降至渗流阈值的Bethe-Hessian矩阵
The Bethe-Hessian down to the Percolation Threshold
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中文总结 AI 辅助
该研究将Bethe-Hessian矩阵负特征值数量与植入模型特征值对应关系的结论，从d≥2推广至最优条件d>1，通过2-核测试向量构造与马尔可夫随机场各向同性基的方法实现稀疏情形下的波动控制。
中文摘要 AI 辅助
Bethe-Hessian是一种对称矩阵，人们已观察到其负谱可编码稀疏随机块模型的信息结构。我们证明，在所有顶点期望度为$d>1$的随机块模型中，Bethe-Hessian的负特征值数量恰好等于植入模型位于体谱外的特征值所预测的数量。$d>1$这一条件是最优的，且与现有基于更大非埃尔米特矩阵的谱方法适用的情形相匹配。我们的结果推广了Stephan和Zhu的定理，他们在$d\bgeq2$的假设下得到了相同结论。\n我们的改进基于两个核心思路。第一，我们在2-核上构造测试向量（2-核上的度波动显著更小），随后在控制二次型的前提下将其扩展到整个图。第二，我们使用底层马尔可夫随机场的各向同性基构造测试向量，其系数适配每个相关的植入特征值。这使得我们能够在整个稀疏情形下控制测试向量的波动。
英文摘要
The Bethe-Hessian is a symmetric matrix for which the negative spectrum has been observed to encode the informative structure of sparse stochastic block models. We prove that, in the stochastic block model where all vertices have expected degree $d>1$, the number of negative eigenvalues of the Bethe-Hessian is exactly the number predicted by the eigenvalues of the planted model lying outside the bulk spectrum. The condition $d>1$ is optimal, and matches a regime in which existing spectral approaches based on larger non-Hermitian matrices apply. Our result extends a theorem of Stephan and Zhu, who established the same conclusion under the assumption $d\geq 2$. Our improvement relies on two main ideas. First, we construct test vectors on the $2$-core, where degree fluctuations are substantially smaller, and then extend them to the entire graph while controlling the quadratic form. Second, we construct the test vectors using an isotropic basis of the underlying Markov random field, with coefficients adapted to each relevant planted eigenvalue. This allows us to control the fluctuations of the test vectors throughout the sparse regime.
