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Message Passing on the Edge: Towards Scalable and Expressive GNNs

Pablo Barceló, Fabian Jogl, Alexander Kozachinskiy, Matthias Lanzinger, Stefan Neumann, Cristóbal Rojas

TL;DR

This work introduces EB-1WL, an edge-based refinement test, and EB-GNN, an edge-centric GNN, to bridge the gap between the expressive power of higher-order WL tests and the practical efficiency of 1WL-based methods. EB-1WL achieves strictly greater expressivity than 1-WL/NC-1WL while maintaining near-linear time per iteration on sparse graphs, supported by a CFOC^3 logical characterization and a lower bound via homomorphism counts from chordal graphs of treewidth 2. EB-GNN exactly matches the expressive power of EB-1WL and demonstrates strong empirical performance across synthetic benchmarks, molecular datasets, and large cybersecurity graphs, often outperforming standard MPNNs and approaching task-specific models with substantially better efficiency. The results establish EB-GNN as a scalable, expressive general-purpose architecture for graph learning, with practical implications for large-scale real-world graphs where higher-order tests are typically prohibitive.

Abstract

We propose EB-1WL, an edge-based color-refinement test, and a corresponding GNN architecture, EB-GNN. Our architecture is inspired by a classic triangle counting algorithm by Chiba and Nishizeki, and explicitly uses triangles during message passing. We achieve the following results: (1)~EB-1WL is significantly more expressive than 1-WL. Further, we provide a complete logical characterization of EB-1WL based on first-order logic, and matching distinguishability results based on homomorphism counting. (2)~In an important distinction from previous proposals for more expressive GNN architectures, EB-1WL and EB-GNN require near-linear time and memory on practical graph learning tasks. (3)~Empirically, we show that EB-GNN is a highly-efficient general-purpose architecture: It substantially outperforms simple MPNNs, and remains competitive with task-specialized GNNs while being significantly more computationally efficient.

Message Passing on the Edge: Towards Scalable and Expressive GNNs

TL;DR

This work introduces EB-1WL, an edge-based refinement test, and EB-GNN, an edge-centric GNN, to bridge the gap between the expressive power of higher-order WL tests and the practical efficiency of 1WL-based methods. EB-1WL achieves strictly greater expressivity than 1-WL/NC-1WL while maintaining near-linear time per iteration on sparse graphs, supported by a CFOC^3 logical characterization and a lower bound via homomorphism counts from chordal graphs of treewidth 2. EB-GNN exactly matches the expressive power of EB-1WL and demonstrates strong empirical performance across synthetic benchmarks, molecular datasets, and large cybersecurity graphs, often outperforming standard MPNNs and approaching task-specific models with substantially better efficiency. The results establish EB-GNN as a scalable, expressive general-purpose architecture for graph learning, with practical implications for large-scale real-world graphs where higher-order tests are typically prohibitive.

Abstract

We propose EB-1WL, an edge-based color-refinement test, and a corresponding GNN architecture, EB-GNN. Our architecture is inspired by a classic triangle counting algorithm by Chiba and Nishizeki, and explicitly uses triangles during message passing. We achieve the following results: (1)~EB-1WL is significantly more expressive than 1-WL. Further, we provide a complete logical characterization of EB-1WL based on first-order logic, and matching distinguishability results based on homomorphism counting. (2)~In an important distinction from previous proposals for more expressive GNN architectures, EB-1WL and EB-GNN require near-linear time and memory on practical graph learning tasks. (3)~Empirically, we show that EB-GNN is a highly-efficient general-purpose architecture: It substantially outperforms simple MPNNs, and remains competitive with task-specialized GNNs while being significantly more computationally efficient.
Paper Structure (31 sections, 10 theorems, 35 equations, 3 figures, 7 tables)

This paper contains 31 sections, 10 theorems, 35 equations, 3 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Further related work.
  3. Preliminaries
  4. Graphs.
  5. WL test.
  6. Neighbor-communication WL test.
  7. Edge-based WL test
  8. The distinguishing power of EB-1WL
  9. Expressivity
  10. Logical characterization
  11. Distinguishing power based on homomorphism counts
  12. Edge-based Graph Neural Networks
  13. Algorithmic implementation.
  14. Experimental Evaluation
  15. Experiment setup
  16. ...and 16 more sections

Key Result

Proposition 1

An iteration of EB-1WL can be performed using $O(m)$ space and $O(\alpha m)$ time.

Figures (3)

  • Figure 1: Message passing scheme of EB-1WL and EB-GNN. Colors correspond to aggregation types.
  • Figure 2: The graphs $G$ and $H$ from the proof of Theorem \ref{['theo:versus']}.
  • Figure 3: Graphs distinguishable by 2WL but not by EB-1WL.

Theorems & Definitions (21)

  • Proposition 1
  • Proposition 2
  • proof
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Example 1
  • Theorem 2
  • proof
  • ...and 11 more