Learning 1-Dimensional Submanifolds for Subsequent Inference on Random Dot Product Graphs

Understand

A random dot product graph (RDPG) is a generative model for networks in which vertices correspond to positions in a latent Euclidean space and edge probabilities are determined by the dot products of the latent positions.

• We consider RDPGs for which the latent positions are randomly sampled from an unknown $1$-dimensional submanifold of the latent space.
• In principle, restricted inference, i.e., procedures that exploit the structure of the submanifold, should be more effective than unrestricted inference; however, it is not clear how to conduct restricted inference when the submanifold is unknown.
• We submit that techniques for manifold learning can be used to learn the unknown submanifold well enough to realize benefit from restricted inference.