A Proof Of The Block Model Threshold Conjecture

TLDR: This work proves the rest of the conjecture of Decelle, Krzkala, Moore and Zdeborová by providing an efficient algorithm for clustering in a way that is correlated with the true partition when s2>d.

Abstract: We study a random graph model named the "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 clusters, with a between-class edge probability of $q$ and a within-class edge probability of $p$. A striking conjecture of Decelle, Krzkala, Moore and Zdeborova 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 $p = a/n$ and $q = b/n$, $s=(a-b)/2$ and $p=(a+b)/2$ then Decelle et al.\ conjectured that it is possible to efficiently cluster in a way correlated with the true partition if $s^2 > p$ and impossible if $s^2 C p \ln p$ for some sufficiently large $C$. In a previous work, we proved that indeed it is information theoretically impossible to to cluster if $s^2 p$. A different independent proof of the same result was recently obtained by Laurent Massoulie.