Recovering Unobserved Network Links from Aggregated Relational Data: Bayesian Latent Surface Modeling and Penalized Regression

Aggregated relational data (ARD) record counts of ties to attribute-defined groups while leaving individual edges unobserved. We compare latent-geometry and regularized network estimators through a common observation map. The comparison distinguishes the realized adjacency matrix, conditional edge probabilities, and model parameters. We study roster-based ARD with known node-level group memberships, giving both estimators the same roster and aggregate counts. We relate the aggregate means to a Poisson working likelihood and a Huber loss, and give their derivatives. Overlapping groups, shared edges, and reporting error affect the interpretation of these objectives. Geometry restricts the representation of edge probabilities, while regularization selects among candidate fits. Identification depends on the observation map and model restrictions rather than uniqueness of a numerical optimizer. A reproducible synthetic experiment specifies the data-generating process, estimation algorithms, and evaluation targets under matched information. The matrix estimator gives better realized-edge rankings and aggregate fit, while the geometric estimator gives lower error for generating probabilities. The resulting framework organizes ARD reconstruction around the interaction of observation design, structural assumptions, and computation.