# A Simple Spectral Algorithm for Recovering Planted Partitions

arXiv: https://arxiv.org/abs/1503.00423 · HTML: https://mlsanity.com/paper/1503.00423/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Mar 2015.

## Built on by (most cited first, 1 of 1)

- [Recovering Unbalanced Communities in the Stochastic Block Model With Application to Clustering with a Faulty Oracle](https://mlsanity.com/paper/2202.08522/) (2022)

## Abstract (arXiv)

In this paper, we consider the planted partition model, in which n = ks vertices of a random graph are partitioned into k "clusters," each of size s. Edges between vertices in the same cluster and different clusters are included with constant probability p and q, respectively (where 0 ≤ q < p ≤ 1). We give an efficient algorithm that, with high probability, recovers the clusters as long as the cluster sizes are are least Omega(√n). Informally, our algorithm constructs the projection operator onto the dominant k-dimensional eigenspace of the graph's adjacency matrix and uses it to recover one cluster at a time. To our knowledge, our algorithm is the first purely spectral algorithm which runs in polynomial time and works even when s = Theta(√ n), though there have been several non-spectral algorithms which accomplish this. Our algorithm is also among the simplest of these spectral algorithms, and its proof of correctness illustrates the usefulness of the Cauchy integral formula in this domain.