A Multiple Random Scan Strategy for Latent Space Models

Latent Space (LS) network models project the nodes of a network on a $d$-dimensional latent space to achieve dimensionality reduction of the network while preserving its relevant features. Inference is often carried out within a Markov Chain Monte Carlo (MCMC) framework. Nonetheless, it is well-known that the computational time for this set of models increases quadratically with the number of nodes. In this work, we build on the Random-Scan (RS) approach to propose an MCMC strategy that alleviates the computational burden for LS models while maintaining the benefits of a general-purpose technique. We call this novel strategy Multiple RS (MRS). This strategy is effective in reducing the computational cost by a factor without severe consequences on the MCMC draws. Moreover, we introduce a novel adaptation strategy that consists of a probabilistic update of the set of latent coordinates of each node. Our Adaptive MRS adapts the acceptance rate of the Metropolis step to adjust the probability of updating the latent coordinates. We show via simulation that the Adaptive MRS approach performs better than MRS in terms of mixing. Finally, we apply our algorithm to a multi-layer temporal LS model and show how our adaptive strategy may be beneficial to empirical applications.