Unifying Graph Neural Networks Through a Common Layer Equation

2026-08-17

AI Summary

This study addresses the issue where inconsistent notation in Graph Neural Networks (GNNs) obscures underlying structural differences. We propose a universal layer equation comprising seven components that decouples information flow from content transformation. This formulation unifies over two hundred architectures, enabling verifiable cross-family reductions and establishing a mapping between component selection and task characteristics. Through component-level comparison, generative analysis, and theoretical investigation, we reveal intrinsic connections between propagation mechanisms and critical issues such as oversmoothing. Consequently, this work provides a standardized theoretical framework and a systematic research paradigm for GNN design, facilitating rigorous comparison and principled architectural development across diverse model families.

Abstract

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.

Problem

• Graph Neural Networks
• Unification
• Common Layer Equation
• Architectural Taxonomy
• Design Space

Innovation

• Unified Layer Equation
• Propagation-Message Factorization
• Component-wise Decomposition
• Design Space Unification
• Theoretical Insights