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Local Vertex Colouring Graph Neural Networks

Shouheng Li, Dongwoo Kim, Qing Wang

TL;DR

The paper tackles the limits of graph neural networks constrained by the Weisfeiler-Lehman ($1$-WL) test by introducing Local Vertex Colouring (LVC), which leverages BFS- and DFS-guided colour updates to capture richer graph structure. It formalizes a search-guided Graph Neural Network (SGN) that propagates embeddings along search trees, enabling detection of properties like biconnectivity and ego shortest-path graphs beyond $1$-WL, with a theoretical expressivity analysis relative to $1$-WL and $3$-WL. The authors prove an expressivity hierarchy for BFS-based (BFC) and DFS-based (DFC) colourings, showing distinct relations to existing WL-based frameworks, and demonstrate SGN's strong performance on vertex and graph classification across homophilic and heterophilic datasets. The work provides theoretical foundations and empirical evidence for integrating graph search principles into GNNs, offering a scalable path to more expressive graph representations, with code available at the cited repository.

Abstract

In recent years, there has been a significant amount of research focused on expanding the expressivity of Graph Neural Networks (GNNs) beyond the Weisfeiler-Lehman (1-WL) framework. While many of these studies have yielded advancements in expressivity, they have frequently come at the expense of decreased efficiency or have been restricted to specific types of graphs. In this study, we investigate the expressivity of GNNs from the perspective of graph search. Specifically, we propose a new vertex colouring scheme and demonstrate that classical search algorithms can efficiently compute graph representations that extend beyond the 1-WL. We show the colouring scheme inherits useful properties from graph search that can help solve problems like graph biconnectivity. Furthermore, we show that under certain conditions, the expressivity of GNNs increases hierarchically with the radius of the search neighbourhood. To further investigate the proposed scheme, we develop a new type of GNN based on two search strategies, breadth-first search and depth-first search, highlighting the graph properties they can capture on top of 1-WL. Our code is available at https://github.com/seanli3/lvc.

Local Vertex Colouring Graph Neural Networks

TL;DR

The paper tackles the limits of graph neural networks constrained by the Weisfeiler-Lehman (-WL) test by introducing Local Vertex Colouring (LVC), which leverages BFS- and DFS-guided colour updates to capture richer graph structure. It formalizes a search-guided Graph Neural Network (SGN) that propagates embeddings along search trees, enabling detection of properties like biconnectivity and ego shortest-path graphs beyond -WL, with a theoretical expressivity analysis relative to -WL and -WL. The authors prove an expressivity hierarchy for BFS-based (BFC) and DFS-based (DFC) colourings, showing distinct relations to existing WL-based frameworks, and demonstrate SGN's strong performance on vertex and graph classification across homophilic and heterophilic datasets. The work provides theoretical foundations and empirical evidence for integrating graph search principles into GNNs, offering a scalable path to more expressive graph representations, with code available at the cited repository.

Abstract

In recent years, there has been a significant amount of research focused on expanding the expressivity of Graph Neural Networks (GNNs) beyond the Weisfeiler-Lehman (1-WL) framework. While many of these studies have yielded advancements in expressivity, they have frequently come at the expense of decreased efficiency or have been restricted to specific types of graphs. In this study, we investigate the expressivity of GNNs from the perspective of graph search. Specifically, we propose a new vertex colouring scheme and demonstrate that classical search algorithms can efficiently compute graph representations that extend beyond the 1-WL. We show the colouring scheme inherits useful properties from graph search that can help solve problems like graph biconnectivity. Furthermore, we show that under certain conditions, the expressivity of GNNs increases hierarchically with the radius of the search neighbourhood. To further investigate the proposed scheme, we develop a new type of GNN based on two search strategies, breadth-first search and depth-first search, highlighting the graph properties they can capture on top of 1-WL. Our code is available at https://github.com/seanli3/lvc.
Paper Structure (38 sections, 33 theorems, 24 equations, 8 figures, 7 tables)

This paper contains 38 sections, 33 theorems, 24 equations, 8 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Graph isomorphism and colour refinement.
  4. GNNs beyond 1-WL.
  5. GNNs in learning graph algorithms.
  6. Preliminaries
  7. Graph searching.
  8. Visiting order subscripting.
  9. Tree and back edges.
  10. Local Vertex Colouring
  11. Search-guided Vertex Colour Update
  12. Search order permutation.
  13. Design considerations.
  14. BFS-guided Colouring
  15. Distinguishing shortest-path graphs.
  16. ...and 23 more sections

Key Result

Lemma 4.1

Let $(u,v)$ and $(u',v')$ be two pairs of vertices. Then $SPG(u,v)\simeq SPG(u',v')$ if and only if one of the following conditions hold under BFC: (1) $\lambda_v(u)= \lambda_{v'}(u')$ and $\lambda_u(v)= \lambda_{u'}(v')$; (2) $\lambda_v(u)= \lambda_{u'}(v')$ and $\lambda_u(v)= \lambda_{v'}(u')$.

Figures (8)

  • Figure 1: Tree edges (green solid lines) and back edges (black dashed lines) classified by BFS and DFS. The subscripted labels $v_0$, …, $v_5$ denote the visit sequence of each vertex, e.g. $v_1$ is visited after $v_0$.
  • Figure 2: A graph and its vertex colours after the first BFC iteration. \ref{['subfig:bfc1']} shows the uncoloured graph, where the vertex colours obtained from \ref{['subfig:bfc2']} and \ref{['subfig:bfc3']} are shown next to each vertex. \ref{['subfig:bfc2']} and \ref{['subfig:bfc3']} show BFC with two different roots (marked with *), respectively, where each vertex is assigned a new colour by BFC. The colour map is shown in the bottom left corner. The subscripted labels $v_0$, …, $v_5$ denote the visit sequence of each vertex, e.g. $v_1$ is visited after $v_0$. For brevity, we show BFC with only two roots.
  • Figure 3: A pair of non-isomorphic three-regular graphs. Pink edges form ESPGs ($\delta = 2$) for vertices $v$ and $u$.
  • Figure 4: An uneven barbell graph and vertex colours after the first DFC iteration. \ref{['subfig:dfc1']} shows an uncoloured graph, where the vertex colours obtained from \ref{['subfig:dfc2']} and \ref{['subfig:dfc3']} are shown next to each vertex. \ref{['subfig:dfc2']} and \ref{['subfig:dfc3']} show DFC with two different roots (marked with *), where each vertex is assigned a new colour by DFC. The colour map is shown on the left. The subscripted label $v_0$, …, $v_5$ denote the visit sequence of each vertex, e.g. $v_1$ is visited after $v_0$. For brevity, DFC is shown with only two roots.
  • Figure 5: (Left) A graph pair can be distinguished by BFC-2 and DFC-1 but not by BFC-1 and 1-WL. (Right) A graph pair can be distinguished by BFC-3 and DFC-2 but not by BFC-2, DFC-1 and 1-WL.
  • ...and 3 more figures

Theorems & Definitions (55)

  • Definition 4.1
  • Lemma 4.1
  • Definition 4.2
  • Lemma 4.2
  • Lemma 4.3
  • Definition 4.3: Back edge crossover
  • Definition 4.4: Back edge cover
  • Lemma 4.4
  • Lemma 4.5
  • Corollary 4.1
  • ...and 45 more