The Bethe-Hessian matrix, introduced by Saade, Krzakala, and Zdeborov\'a (2014), is a Hermitian matrix designed for applying spectral clustering algorithms to sparse networks. Rather than employing a non-symmetric and high-dimensional non-backtracking operator, a spectral method based on the Bethe-Hessian matrix is conjectured to also reach the Kesten-Stigum detection threshold in the sparse stochastic block model (SBM). We provide the first rigorous analysis of the Bethe-Hessian spectral method in the SBM under both the bounded expected degree and the growing degree regimes. Specifically, we demonstrate that: (i) When the expected degree $d\geq 2$, the number of negative outliers of the Bethe-Hessian matrix can consistently estimate the number of blocks above the Kesten-Stigum threshold, thus confirming a conjecture from Saade, Krzakala, and Zdeborov\'a (2014) for $d\geq 2$. (ii) For sufficiently large $d$, its eigenvectors can be used to achieve weak recovery. (iii) As $d\to\infty$, we establish the concentration of the locations of its negative outlier eigenvalues, and weak consistency can be achieved via a spectral method based on the Bethe-Hessian matrix.

翻译：Bethe-Hessian 矩阵由 Saade、Krzakala 和 Zdeborová (2014) 提出，是一种为稀疏网络应用谱聚类算法而设计的厄米矩阵。该方法并非使用非对称且高维的非回溯算子，而是基于 Bethe-Hessian 矩阵的谱方法，据推测同样能够在稀疏随机块模型（SBM）中达到 Kesten-Stigum 检测阈值。我们首次在 SBM 中，针对有界期望度和增长度两种机制，对 Bethe-Hessian 谱方法进行了严格分析。具体而言，我们证明：(i) 当期望度 $d \geq 2$ 时，Bethe-Hessian 矩阵的负离群值数量能够一致地估计出超过 Kesten-Stigum 阈值的块数，从而证实了 Saade、Krzakala 和 Zdeborová (2014) 对于 $d \geq 2$ 情形的猜想。(ii) 对于足够大的 $d$，其特征向量可用于实现弱恢复。(iii) 当 $d \to \infty$ 时，我们确立了其负离群特征值位置的集中性，并且通过基于 Bethe-Hessian 矩阵的谱方法可以实现弱一致性。