AI 中文总结
本文证明从聚合关系数据中一致恢复边概率是不可能的，即使总体可识别；通过极小极大下界和混合累积量分析，指出障碍在于单网络估计而非总体识别。
AI 中文摘要
我们证明了一个不可能性定理，即从完整的聚合关系数据中一致地恢复带标签的边概率是不可能的，即使总体计数定律能识别出每个概率。对于任意已知划分为两个相等特征组的划分，我们考虑一个具有平衡秩一信号和未知活动效应（具有有界总二元能量）的独立边逻辑斯蒂网络。信号幅度已知且固定为小的正值。每个节点向两个组报告其计数。概率矩阵的极小极大均方误差在网络增长时仍远离零，而在相同参数类下，在完全邻接矩阵下该误差趋于零。下界通过一个两节点预言机来考虑整个计数数组上的依赖性，所有观测计数都可以从该预言机重建。条件二项平滑限制了关于局部符号方向的信息，而锚定的多比特构造将这些模糊性转化为非消失的归一化矩阵损失。一个混合累积量恒等式从总体度定律中恢复每个边概率，将障碍定位在从单个聚合网络进行估计，而非总体识别。
英文摘要
We prove an impossibility theorem for uniformly consistent recovery of labeled edge probabilities from complete aggregated relational data, even when the population count law identifies every probability. For any known partition into two equal trait groups, we consider an independent-edge logistic network with a balanced rank-one signal and unknown activity effects with bounded total dyadic energy. The signal amplitude is known and fixed at a small positive value. Every node reports its counts to both groups. The minimax mean squared error for the probability matrix remains bounded away from zero as the network grows, whereas it tends to zero under full adjacency on the same parameter class. The lower bound accounts for the dependence across the entire count array through a two-node oracle from which all observed counts can be reconstructed. Conditional binomial smoothing bounds the information about a local sign orientation, and an anchored many-bit construction converts these ambiguities into a nonvanishing normalized matrix loss. A mixed-cumulant identity recovers every edge probability from the population degree law, locating the obstruction in estimation from a single aggregated network rather than in population identification.
Comments12 pages
