Computational and statistical thresholds in multi-layer stochastic block models
研究了多层随机块模型中社区恢复与检测的临界网络密度阈值，发现无计算限制时阈值与层数线性相关，而多项式时间算法下阈值与层数平方根相关，部分解决了相关开放问题。
We study the problem of community recovery and detection in multi-layer stochastic block models, focusing on the critical network density threshold for consistent community structure inference. Using a prototypical two-block model, we reveal a computational barrier for such multilayer stochastic block models that does not exist for its single-layer counterpart: When there are no computational constraints, the density threshold depends linearly on the number of layers. However, when restricted to polynomial-time algorithms, the density threshold scales with the square root of the number of layers, assuming correctness of a low-degree polynomial hardness conjecture. Our results provide a nearly complete picture of the optimal inference in multiple-layer stochastic block models and partially settle the open question in (J. Amer. Statist. Assoc. 118 (2023) 2433–2445) regarding the optimality of the bias-adjusted spectral method.