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GraphMinNet: Learning Dependencies in Graphs with Light Complexity Minimal Architecture

Md Atik Ahamed, Andrew Cheng, Qiang Ye, Qiang Cheng

TL;DR

GraphMinNet tackles the challenge of modeling long-range dependencies in graphs without sacrificing scalability. It generalizes the minimal GRU idea to graphs by integrating node features with Laplacian-based positional encodings into a content–position embedding and updates node representations through a global, permutation-equivariant inner-product mechanism, achieving linear time complexity. The authors prove permutation equivariance, Lipschitz stability, and a non-decaying gradient property, situating the method’s expressiveness between 1-WL and 3-WL. Empirically, GraphMinNet delivers state-of-the-art or competitive results across ten diverse datasets while maintaining efficiency, validating its capacity to capture both local and global graph structures at scale.

Abstract

Graph Neural Networks (GNNs) have demonstrated remarkable success in various applications, yet they often struggle to capture long-range dependencies (LRD) effectively. This paper introduces GraphMinNet, a novel GNN architecture that generalizes the idea of minimal Gated Recurrent Units to graph-structured data. Our approach achieves efficient LRD modeling with linear computational complexity while maintaining permutation equivariance and stability. The model incorporates both structural and positional information through a unique combination of feature and positional encodings, leading to provably stronger expressiveness than the 1-WL test. Theoretical analysis establishes that GraphMinNet maintains non-decaying gradients over long distances, ensuring effective long-range information propagation. Extensive experiments on ten diverse datasets, including molecular graphs, image graphs, and synthetic networks, demonstrate that GraphMinNet achieves state-of-the-art performance while being computationally efficient. Our results show superior performance on 6 out of 10 datasets and competitive results on the others, validating the effectiveness of our approach in capturing both local and global graph structures.

GraphMinNet: Learning Dependencies in Graphs with Light Complexity Minimal Architecture

TL;DR

GraphMinNet tackles the challenge of modeling long-range dependencies in graphs without sacrificing scalability. It generalizes the minimal GRU idea to graphs by integrating node features with Laplacian-based positional encodings into a content–position embedding and updates node representations through a global, permutation-equivariant inner-product mechanism, achieving linear time complexity. The authors prove permutation equivariance, Lipschitz stability, and a non-decaying gradient property, situating the method’s expressiveness between 1-WL and 3-WL. Empirically, GraphMinNet delivers state-of-the-art or competitive results across ten diverse datasets while maintaining efficiency, validating its capacity to capture both local and global graph structures at scale.

Abstract

Graph Neural Networks (GNNs) have demonstrated remarkable success in various applications, yet they often struggle to capture long-range dependencies (LRD) effectively. This paper introduces GraphMinNet, a novel GNN architecture that generalizes the idea of minimal Gated Recurrent Units to graph-structured data. Our approach achieves efficient LRD modeling with linear computational complexity while maintaining permutation equivariance and stability. The model incorporates both structural and positional information through a unique combination of feature and positional encodings, leading to provably stronger expressiveness than the 1-WL test. Theoretical analysis establishes that GraphMinNet maintains non-decaying gradients over long distances, ensuring effective long-range information propagation. Extensive experiments on ten diverse datasets, including molecular graphs, image graphs, and synthetic networks, demonstrate that GraphMinNet achieves state-of-the-art performance while being computationally efficient. Our results show superior performance on 6 out of 10 datasets and competitive results on the others, validating the effectiveness of our approach in capturing both local and global graph structures.

Paper Structure

This paper contains 27 sections, 9 theorems, 21 equations, 5 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Related Works
  3. Methods
  4. Preliminary
  5. GraphMinNet for Graph Learning
  6. Properties of GraphMinNet
  7. Experiments
  8. Robustness Analysis
  9. Ablation Studies
  10. Optional Self-term
  11. Embedding Representation
  12. Effectiveness of Dropout Regularization
  13. Local Method
  14. Nonlinear $W$ and $B$
  15. Projection Strategy on $z_u$ and $\tilde{h}_u$
  16. ...and 12 more sections

Key Result

Proposition 3.3

The formulation of GraphMinNet in Equations (eq_graph_expanded) (or (eq_GraphMinNet_with_self)) through (eq_tilde_h) is permutation equivariant. Moreover, if the functions $\phi_i(\cdot)$ used in forming $A_u$ and $C_u$ (Equations (eq_Au) and (eq_Cu)) are Lipschitz, then GraphMinNet is Lipschitz sta

Figures (5)

  • Figure 1: Schematic diagram of our proposed method. Here $u\in\mathbb{R}^{l}$ represents each node. $\sigma$ denotes the sigmoid activation function. $\beta$ represents a learnable parameter to focus on the specific part.
  • Figure 2: Run time comparison per epochs including train, validation, and test phases.
  • Figure 3: Scalability analysis of GraphMinNet. (a) shows the linear growth of FLOPs, reflecting computational efficiency with increasing graph size, and (b) depicts the linear growth of maximum memory usage, demonstrating feasible memory requirements.
  • Figure 4: Robustness analysis of our model under varying noise levels. SD: signal-dependent; WN: white noise.
  • Figure 5: Effectiveness of dropouts.

Theorems & Definitions (15)

  • Definition 3.1: Permutation Equivariance
  • Definition 3.2: (Lipschitz) Stability of a GNN
  • Proposition 3.3
  • Corollary 3.4
  • Proposition 3.5: Long-range Dependency
  • Proposition 3.6: Complexity and Scalability
  • Proposition 3.7: Expressiveness
  • Proposition 7.1
  • proof
  • Proposition 7.2
  • ...and 5 more