On estimation and inference in latent structure random graphs

TL;DR

This paper introduces a latent structure model for random graphs, combining probabilistic and geometric constraints, and demonstrates how spectral methods can estimate parameters and structural support, with applications to biological connectomes.

Contribution

It extends random dot product graphs by incorporating structural support and provides methods for parameter estimation and support learning, including applications to biological data.

Findings

Spectral estimates effectively recover latent positions.

Methods successfully estimate structural support when unknown.

Application to Drosophila connectome demonstrates practical utility.

Abstract

We define a latent structure model (LSM) 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 the latent positions for the graph are drawn. For a one-dimensional latent structure model with known structural support, we show how spectral estimates of the latent positions of an RDPG can be used for efficient estimation of the paramaters of the LSM. We describe how to estimate or learn the structural support in cases where it is unknown, with an illustrative focus on graphs with latent positions along the Hardy-Weinberg curve. Finally, we use the latent structure model formulation to test bilateral homology in the Drosophila connectome.