Unifying Graph Neural Networks Through a Common Layer Equation

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graph-neural-networks unification layer-equation architecture-comparison oversmoothing expressivity theoretical-framework

Summary

The paper introduces a common layer equation that unifies graph neural networks into seven components, enabling architectural comparison, theoretical analysis, and insights into issues like oversmoothing and expressivity.

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.
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Abstract

A unified layer equation decomposes graph neural networks into seven components to compare architectures, derive theoretical bounds, and expose design choices linked to oversmoothing and expressivity.

Graph neural networksare 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-channelmessage maps, channel-fusion operator, ego/residual map, and update map. The central factorization separates where information moves, encoded by thepropagation bank, from what moves, encoded by themessage maps. Function-valued fillings extend the same equation across local message passing, attention,spectral filtering,global communication,relation-specific channels,higher-order domains, andgeometric 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 tooversmoothing,oversquashing,heterophily, andexpressivity. It further exposes the empirical inverse problem of mapping measurable graph and task properties to validated component choices.

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