On Estimation and Inference in Latent Structure Random Graphs

We define a latent structure random graph as a random dot product graph (RDPG) in which the latent position distribution incorporates both probabilistic and geometric constraints, delineated by a family of underlying distributions on some fixed Euclidean space, and a structural support submanifold from which are drawn the latent positions for the graph. For a one-dimensional latent structure model with known structural support, we extend existing results on the consistency of spectral estimates in RDPGs to demonstrate that the parameters of the underlying distribution can be efficiently estimated. We describe how to estimate or learn the structural support in cases where it is unknown, with a focus on graphs with latent positions along the Hardy-Weinberg curve. Finally, we use the latent structural model formulation to address a hitherto-open question in neuroscience on the bilateral homology of the Drosophila left and right hemisphere connectome.