  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.
