Reconstruction and estimation in the planted partition model

Abstract

The planted partition model (also known as the stochastic blockmodel) is a classical cluster-exhibiting random graph model that has been extensively studied in statistics, physics, and computer science. In its simplest form, the planted partition model is a model for random graphs on $$n$$ nodes with two equal-sized clusters, with an between-class edge probability of $$q$$ and a within-class edge probability of $$p$$ . Although most of the literature on this model has focused on the case of increasing degrees (ie. $$pn, qn \rightarrow \infty $$ as $$n \rightarrow \infty $$ ), the sparse case $$p, q = O(1/n)$$ is interesting both from a mathematical and an applied point of view. 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 $$q = b/n$$ , then Decelle et al. conjectured that it is possible to cluster in a way correlated with the true partition if $$(a - b)^2 > 2(a + b)$$ , and impossible if $$(a - b)^2 < 2(a + b)$$ . By comparison, the best-known rigorous result is that of Coja-Oghlan, who showed that clustering is possible if $$(a - b)^2 > C (a + b)$$ for some sufficiently large $$C$$ . We prove half of their prediction, showing that it is indeed impossible to cluster if $$(a - b)^2 < 2(a + b)$$ . Furthermore we show that it is impossible even to estimate the model parameters from the graph when $$(a - b)^2 < 2(a + b)$$ ; on the other hand, we provide a simple and efficient algorithm for estimating $$a$$ and $$b$$ when $$(a - b)^2 > 2(a + b)$$ . Following Decelle et al, our work establishes a rigorous connection between the clustering problem, spin-glass models on the Bethe lattice and the so called reconstruction problem. This connection points to fascinating applications and open problems.