Activity-Conditioned Residual Association from Aggregated Relational Data

Aggregated relational data (ARD) record how many ties sampled respondents have to predeclared groups without revealing individual dyads. We ask whether such counts can falsify a pure additive-activity network model for one predeclared form of residual cross-group association. When the groups form an exhaustive partition, respondent degree is observed exactly. For arbitrary fixed activity values under an independent-Bernoulli additive-logit null, conditioning two respondents on equal degree yields a finite nonpositive sign restriction for their two disjoint group-count differences. Operational inference is narrower: on a regular repeated-cell array, analyst-randomized pair thinning gives a conservative one-network test with graph-independent sampled respondents. Finite and growing matched-law constructions establish that the target contains bivariate information absent from a collapsed binary count, while a declared rank-one path gives pointwise power. A deterministic marked-profile contract and a fixed-dimensional lattice local-limit argument make the design and conditioning requirements explicit. These results define a target- and design-specific specification diagnostic, not recovery of a labeled probability matrix, latent geometry, or arbitrary residual dependence.