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Weisfeiler and Lehman Go Paths: Learning Topological Features via Path Complexes

Quang Truong, Peter Chin

TL;DR

The paper tackles the expressivity gap of $1$-WL GNNs by introducing a path-centric topological framework. It lifts graphs to path complexes and defines the Path Weisfeiler-Lehman (PWL) test, enabling color refinement on elementary paths, with a neural realization called Path Complex Networks (PCN) through Path Isomorphism Networks (PIN). The authors prove PWL is at least as powerful as SWL and not less than CWL($k$-IC), and not less than $3$-WL, while empirically achieving strong results on TUDataset, ZINC, OGBG-MOLHIV, and SRGs. This approach, which avoids reliance on clique or cycle substructures, demonstrates strong generalization and scalability benefits, offering a versatile, topology-aware alternative for graph representation learning.

Abstract

Graph Neural Networks (GNNs), despite achieving remarkable performance across different tasks, are theoretically bounded by the 1-Weisfeiler-Lehman test, resulting in limitations in terms of graph expressivity. Even though prior works on topological higher-order GNNs overcome that boundary, these models often depend on assumptions about sub-structures of graphs. Specifically, topological GNNs leverage the prevalence of cliques, cycles, and rings to enhance the message-passing procedure. Our study presents a novel perspective by focusing on simple paths within graphs during the topological message-passing process, thus liberating the model from restrictive inductive biases. We prove that by lifting graphs to path complexes, our model can generalize the existing works on topology while inheriting several theoretical results on simplicial complexes and regular cell complexes. Without making prior assumptions about graph sub-structures, our method outperforms earlier works in other topological domains and achieves state-of-the-art results on various benchmarks.

Weisfeiler and Lehman Go Paths: Learning Topological Features via Path Complexes

TL;DR

The paper tackles the expressivity gap of -WL GNNs by introducing a path-centric topological framework. It lifts graphs to path complexes and defines the Path Weisfeiler-Lehman (PWL) test, enabling color refinement on elementary paths, with a neural realization called Path Complex Networks (PCN) through Path Isomorphism Networks (PIN). The authors prove PWL is at least as powerful as SWL and not less than CWL(-IC), and not less than -WL, while empirically achieving strong results on TUDataset, ZINC, OGBG-MOLHIV, and SRGs. This approach, which avoids reliance on clique or cycle substructures, demonstrates strong generalization and scalability benefits, offering a versatile, topology-aware alternative for graph representation learning.

Abstract

Graph Neural Networks (GNNs), despite achieving remarkable performance across different tasks, are theoretically bounded by the 1-Weisfeiler-Lehman test, resulting in limitations in terms of graph expressivity. Even though prior works on topological higher-order GNNs overcome that boundary, these models often depend on assumptions about sub-structures of graphs. Specifically, topological GNNs leverage the prevalence of cliques, cycles, and rings to enhance the message-passing procedure. Our study presents a novel perspective by focusing on simple paths within graphs during the topological message-passing process, thus liberating the model from restrictive inductive biases. We prove that by lifting graphs to path complexes, our model can generalize the existing works on topology while inheriting several theoretical results on simplicial complexes and regular cell complexes. Without making prior assumptions about graph sub-structures, our method outperforms earlier works in other topological domains and achieves state-of-the-art results on various benchmarks.
Paper Structure (39 sections, 13 theorems, 20 equations, 4 figures, 10 tables)

This paper contains 39 sections, 13 theorems, 20 equations, 4 figures, 10 tables.

Table of Contents

  1. Introduction
  2. Main Contributions
  3. Preliminaries
  4. Path Complex
  5. Complex Lifting Transformations
  6. Weisfeiler-Lehman Tests
  7. Path Weisfeiler-Lehman Test
  8. Procedure
  9. Theoretical Results
  10. Realization of PWL Test via Message-passing Neural Networks
  11. Experiments
  12. TUDataset Benchmarks
  13. ZINC
  14. OGBG-MOLHIV
  15. Strongly Regular Graphs
  16. ...and 24 more sections

Key Result

Corollary 10

Consider two path colorings $c$ and $d$ on path complexes $X$ and $Y$ respectively such that $c \sqsubseteq d$. If $d^X \neq d^Y$, then $c^X \neq c^Y$.

Figures (4)

  • Figure 1: An illustration of different $2$-path spaces of a path complex arising from the graph on the left. Each element in each space is an elementary $2$-path that spans the corresponding space.
  • Figure 2: (a) Original graph; (b) Simplicial complex, which contains a 2-simplex, 4 1-simplices, and 4 0-simplices, arising from the original graph. Regular cell complex coincides with the simplicial complex in this case; (c) Simple path spaces $\mathcal{S}_2$ and $\mathcal{S}_3$ corresponding to the path complex arising from the original graph. Elementary paths of $\mathcal{S}_0$ and $\mathcal{S}_1$ are indeed 0-simplices (0-cells) and 1-simplices (1-cells) of the simplicial complex (regular cell complex).
  • Figure 3: Examples of path complexes arising from (a) a simple path with length of 3 and (b) a ring with size of 4. Blue arrows demonstrate upper-adjacent relations, while orange arrows demonstrate boundary relations.
  • Figure 4: Failure rate comparison between CWN and PCN on SRG Families over 10 different seeds. (a) 3 message-passing layers. (b) 4 message-passing layers. (c) 5 message-passing layers. (d) 6 message-passing layers.

Theorems & Definitions (33)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4: Boundary incidence relation
  • Definition 5: Relations between members
  • Definition 6: Color representations of relations
  • Definition 7
  • Definition 8
  • Definition 9: Color refinement on elementary paths
  • Corollary 10
  • ...and 23 more