Unifying Graph Neural Networks Through a Common Layer Equation

Graph neural networks are commonly described through family-specific equations whose notation obscures shared computations and structural differences. We introduce a common layer equation that represents covered architectures through seven components: an update domain, channel set, propagation bank, per-channel message maps, channel-fusion operator, ego/residual map, and update map. The central factorization separates where information moves, encoded by the propagation bank, from what moves, encoded by the message maps. Function-valued fillings extend the same equation across local message passing, attention, spectral filtering, global communication, relation-specific channels, higher-order domains, and geometric messages. We make this unification explicit and checkable through worked reductions of canonical layers and component assignments spanning seven nonexclusive architectural families. A fixed slot discipline assigns operations by computational role and defines the framework's coverage boundary. The decomposition also yields component-level theoretical insights: under endpoint-local messages and node-local updates, operator support bounds one-layer dependencies, and one-layer global mixing requires a full effective operator row under the stated hypotheses. The resulting framework organizes more than 200 architectures in a common design space, enables component-wise comparison and generation of structurally consistent architectures, and connects propagation choices to oversmoothing, oversquashing, heterophily, and expressivity. It further exposes the empirical inverse problem of mapping measurable graph and task properties to validated component choices.

AEGIS: A Multi-Task Joint-Embedding Predictive Architecture for Mammography

We present Aegis, a joint-embedding predictive architecture for breast cancer detection and density assessment in mammography. We train three Vision Transformer variants (Small/Base/Large) using self-supervised joint-embedding predictive architecture (JEPA) pre-training on 71,103 studies from 14 clinical sites, followed by supervised fine-tuning with progressive resolution scaling up to 2048x1536. On a curated 785-study test set, our largest model achieves area under the receiver operating characteristic curve (AUC) 0.949 for breast cancer triage with 93% sensitivity and 75% specificity at the optimal operating point. An ensemble combining our model with a U.S. Food and Drug Administration-cleared baseline further improves discrimination to 0.952 AUC. For breast density classification, the model achieves 0.953 AUC for binary (dense vs. non-dense) classification and 62.6% exact accuracy across four Breast Imaging Reporting and Data System (BI-RADS) categories, with 98.8% adjacent accuracy comparable to reported human inter-reader agreement. External validation on the public VinDr-Mammo dataset provides evidence of cross-population transfer under a different reference standard, with the largest model achieving 0.871 AUC for triage in a zero-shot setting.