Theoretical Insights into Line Graph Transformation on Graph Learning

TL;DR

This study focuses on two types of graphs known to be challenging to the Weisfeiler-Leman tests: Cai-F\"urer-Immerman (CFI) graphs and strongly regular graphs, and shows that applying line graph transformation helps exclude these challenging graph properties, thus potentially assist WL tests in distinguishing these graphs.

Abstract

Line graph transformation has been widely studied in graph theory, where each node in a line graph corresponds to an edge in the original graph. This has inspired a series of graph neural networks (GNNs) applied to transformed line graphs, which have proven effective in various graph representation learning tasks. However, there is limited theoretical study on how line graph transformation affects the expressivity of GNN models. In this study, we focus on two types of graphs known to be challenging to the Weisfeiler-Leman (WL) tests: Cai-Fürer-Immerman (CFI) graphs and strongly regular graphs, and show that applying line graph transformation helps exclude these challenging graph properties, thus potentially assist WL tests in distinguishing these graphs. We empirically validate our findings by conducting a series of experiments that compare the accuracy and efficiency of graph isomorphism tests and GNNs on both line-transformed and original graphs across these graph structure types.

Figures (9)

• Figure 1: An example of converting a graph $G$ to its line graph $L(G)$.
• Figure 2: Relationships between WL tests and challenging graph types.
• Figure 3: Examples of graph structures.
• Figure 4: Example of a pair of CFI graphs and a pair of strongly regular graphs.
• Figure 5: An CFI graph substructure $X_3$ and its induced subgraph $K_{1,3}$ in red.

Theorems & Definitions (23)

• Lemma 1
• Theorem 2: Whitney's Isomorphism Theorem whitney1992congruent
• Corollary 3
• Theorem 4
• Corollary 5
• Theorem 6
• Theorem 7
• Theorem 8
• Corollary 9
• Corollary 10