  Latent space models (LSMs) are frequently used to model network data by
embedding a network's nodes into a low-dimensional latent space; however,
choosing the dimension of this space remains a challenge. To this end, we begin
by formalizing a class of LSMs we call generalized linear network eigenmodels
(GLNEMs) that can model various edge types (binary, ordinal, non-negative
continuous) found in scientific applications. This model class subsumes the
traditional eigenmodel by embedding it in a generalized linear model with an
exponential dispersion family random component and fixes identifiability issues
that hindered interpretability. Next, we propose a Bayesian approach to
dimension selection for GLNEMs based on an ordered spike-and-slab prior that
provides improved dimension estimation and satisfies several appealing
theoretical properties. In particular, we show that the model's posterior
concentrates on low-dimensional models near the truth. We demonstrate our
approach's consistent dimension selection on simulated networks. Lastly, we use
GLNEMs to study the effect of covariates on the formation of networks from
biology, ecology, and economics and the existence of residual latent structure.
