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Permutation-Invariant Graph Partitioning:How Graph Neural Networks Capture Structural Interactions?

Asela Hevapathige, Qing Wang

TL;DR

The paper investigates how Graph Neural Networks (GNNs) can better capture structural interactions by employing permutation-invariant graph partitioning. It introduces Graph Partitioning Neural Networks (GPNNs), which integrate partition-colored embeddings and interaction embeddings to encode intra- and inter-partition structure, and establishes a theoretical link between partitioning schemes and graph isomorphism through partition- and interaction-isomorphism. The expressivity of GPNNs is analyzed, showing they lie between $1$-WL and $3$-WL and can be tuned via partition colouring and interaction types; the workload achieves a practical balance between expressivity and computational cost. Empirically, GPNNs outperform strong baselines on graph classification, regression, and node classification across diverse datasets, with core-degree and degree-based partition schemes delivering the best performance while maintaining efficiency. The work highlights a pathway to more powerful, scalable GNNs by leveraging permutation-invariant partitioning to reveal and encode structural interactions, though it leaves open learning of partitioning schemes for future research.

Abstract

Graph Neural Networks (GNNs) have paved the way for being a cornerstone in graph-related learning tasks. Yet, the ability of GNNs to capture structural interactions within graphs remains under-explored. In this work, we address this gap by drawing on the insight that permutation invariant graph partitioning enables a powerful way of exploring structural interactions. We establish theoretical connections between permutation invariant graph partitioning and graph isomorphism, and then propose Graph Partitioning Neural Networks (GPNNs), a novel architecture that efficiently enhances the expressive power of GNNs in learning structural interactions. We analyze how partitioning schemes and structural interactions contribute to GNN expressivity and their trade-offs with complexity. Empirically, we demonstrate that GPNNs outperform existing GNN models in capturing structural interactions across diverse graph benchmark tasks.

Permutation-Invariant Graph Partitioning:How Graph Neural Networks Capture Structural Interactions?

TL;DR

The paper investigates how Graph Neural Networks (GNNs) can better capture structural interactions by employing permutation-invariant graph partitioning. It introduces Graph Partitioning Neural Networks (GPNNs), which integrate partition-colored embeddings and interaction embeddings to encode intra- and inter-partition structure, and establishes a theoretical link between partitioning schemes and graph isomorphism through partition- and interaction-isomorphism. The expressivity of GPNNs is analyzed, showing they lie between -WL and -WL and can be tuned via partition colouring and interaction types; the workload achieves a practical balance between expressivity and computational cost. Empirically, GPNNs outperform strong baselines on graph classification, regression, and node classification across diverse datasets, with core-degree and degree-based partition schemes delivering the best performance while maintaining efficiency. The work highlights a pathway to more powerful, scalable GNNs by leveraging permutation-invariant partitioning to reveal and encode structural interactions, though it leaves open learning of partitioning schemes for future research.

Abstract

Graph Neural Networks (GNNs) have paved the way for being a cornerstone in graph-related learning tasks. Yet, the ability of GNNs to capture structural interactions within graphs remains under-explored. In this work, we address this gap by drawing on the insight that permutation invariant graph partitioning enables a powerful way of exploring structural interactions. We establish theoretical connections between permutation invariant graph partitioning and graph isomorphism, and then propose Graph Partitioning Neural Networks (GPNNs), a novel architecture that efficiently enhances the expressive power of GNNs in learning structural interactions. We analyze how partitioning schemes and structural interactions contribute to GNN expressivity and their trade-offs with complexity. Empirically, we demonstrate that GPNNs outperform existing GNN models in capturing structural interactions across diverse graph benchmark tasks.
Paper Structure (20 sections, 14 theorems, 12 equations, 4 figures, 8 tables)

This paper contains 20 sections, 14 theorems, 12 equations, 4 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. GNNs beyond 1-WL.
  4. Partitioning-based GNNs.
  5. Graph Partitioning Scheme
  6. Proposed GNN Architecture
  7. Expressivity Analysis
  8. Complexity Analysis
  9. Practical Choices of Partitioning Schemes
  10. Experiments
  11. Results and Discussion
  12. Conclusion, Limitations and Future Work
  13. Model Architecture
  14. Proofs
  15. Supplementary Experimental Details
  16. ...and 5 more sections

Key Result

Theorem 3.5

Fix a graph partitioning scheme. We have: (a) If $G \simeq G^{'}$, then $G\stackrel{II}{\simeq}G^{'}$, but not vice versa; (b) If $G\stackrel{II}{\simeq}G^{'}$, then $G\stackrel{PI}{\simeq}G^{'}$, but not vice versa.

Figures (4)

  • Figure 1: The expressivity of GPNN variants is analysed in terms of (1) interaction type: $E^{\star}$ (intra-edges), $E^{\diamond}$ (intra-edges and inter-edges), and $E^{\dagger}$ (all edges); (2) expressive power: $1$-WL and $3$-WL, under different permutation-invariant graph partitioning schemes $\lambda_{\bot}$, $\lambda_{\leq1\text{-}WL}$, and $\lambda_{\geq3\text{-}WL}$. Detailed theoretical results are provided in the section "Expressivity Analysis".
  • Figure 2: Two pairs of non-isomorphic graphs, $(G_1, H_1)$ and $(G_2, H_2)$, are partitioned by node degrees, grouping nodes with the same degree. Boundary and partitioned subgraphs are shown, with different subgraphs highlighted in red and blue. (a) $G_1 \stackrel{PI}{\simeq} H_1$ and $G_1 \stackrel{II}{\simeq} H_1$; (b) $G_2 \stackrel{PI}{\simeq} H_2$, but $G_2 \not\stackrel{II}{\simeq} H_2$.
  • Figure 3: A high-level workflow of $\lambda_\text{core}$-GPNN for an input graph. The intra-edges are marked using grey color. GPNN generates vertex representations by considering interactions within and between partitions.
  • Figure 4: Node distribution percentage (y-axis ) concerning partitions (x-axis) under different graph partitioning schemes for OGB Datasets.

Theorems & Definitions (31)

  • Definition 3.1: Graph Partitioning Scheme
  • Definition 3.2: Partition-Isomorphism
  • Definition 3.3: Boundary Subgraph
  • Definition 3.4: Interaction-Isomorphism
  • Theorem 3.5
  • Definition 4.1: Partition Colouring
  • Definition 4.2: Coloured Neighbourhood
  • Remark 4.3
  • Definition 5.1: $\lambda$-Equivalence
  • Proposition 5.1
  • ...and 21 more