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Edged Weisfeiler-Lehman Algorithm

Xiao Yue, Bo Liu, Feng Zhang, Guangzhi Qu

TL;DR

The paper extends the classic Weisfeiler-Lehman test by incorporating edge features into node color refinements (E-WL) and builds an Edged Graph Isomorphism Network (EGIN) that leverages these edge-aware representations. It introduces two variants, EGIN-C (cross updating) and EGIN-E (edge embedding), to address feature-dimension issues and enhance expressiveness. Through experiments on 12 edge-featured datasets, EGIN and its variants generally outperform strong baselines, demonstrating the practical value of exploiting edge information in graph classification. The work also discusses an edge-aggregation variant (E-WL-EA) and proves that edge aggregation does not surpass E-WL in discriminative power, reinforcing the sufficiency of the edge-aware WL approach for graph discrimination tasks.

Abstract

As a classical approach on graph learning, the propagation-aggregation methodology is widely exploited by many of Graph Neural Networks (GNNs), wherein the representation of a node is updated by aggregating representations from itself and neighbor nodes recursively. Similar to the propagation-aggregation methodology, the Weisfeiler-Lehman (1-WL) algorithm tests isomorphism through color refinement according to color representations of a node and its neighbor nodes. However, 1-WL does not leverage any edge features (labels), presenting a potential improvement on exploiting edge features in some fields. To address this limitation, we proposed a novel Edged-WL algorithm (E-WL) which extends the original 1-WL algorithm to incorporate edge features. Building upon the E-WL algorithm, we also introduce an Edged Graph Isomorphism Network (EGIN) model for further exploiting edge features, which addresses one key drawback in many GNNs that do not utilize any edge features of graph data. We evaluated the performance of proposed models using 12 edge-featured benchmark graph datasets and compared them with some state-of-the-art baseline models. Experimental results indicate that our proposed EGIN models, in general, demonstrate superior performance in graph learning on graph classification tasks.

Edged Weisfeiler-Lehman Algorithm

TL;DR

The paper extends the classic Weisfeiler-Lehman test by incorporating edge features into node color refinements (E-WL) and builds an Edged Graph Isomorphism Network (EGIN) that leverages these edge-aware representations. It introduces two variants, EGIN-C (cross updating) and EGIN-E (edge embedding), to address feature-dimension issues and enhance expressiveness. Through experiments on 12 edge-featured datasets, EGIN and its variants generally outperform strong baselines, demonstrating the practical value of exploiting edge information in graph classification. The work also discusses an edge-aggregation variant (E-WL-EA) and proves that edge aggregation does not surpass E-WL in discriminative power, reinforcing the sufficiency of the edge-aware WL approach for graph discrimination tasks.

Abstract

As a classical approach on graph learning, the propagation-aggregation methodology is widely exploited by many of Graph Neural Networks (GNNs), wherein the representation of a node is updated by aggregating representations from itself and neighbor nodes recursively. Similar to the propagation-aggregation methodology, the Weisfeiler-Lehman (1-WL) algorithm tests isomorphism through color refinement according to color representations of a node and its neighbor nodes. However, 1-WL does not leverage any edge features (labels), presenting a potential improvement on exploiting edge features in some fields. To address this limitation, we proposed a novel Edged-WL algorithm (E-WL) which extends the original 1-WL algorithm to incorporate edge features. Building upon the E-WL algorithm, we also introduce an Edged Graph Isomorphism Network (EGIN) model for further exploiting edge features, which addresses one key drawback in many GNNs that do not utilize any edge features of graph data. We evaluated the performance of proposed models using 12 edge-featured benchmark graph datasets and compared them with some state-of-the-art baseline models. Experimental results indicate that our proposed EGIN models, in general, demonstrate superior performance in graph learning on graph classification tasks.

Paper Structure

This paper contains 18 sections, 3 theorems, 19 equations, 2 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Weisfeiler-Lehman Algorithm
  5. Propagation-Aggregation Strategy in Graph Neural Networks
  6. Related work
  7. Improvements on 1-WL Algorithm
  8. Graph Neural Networks
  9. Edged Weisfeiler-Lehman Algorithm
  10. An Edged Graph Isomorphism Network Model
  11. EGIN-C Model
  12. EGIN-E Model
  13. Experiments
  14. Datasets
  15. Experimental Settings
  16. ...and 3 more sections

Key Result

Theorem 1

The Discriminative power of the E-WL algorithm is either equal to or greater than that of the 1-WL algorithm.

Figures (2)

  • Figure 1: Apply 1-WL algorithm on three example graphs
  • Figure 2: Apply E-WL algorithm on three example graphs

Theorems & Definitions (10)

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