Network recovery from aggregated relational data: An impossibility theorem

Abstract

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.