Achieving Exact Cluster Recovery Threshold via Semidefinite Programming

TL;DR

This paper proves that semidefinite programming can exactly recover clusters in stochastic block models at the optimal threshold, confirming a longstanding conjecture and extending results to planted dense subgraph models.

Contribution

It demonstrates that semidefinite programming relaxations achieve the optimal recovery threshold in stochastic block models and planted dense subgraph models, resolving a key conjecture.

Findings

SDP achieves the optimal exact recovery threshold in stochastic block models.

SDP also attains the optimal threshold in planted dense subgraph models.

The results confirm the conjecture of Abbe et al. regarding SDP performance.

Abstract

The binary symmetric stochastic block model deals with a random graph of $n$ vertices partitioned into two equal-sized clusters, such that each pair of vertices is connected independently with probability $p$ within clusters and $q$ across clusters. In the asymptotic regime of $p=a \log n/n$ and $q=b \log n/n$ for fixed $a,b$ and $n \to \infty$, we show that the semidefinite programming relaxation of the maximum likelihood estimator achieves the optimal threshold for exactly recovering the partition from the graph with probability tending to one, resolving a conjecture of Abbe et al. \cite{Abbe14}. Furthermore, we show that the semidefinite programming relaxation also achieves the optimal recovery threshold in the planted dense subgraph model containing a single cluster of size proportional to $n$.