Identifying the latent space geometry of network models through analysis of curvature
This work addresses the problem of characterizing the underlying geometric structure of complex networks, which is important for researchers modeling network data.
This paper proposes a method to identify the latent space geometry of network models by estimating manifold type, dimension, and curvature. It represents graphs as noisy distance matrices and uses hypothesis tests to determine plausible isometric embeddings in candidate geometries, applying the approach to economics and neuroscience datasets.
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.