  A common approach to modeling networks assigns each node to a position on a
low-dimensional manifold where distance is inversely proportional to connection
likelihood. More positive manifold curvature encourages more and tighter
communities; negative curvature induces repulsion. We consistently estimate
manifold type, dimension, and curvature from simply connected, complete
Riemannian manifolds of constant curvature. We represent the graph as a noisy
distance matrix based on the ties between cliques, then develop hypothesis
tests to determine whether the observed distances could plausibly be embedded
isometrically in each of the candidate geometries. We apply our approach to
data-sets from economics and neuroscience.
