A Simple Spectral Algorithm for Recovering Planted Partitions

TL;DR

This paper introduces a simple, efficient spectral algorithm for recovering planted partitions in random graphs, effective even when cluster sizes are as small as proportional to the square root of the total number of vertices.

Contribution

It presents the first polynomial-time spectral algorithm capable of recovering clusters of size ()\u221a{n}, demonstrating simplicity and effectiveness in the planted partition model.

Findings

Recovers clusters with size ()() () \

Operates efficiently in polynomial time

Uses spectral projection and Cauchy integral formula for correctness

Abstract

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 $0 \le q < p \le 1$). We give an efficient algorithm that, with high probability, recovers the clusters as long as the cluster sizes are are least $\Omega(\sqrt{n})$. 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 $s = \Theta(\sqrt n)$, 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.