Identifying the latent space geometry of network models through analysis of curvature.

This article develops a consistent statistical methodology to estimate the latent geometric space underlying network data, focusing on simply connected, complete Riemannian manifolds of constant curvature—specifically Euclidean, spherical, and hyperbolic spaces. The approach transforms observed network connections into a noisy distance matrix between groups of nodes (cliques) and uses spectral properties of a test matrix derived from these distances to identify the manifold type, curvature, and dimension without requiring prior assumptions or direct estimation of node locations or fixed effects. Theoretical results establish consistency of the estimators under mild assumptions on node distributions and clique structures, supported by simulation studies demonstrating controlled Type 1 error and increasing power with clique size and number. Empirical applications include Indian village social networks, where most networks are classified as spherical and microfinance introduction is associated with shifts toward positive curvature, and the neural network of the C. elegans worm, which rejects spherical geometry, suggesting non-positive curvature; these examples illustrate the method's ability to capture meaningful latent geometric features across diverse domains.