Consistency of the Bethe-Hessian estimator for small average degree (1 < d < 2)

Establish the consistency of the Bethe-Hessian estimator—i.e., consistently estimating the numbers of assortative and disassortative communities by counting the negative outlier eigenvalues of the Bethe-Hessian matrices H(±√d), where H(t) = t^2 I − t A + (D − I)—in the stochastic block model under the bounded expected degree regime when the average expected degree satisfies d ∈ (1, 2).

Background

The paper rigorously analyzes the Bethe-Hessian spectral method for community detection in the stochastic block model (SBM). It confirms that when the expected average degree d ≥ 2, counting negative outlier eigenvalues of H(±√d) consistently estimates the number of communities above the Kesten-Stigum threshold, thereby validating a conjecture from Saade, Krzakala, and Zdeborová for this degree range.

The authors further show eigenvalue concentration and eigenvector approximation results in growing-degree regimes and provide a parameter-free spectral algorithm achieving weak recovery and weak consistency without degree regularization. However, despite these advances, the consistency of the Bethe-Hessian estimator for the smaller bounded-degree range d ∈ (1, 2) is not established and is explicitly highlighted as remaining open.