Asymptotically Normal Estimation of Local Latent Network Curvature

Network data, commonly used throughout the physical, social, and biological sciences, consist of nodes (individuals) and the edges (interactions) between them. One way to represent the complex, high-dimensional structure in network data is to embed the graph into a low-dimensional geometric space. Curvature of this space, in particular, provides insights about structure in the graph, such as the propensity to form triangles or present tree-like structure. We derive an estimating function for curvature based on triangle side lengths and the midpoints between sides where the only input is a distance matrix and also establish asymptotic normality. We next introduce a novel latent distance matrix estimator for networks as well as an efficient algorithm to compute the estimate via solving iterative quadratic programs. We apply this method to the Los Alamos National Laboratory Unified Network and Host dataset and show how curvature estimates can be used to detect a red-team attack faster than naive methods, as well as discover non-constant latent curvature in coauthorship networks in physics.