Unifying Graph Neural Networks Through a Common Layer Equation

Summary

This paper introduces a common layer equation that unifies many graph neural network architectures via seven components: 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 (propagation bank) from what moves (message maps). Function-valued fillings extend the equation to local message passing, attention, spectral filtering, global communication, relation-specific channels, higher-order domains, and geometric messages. Worked reductions and component assignments cover seven nonexclusive architectural families, organizing over 200 architectures in a common design space. The framework enables component-wise comparison and generation of structurally consistent architectures, and connects propagation choices to oversmoothing, oversquashing, heterophily, and expressivity. It also exposes the empirical inverse problem of mapping measurable graph and task properties to validated component choices. Theoretical insights include operator support bounds and conditions for one-layer global mixing under stated hypotheses.