  Interactions between actors are frequently represented using a network. The
latent position model is widely used for analysing network data, whereby each
actor is positioned in a latent space. Inferring the dimension of this space is
challenging. Often, for simplicity, two dimensions are used or model selection
criteria are employed to select the dimension, but this requires choosing a
criterion and the computational expense of fitting multiple models. Here the
latent shrinkage position model (LSPM) is proposed which intrinsically infers
the effective dimension of the latent space. The LSPM employs a Bayesian
nonparametric multiplicative truncated gamma process prior that ensures
shrinkage of the variance of the latent positions across higher dimensions.
Dimensions with non-negligible variance are deemed most useful to describe the
observed network, inducing automatic inference on the latent space dimension.
While the LSPM is applicable to many network types, logistic and Poisson LSPMs
are developed here for binary and count networks respectively. Inference
proceeds via a Markov chain Monte Carlo algorithm, where novel surrogate
proposal distributions reduce the computational burden. The LSPM's properties
are assessed through simulation studies, and its utility is illustrated through
application to real network datasets. Open source software assists wider
implementation of the LSPM.
