  Latent space models (LSMs) are often used to analyze dynamic (time-varying)
networks that evolve in continuous time. Existing approaches to Bayesian
inference for these models rely on Markov chain Monte Carlo algorithms, which
cannot handle modern large-scale networks. To overcome this limitation, we
introduce a new prior for continuous-time LSMs based on Bayesian P-splines that
allows the posterior to adapt to the dimension of the latent space and the
temporal variation in each latent position. We propose a stochastic variational
inference algorithm to estimate the model parameters. We use stochastic
optimization to subsample both dyads and observed time points to design a fast
algorithm that is linear in the number of edges in the dynamic network.
Furthermore, we establish non-asymptotic error bounds for point estimates
derived from the variational posterior. To our knowledge, this is the first
such result for Bayesian estimators of continuous-time LSMs. Lastly, we use the
method to analyze a large data set of international conflicts consisting of
4,456,095 relations from 2018 to 2022.
