Spectral detection in the censored block model

Abstract

Abstract-We consider the problem of partially recovering hidden binary variables from the observation of (few) censored edge weights, a problem with applications in community detection, correlation clustering and synchronization. We describe two spectral algorithms for this task based on the non-backtracking and the Bethe Hessian operators. These algorithms are shown to be asymptotically optimal for the partial recovery problem, in that they detect the hidden assignment as soon as it is information theoretically possible to do so. A. IntroductionIn many inference problems, the available data can be represented on a weighted graph. Given the knowledge of the edge weights, the task is to infer latent variables carried by the nodes. Here, we shall consider the problem of recovering binary node labels from censored edge measurements [1], [2]. Specifically, given an Erdős-Rényi random graph G = (V, E) ∈ G(n, α/n) with n nodes carrying latent variables σ i = ±1, 1 ≤ i ≤ n, we draw the edge labels J ij = ±1, (ij) ∈ E from the following distribution:where is a noise parameter. In the noiseless case = 0, we have σ i σ j = J ij and one can easily recover the communities in each connected component along a spanning tree. When = 1/2, on the other hand, the graph doesn't contain any information about the latent variables σ i , and recovery is impossible. What happens in between? The problem of exactly recovering the latent variables σ i has been studied in [1]. It turns out that, asymptotically in the large n limit, exact recovery is shown to be possible if and only ifwhere α is the average degree of the graph. Note that the variable of an isolated vertex cannot be recovered so that the average degree has to grow at least like log n, as in the Coupon collector's problem, to ensure that the graph is connected.We consider in this paper the case where the average degree α will remain fixed as n tends to infinity. In this setting, we cannot ask for exact recovery and we consider here a different question: is it possible to infer an assignmentσ i of the latent variables that is positively correlated with the planted variables σ i ? We call positively correlated an assignmentσ i such that the following quantity, called overlap, is strictly positive:In the limit n → ∞, this overlap vanishes for a random guessσ i , and is equal to unity if the recovery is exact. We will refer to the task of finding a positively correlated assignment σ i as partial recovery. This task has been shown [3], [4] to be possible only ifTo the best of our knowledge, there is no rigorous proof that this bound is also sufficient. In [3], the same authors also showed that belief propagation (BP) allows to saturate this bound. However, there is no rigorous analysis of BP for this problem and the fact that condition (4) is necessary and sufficient was left as a conjecture in [3] and only the necessary part was proved in [4]. Moreover, from a practical point of view, BP requires the knowledge of the noise parameter .In this contribution, we describe two simple sp...

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