A Proof of the Block Model Threshold Conjecture

We study a random graph model called the “stochastic block model” in statistics and the “planted partition model” in theoretical computer science. In its simplest form, this is a random graph with two equal-sized classes of vertices, with a within-class edge probability of q and a between-class edge probability of q′. A striking conjecture of Decelle, Krzkala, Moore and Zdeborová [9], based on deep, non-rigorous ideas from statistical physics, gave a precise prediction for the algorithmic threshold of clustering in the sparse planted partition model. In particular, if q=a/n and q′=b/n, s=(a−b)/2 and d=(a+b)/2, then Decelle et al. conjectured that it is possible to efficiently cluster in a way correlated with the true partition if s²>d and impossible if s²<d. By comparison, until recently the best-known rigorous result showed that clustering is possible if s²>Cdlnd for sufficiently large C. In a previous work, we proved that indeed it is information theoretically impossible to cluster if s² ≤ d and moreover that it is information theoretically impossible to even estimate the model parameters from the graph when s² < d. Here we prove the rest of the conjecture by providing an efficient algorithm for clustering in a way that is correlated with the true partition when s²>d. A different independent proof of the same result was recently obtained by Massoulié [20].