度修正泊松随机块模型中精确社区恢复的一个阈值常数
A Threshold Constant for Exact Community Recovery in the Degree-Corrected Poisson Stochastic Block Model
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中文总结 AI 辅助
本研究通过最大剖面似然估计器，为度修正泊松随机块模型确定了精确社区恢复的显式阈值常数，证明了其大于1时实现强相合恢复，并推测其为信息论尖锐阈值。
中文摘要 AI 辅助
我们通过最大剖面似然估计器研究度修正泊松随机块模型（DC-SBM）中的精确社区恢复问题。我们的主要贡献是确定了一个显式的复杂度常数 $C(\pi,S)$，它紧密地刻画了精确恢复的阈值。该常数将社区之间的加权Chernoff--Hellinger分离与由度归一化块体积引起的Kullback--Leibler修正相结合。我们严格证明了 $C(\pi,S)>1$ 确保了最大剖面似然估计器的强相合性，从而在直到对数稀疏尺度上实现精确社区恢复。我们推测 $C(\pi,S)=1$ 代表了尖锐的信息论阈值，为精确恢复转变提供了一个上界，并讨论了与二元SBM和二元DC-SBM情形的类比。此外，当平均度发散时，我们建立了弱相合性，为稀疏和稠密区域提供了一个统一的图景。证明需要在对异质度参数存在的情况下对剖面似然进行新的控制，这些参数的节点特定权重从根本上改变了恢复问题的几何结构和集中性。
英文摘要
We study exact community recovery in the Degree-Corrected Poisson Stochastic Block Model (DC-SBM) through the maximum profile likelihood estimator. Our main contribution is identifying an explicit complexity constant $C(π,S)$, which closely captures the exact recovery threshold. This constant combines a weighted Chernoff--Hellinger separation between communities with a Kullback--Leibler correction induced by the degree-normalized block volumes. We rigorously prove that $C(π,S)>1$ ensures the strong consistency of the maximum profile likelihood estimator, yielding exact community recovery down to the logarithmic sparsity scale. We conjecture that $C(π,S)=1$ represents the sharp information-theoretic threshold, providing an upper bound on the exact-recovery transition, and we discuss analogies with the binary SBM and binary DC-SBM cases. Additionally, we establish weak consistency when the average degree diverges, offering a unified picture across sparse and dense regimes. The proofs require new control of the profile likelihood in the presence of heterogeneous degree parameters, whose node-specific weights fundamentally alter the geometry and concentration of the recovery problem.
