  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.
