# Optimal Bayesian Estimation for Random Dot Product Graphs

arXiv: https://arxiv.org/abs/1904.12070 · HTML: https://mlsanity.com/paper/1904.12070/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Apr 2019.

## Built on by (most cited first, 4 of 4)

- [Community Detection in General Hypergraph via Graph Embedding](https://mlsanity.com/paper/2103.15035/) (2021)
- [Transfer Learning for Latent Variable Network Models](https://mlsanity.com/paper/2406.03437/) (2024)
- [Hypothesis testing on invariant subspaces of non-diagonalizable matrices with applications to network statistics](https://mlsanity.com/paper/2303.18233/) (2023)
- [Random Dot Product Graphs as Dynamical Systems: Limitations and Opportunities](https://mlsanity.com/paper/2603.05703/) (2026)

## Abstract (arXiv)

We propose a Bayesian approach, called the posterior spectral embedding, for estimating the latent positions in random dot product graphs, and prove its optimality. Unlike the classical spectral-based adjacency/Laplacian spectral embedding, the posterior spectral embedding is a fully-likelihood based graph estimation method taking advantage of the Bernoulli likelihood information of the observed adjacency matrix. We develop a minimax-lower bound for estimating the latent positions, and show that the posterior spectral embedding achieves this lower bound since it both results in a minimax-optimal posterior contraction rate, and yields a point estimator achieving the minimax risk asymptotically. The convergence results are subsequently applied to clustering in stochastic block models, the result of which strengthens an existing result concerning the number of mis-clustered vertices. We also study a spectral-based Gaussian spectral embedding as a natural Bayesian analogy of the adjacency spectral embedding, but the resulting posterior contraction rate is sub-optimal with an extra logarithmic factor. The practical performance of the proposed methodology is illustrated through extensive synthetic examples and the analysis of a Wikipedia graph data.