Decoding Binary Node Labels from Censored Edge Measurements: Phase Transition and Efficient Recovery

Abstract

We consider the problem of clustering a graph G into two communities by observing a subset of the vertex correlations. Specifically, we consider the inverse problem with observed variables Y = B <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G</sub> x ⊕ Z, where B <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G</sub> is the incidence matrix of a graph G, x is the vector of unknown vertex variables (with a uniform prior), and Z is a noise vector with Bernoulli (ε) i.i.d. entries. All variables and operations are Boolean. This model is motivated by coding, synchronization, and community detection problems. In particular, it corresponds to a stochastic block model or a correlation clustering problem with two communities and censored edges. Without noise, exact recovery (up to global flip) of x is possible if and only the graph G is connected, with a sharp threshold at the edge probability log (n)/n for Erdos-Renyi random graphs. The first goal of this paper is to determine how the edge probabilityp needs to scale to allow exact recovery in the presence of noise. Defining the degree rate of the graph by α = np/log(n), it is shown that exact recovery is possible if and only if α > 2/(1 - 2ε) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> + o(1/(1 - 2ε) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ). In other words, 2/(1 - 2ε) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> is the information theoretic threshold for exact recovery at low-SNR. In addition, an efficient recovery algorithm based on semidefinite programming is proposed and shown to succeed in the threshold regime up to twice the optimal rate. For a deterministic graph G, defining the degree rate as α = d/log(n), where d is the minimum degree of the graph, it is shown that the proposed method achieves the rate α > 4((1 + λ)/(1 - λ) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> /(1 - 2ε) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> + o(1/(1 - 2ε) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ), where 1-λ is the spectral gap of the graph G.