Stochastic Block Models and Reconstruction

TL;DR

This paper rigorously analyzes the stochastic block model in the sparse regime, proving part of a conjecture about the threshold for successful clustering and parameter estimation, and establishing connections to spin-glass models and reconstruction problems.

Contribution

It proves the impossibility of clustering below the conjectured threshold and provides an efficient method for parameter estimation above it, confirming half of the non-rigorous predictions.

Findings

Clustering is impossible if (a - b)^2 < 2(a + b).

Parameter estimation is feasible when (a - b)^2 > 2(a + b).

Established a connection between clustering, spin-glass models, and reconstruction problems.

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.\ $pn, qn \to \infty$ as $n \to \infty$), 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 Zdeborov\'a 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.