  We review the class of continuous latent space (statistical) models for
network data, paying particular attention to the role of the geometry of the
latent space. In these models, the presence/absence of network dyadic ties are
assumed to be conditionally independent given the dyads? unobserved positions
in a latent space. In this way, these models provide a probabilistic framework
for embedding network nodes in a continuous space equipped with a geometry that
facilitates the description of dependence between random dyadic ties.
Specifically, these models naturally capture homophilous tendencies and triadic
clustering, among other common properties of observed networks. In addition to
reviewing the literature on continuous latent space models from a geometric
perspective, we highlight the important role the geometry of the latent space
plays on properties of networks arising from these models via intuition and
simulation. Finally, we discuss results from spectral graph theory that allow
us to explore the role of the geometry of the latent space, independent of
network size. We conclude with conjectures about how these results might be
used to infer the appropriate latent space geometry from observed networks.
