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Empowering GNNs via Edge-Aware Weisfeiler-Leman Algorithm

Meng Liu, Haiyang Yu, Shuiwang Ji

TL;DR

The paper addresses the limited expressiveness of standard $1$-WL-based GNNs by introducing NC-1-WL, which accounts for edges among a node's neighbors, yielding a theoretical expressiveness niche between $1$-WL and $3$-WL. Building on this, the authors propose NC-GNN, a differentiable neural framework whose one-layer design preserves the same expressive power as NC-1-WL under injectivity assumptions. They demonstrate both theoretical (injectivity-based) and empirical gains: NC-GNN outperforms GIN on multiple benchmarks with competitive efficiency, and can count 3-cycles and distinguish certain graph structures beyond 1-WL. The work positions NC-GNN as a scalable, powerful base model for graph learning, particularly suitable for social networks and large graphs.

Abstract

Message passing graph neural networks (GNNs) are known to have their expressiveness upper-bounded by 1-dimensional Weisfeiler-Leman (1-WL) algorithm. To achieve more powerful GNNs, existing attempts either require ad hoc features, or involve operations that incur high time and space complexities. In this work, we propose a general and provably powerful GNN framework that preserves the scalability of the message passing scheme. In particular, we first propose to empower 1-WL for graph isomorphism test by considering edges among neighbors, giving rise to NC-1-WL. The expressiveness of NC-1-WL is shown to be strictly above 1-WL and below 3-WL theoretically. Further, we propose the NC-GNN framework as a differentiable neural version of NC-1-WL. Our simple implementation of NC-GNN is provably as powerful as NC-1-WL. Experiments demonstrate that our NC-GNN performs effectively and efficiently on various benchmarks.

Empowering GNNs via Edge-Aware Weisfeiler-Leman Algorithm

TL;DR

The paper addresses the limited expressiveness of standard -WL-based GNNs by introducing NC-1-WL, which accounts for edges among a node's neighbors, yielding a theoretical expressiveness niche between -WL and -WL. Building on this, the authors propose NC-GNN, a differentiable neural framework whose one-layer design preserves the same expressive power as NC-1-WL under injectivity assumptions. They demonstrate both theoretical (injectivity-based) and empirical gains: NC-GNN outperforms GIN on multiple benchmarks with competitive efficiency, and can count 3-cycles and distinguish certain graph structures beyond 1-WL. The work positions NC-GNN as a scalable, powerful base model for graph learning, particularly suitable for social networks and large graphs.

Abstract

Message passing graph neural networks (GNNs) are known to have their expressiveness upper-bounded by 1-dimensional Weisfeiler-Leman (1-WL) algorithm. To achieve more powerful GNNs, existing attempts either require ad hoc features, or involve operations that incur high time and space complexities. In this work, we propose a general and provably powerful GNN framework that preserves the scalability of the message passing scheme. In particular, we first propose to empower 1-WL for graph isomorphism test by considering edges among neighbors, giving rise to NC-1-WL. The expressiveness of NC-1-WL is shown to be strictly above 1-WL and below 3-WL theoretically. Further, we propose the NC-GNN framework as a differentiable neural version of NC-1-WL. Our simple implementation of NC-GNN is provably as powerful as NC-1-WL. Experiments demonstrate that our NC-GNN performs effectively and efficiently on various benchmarks.
Paper Structure (25 sections, 5 theorems, 14 equations, 2 figures, 10 tables, 2 algorithms)

This paper contains 25 sections, 5 theorems, 14 equations, 2 figures, 10 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The NC-1-WL Algorithm
  4. The NC-GNN Framework
  5. Related Work
  6. Experiments
  7. Conclusions
  8. Proofs of Theorems and Lemmas
  9. Proof of Theorem 1
  10. Proof of Theorem 2
  11. Proof of Theorem 3
  12. Proof of Lemma 4
  13. Proof of Lemma 5
  14. More Comparisons with Related Work
  15. NC-GNN vs. Counting 3-Cycles
  16. ...and 10 more sections

Key Result

Theorem 1

NC-1-WL is strictly more powerful than 1-WL in distinguishing non-isomorphic graphs.

Figures (2)

  • Figure 1: (a) Several example pairs of non-isomorphic graphs, partially adapted from sato2020survey, that cannot be distinguished by 1-WL. Colors represent initial node labels or features. Our NC-1-WL can distinguish them. (b) A comparison between the executions of 1-WL and NC-1-WL on two non-isomorphic graphs.
  • Figure 2: Two graphs, adapted from sato2020survey, that cannot be distinguished by NC-1-WL but can be distinguished by 3-WL.

Theorems & Definitions (10)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 4
  • Lemma 5
  • proof
  • proof
  • proof
  • proof
  • proof