Semisupervised regression in latent structure networks on unknown manifolds
This work addresses regression prediction in network data with latent geometric structures, but it is incremental as it builds on existing latent position models with specific assumptions.
The paper tackled the problem of predicting response variables on out-of-sample nodes in random dot product graphs where latent positions lie on an unknown one-dimensional curve, by proposing a manifold learning and graph embedding technique, and established convergence guarantees supported by simulations and Drosophila brain data.
Random graphs are increasingly becoming objects of interest for modeling networks in a wide range of applications. Latent position random graph models posit that each node is associated with a latent position vector, and that these vectors follow some geometric structure in the latent space. In this paper, we consider random dot product graphs, in which an edge is formed between two nodes with probability given by the inner product of their respective latent positions. We assume that the latent position vectors lie on an unknown one-dimensional curve and are coupled with a response covariate via a regression model. Using the geometry of the underlying latent position vectors, we propose a manifold learning and graph embedding technique to predict the response variable on out-of-sample nodes, and we establish convergence guarantees for these responses. Our theoretical results are supported by simulations and an application to Drosophila brain data.