  Statistical analysis of a graph often starts with embedding, the process of
representing its nodes as points in space. How to choose the embedding
dimension is a nuanced decision in practice, but in theory a notion of true
dimension is often available. In spectral embedding, this dimension may be very
high. However, this paper shows that existing random graph models, including
graphon and other latent position models, predict the data should live near a
much lower-dimensional set. One may therefore circumvent the curse of
dimensionality by employing methods which exploit hidden manifold structure.
