# Exact Recovery in the Stochastic Block Model

arXiv: https://arxiv.org/abs/1405.3267 · HTML: https://mlsanity.com/paper/1405.3267/

## Key facts

- Cited by 78 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 May 2014.

## Built on by (most cited first, 30 of 78)

- [Graph Convolution for Semi-Supervised Classification: Improved Linear Separability and Out-of-Distribution Generalization](https://mlsanity.com/paper/2102.06966/) (2021)
- [Effects of Graph Convolutions in Multi-layer Networks](https://mlsanity.com/paper/2204.09297/) (2022)
- [Correlated Stochastic Block Models: Exact Graph Matching with Applications to Recovering Communities](https://mlsanity.com/paper/2107.06767/) (2021)
- [Private estimation algorithms for stochastic block models and mixture models](https://mlsanity.com/paper/2301.04822/) (2023)
- [A Neural Collapse Perspective on Feature Evolution in Graph Neural Networks](https://mlsanity.com/paper/2307.01951/) (2023)
- [Differentially Private Community Detection for Stochastic Block Models](https://mlsanity.com/paper/2202.00636/) (2022)
- [Exact Recovery in the General Hypergraph Stochastic Block Model](https://mlsanity.com/paper/2105.04770/) (2021)
- [Minimax Rates for Robust Community Detection](https://mlsanity.com/paper/2207.11903/) (2022)
- [Quantifying the Optimization and Generalization Advantages of Graph Neural Networks Over Multilayer Perceptrons](https://mlsanity.com/paper/2306.13926/) (2023)
- [A Comprehensive Review of Community Detection in Graphs](https://mlsanity.com/paper/2309.11798/) (2023)
- [An Inexact Projected Gradient Method with Rounding and Lifting by Nonlinear Programming for Solving Rank-One Semidefinite Relaxation of Polynomial Optimization](https://mlsanity.com/paper/2105.14033/) (2021)
- [Exact Community Recovery in Correlated Stochastic Block Models](https://mlsanity.com/paper/2203.15736/) (2022)
- [Partial recovery and weak consistency in the non-uniform hypergraph Stochastic Block Model](https://mlsanity.com/paper/2112.11671/) (2021)
- [DeepWalking Backwards: From Embeddings Back to Graphs](https://mlsanity.com/paper/2102.08532/) (2021)
- [Estimating Graph Dimension with Cross-validated Eigenvalues](https://mlsanity.com/paper/2108.03336/) (2021)
- [Computational Lower Bounds for Graphon Estimation via Low-degree Polynomials](https://mlsanity.com/paper/2308.15728/) (2023)
- [Matrix Completion with Hierarchical Graph Side Information](https://mlsanity.com/paper/2201.01728/) (2022)
- [Sparse random hypergraphs: Non-backtracking spectra and community detection](https://mlsanity.com/paper/2203.07346/) (2022)
- [Joint Community Detection and Rotational Synchronization via Semidefinite Programming](https://mlsanity.com/paper/2105.06031/) (2021)
- [Bipartite mixed membership distribution-free model. A novel model for community detection in overlapping bipartite weighted networks](https://mlsanity.com/paper/2211.00912/) (2022)
- [DiSC: Differential Spectral Clustering of Features](https://mlsanity.com/paper/2211.05314/) (2022)
- [Neural-prior stochastic block model](https://mlsanity.com/paper/2303.09995/) (2023)
- [Differentially private exact recovery for stochastic block models](https://mlsanity.com/paper/2406.02644/) (2024)
- [Community Detection with a Subsampled Semidefinite Program](https://mlsanity.com/paper/2102.01419/) (2021)
- [Semidefinite Programming for Community Detection with Side Information](https://mlsanity.com/paper/2105.02816/) (2021)
- [A useful criterion on studying consistent estimation in community detection](https://mlsanity.com/paper/2109.14950/) (2021)
- [A Spectral Method for Joint Community Detection and Orthogonal Group Synchronization](https://mlsanity.com/paper/2112.13199/) (2021)
- [Exact Community Recovery over Signed Graphs](https://mlsanity.com/paper/2202.12255/) (2022)
- [Principles for Initialization and Architecture Selection in Graph Neural Networks with ReLU Activations](https://mlsanity.com/paper/2306.11668/) (2023)
- [Low degree conjecture implies sharp computational thresholds in stochastic block model](https://mlsanity.com/paper/2502.15024/) (2025)

## Abstract (arXiv)

The stochastic block model (SBM) with two communities, or equivalently the planted bisection model, is a popular model of random graph exhibiting a cluster behaviour. In the symmetric case, the graph has two equally sized clusters and vertices connect with probability p within clusters and q across clusters. In the past two decades, a large body of literature in statistics and computer science has focused on providing lower-bounds on the scaling of |p-q| to ensure exact recovery. In this paper, we identify a sharp threshold phenomenon for exact recovery: if α=pn/log(n) and β=qn/log(n) are constant (with α>β), recovering the communities with high probability is possible if fracα+β2 - √α β>1 and impossible if fracα+β2 - √α β<1. In particular, this improves the existing bounds. This also sets a new line of sight for efficient clustering algorithms. While maximum likelihood (ML) achieves the optimal threshold (by definition), it is in the worst-case NP-hard. This paper proposes an efficient algorithm based on a semidefinite programming relaxation of ML, which is proved to succeed in recovering the communities close to the threshold, while numerical experiments suggest it may achieve the threshold. An efficient algorithm which succeeds all the way down to the threshold is also obtained using a partial recovery algorithm combined with a local improvement procedure.