# Consistency of adjacency spectral embedding for the mixed membership stochastic blockmodel

arXiv: https://arxiv.org/abs/1705.04518 · HTML: https://mlsanity.com/paper/1705.04518/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 May 2017.

## Built on by (most cited first, 2 of 2)

- [Spectral embedding and the latent geometry of multipartite networks](https://mlsanity.com/paper/2202.03945/) (2022)
- [Uncovering Locally Low-dimensional Structure in Networks by Locally Optimal Spectral Embedding](https://mlsanity.com/paper/2603.11965/) (2026)

## Abstract (arXiv)

The mixed membership stochastic blockmodel is a statistical model for a graph, which extends the stochastic blockmodel by allowing every node to randomly choose a different community each time a decision of whether to form an edge is made. Whereas spectral analysis for the stochastic blockmodel is increasingly well established, theory for the mixed membership case is considerably less developed. Here we show that adjacency spectral embedding into ℝk, followed by fitting the minimum volume enclosing convex k-polytope to the k-1 principal components, leads to a consistent estimate of a k-community mixed membership stochastic blockmodel. The key is to identify a direct correspondence between the mixed membership stochastic blockmodel and the random dot product graph, which greatly facilitates theoretical analysis. Specifically, a 2 rightarrow ∞ norm and central limit theorem for the random dot product graph are exploited to respectively show consistency and partially correct the bias of the procedure.