Consistency Thresholds for the Planted Bisection Model

TL;DR

This paper characterizes the exact thresholds for reliably recovering the planted bisection in a random graph model, providing an efficient algorithm that works near these thresholds and establishing conditions for asymptotic recoverability.

Contribution

It establishes necessary and sufficient conditions for asymptotic recoverability and introduces an efficient, near-linear time algorithm for recovering the planted bisection.

Findings

Recovery is possible if and only if nodes align with the majority of neighbors.

The algorithm combines spectral clustering, a 'replica' stage, and local moves.

Conditions for recoverability are tight and match theoretical thresholds.

Abstract

The planted bisection model is a random graph model in which the nodes are divided into two equal-sized communities and then edges are added randomly in a way that depends on the community membership. We establish necessary and sufficient conditions for the asymptotic recoverability of the planted bisection in this model. When the bisection is asymptotically recoverable, we give an efficient algorithm that successfully recovers it. We also show that the planted bisection is recoverable asymptotically if and only if with high probability every node belongs to the same community as the majority of its neighbors. Our algorithm for finding the planted bisection runs in time almost linear in the number of edges. It has three stages: spectral clustering to compute an initial guess, a "replica" stage to get almost every vertex correct, and then some simple local moves to finish the job. An independent work by Abbe, Bandeira, and Hall establishes similar (slightly weaker) results but only in the case of logarithmic average degree.