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Latent space models (LSMs) are often used to analyze dynamic (time-varying)networks that evolve in continuous time. Existing approaches to Bayesianinference for these models rely on Markov chain Monte Carlo algorithms, whichcannot handle modern large-scale networks. To overcome this limitation, weintroduce a new prior for continuous-time LSMs based on Bayesian P-splines thatallows the posterior to adapt to the dimension of the latent space and thetemporal variation in each latent position. We propose a stochastic variationalinference algorithm to estimate the model parameters. We use stochasticoptimization to subsample both dyads and observed time points to design a fastalgorithm that is linear in the number of edges in the dynamic network.Furthermore, we establish non-asymptotic error bounds for point estimatesderived from the variational posterior. To our knowledge, this is the firstsuch result for Bayesian estimators of continuous-time LSMs. Lastly, we use themethod to analyze a large data set of international conflicts consisting of4,456,095 relations from 2018 to 2022.

Fast Variational Inference of Latent Space Models for Dynamic Networks Using Bayesian P-Splines