Latent position models (LPMs) are a large and popular class of models for random graphs. However, fitting Bayesian LPMs is computationally challenging - computing the likelihood even once takes time that is quadratic in the number of vertices $|V|$ of the observed graph $G = (V,E)$. Many previous papers have introduced approximate MCMC algorithms to speed this up, with the most similar to ours, Rastelli et al (2024), presenting an algorithm that has amortized running time that can be reduced almost to $O(|E|)$ and good empirical performance on reasonable inference problems. The present paper offers two algorithms for solving the same problem: a ``fast" algorithm with running time of the same almost-$O(|E|)$ order as astelli et al and much stronger accuracy guarantees, and a ``faster" algorithm with an improved running time of almost $O(|V|)$, and accuracy guarantees that are slightly improved compared to Rastelli et al (but not sufficient for all tasks). The main improvements come from the introduction of a simple auxiliary data structure that can be cheaply updated during an MCMC run; we suspect that the same ``cheap sketch" may be useful for other MCMC algorithms.

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