductive biases useful for certain real-world data properties, e.g.
scale-free, hierarchical or cyclical. However, the popular graph neural
networks are currently limited in modeling data only via Euclidean geometry and
associated vector space operations. Here, we bridge this gap by proposing
mathematically grounded generalizations of graph convolutional networks (GCN)
to (products of) constant curvature spaces. We do this by i) introducing a
unified formalism that can interpolate smoothly between all geometries of
constant curvature, ii) leveraging gyro-barycentric coordinates that generalize
the classic Euclidean concept of the center of mass. Our class of models
smoothly recover their Euclidean counterparts when the curvature goes to zero
from either side. Empirically, we outperform Euclidean GCNs in the tasks of
node classification and distortion minimization for symbolic data exhibiting
non-Euclidean behavior, according to their discrete curvature.
