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Toward Effective Digraph Representation Learning: A Magnetic Adaptive Propagation based Approach

Xunkai Li, Daohan Su, Zhengyu Wu, Guang Zeng, Hongchao Qin, Rong-Hua Li, Guoren Wang

TL;DR

The paper tackles limitations of MagDGs in digraph learning caused by hand-crafted $q$ and coarse propagation. It introduces MAP, a plug-and-play, weight-free complex-domain propagation strategy that encodes topology- and feature-driven uncertainty into adaptive edge directions, and MAP++, a learnable extension enabling edge-wise propagation and node-wise aggregation for deeper receptive fields. Theoretical analysis links MAP to graph attribute synchronization and angular synchronization, providing robustness guarantees and connections to existing synchronization-based methods. Empirically, MAP improves a range of MagDG baselines across 12 datasets, while MAP++ achieves state-of-the-art results on multiple tasks with scalable training, including large-scale graphs. The work demonstrates that tailoring complex-domain propagation to node context yields significant performance gains and offers a practical path toward scalable, directed-graph representation learning.

Abstract

The $q$-parameterized magnetic Laplacian serves as the foundation of directed graph (digraph) convolution, enabling this kind of digraph neural network (MagDG) to encode node features and structural insights by complex-domain message passing. As a generalization of undirected methods, MagDG shows superior capability in modeling intricate web-scale topology. Despite the great success achieved by existing MagDGs, limitations still exist: (1) Hand-crafted $q$: The performance of MagDGs depends on selecting an appropriate $q$-parameter to construct suitable graph propagation equations in the complex domain. This parameter tuning, driven by downstream tasks, limits model flexibility and significantly increases manual effort. (2) Coarse Message Passing: Most approaches treat all nodes with the same complex-domain propagation and aggregation rules, neglecting their unique digraph contexts. This oversight results in sub-optimal performance. To address the above issues, we propose two key techniques: (1) MAP is crafted to be a plug-and-play complex-domain propagation optimization strategy in the context of digraph learning, enabling seamless integration into any MagDG to improve predictions while enjoying high running efficiency. (2) MAP++ is a new digraph learning framework, further incorporating a learnable mechanism to achieve adaptively edge-wise propagation and node-wise aggregation in the complex domain for better performance. Extensive experiments on 12 datasets demonstrate that MAP enjoys flexibility for it can be incorporated with any MagDG, and scalability as it can deal with web-scale digraphs. MAP++ achieves SOTA predictive performance on 4 different downstream tasks.

Toward Effective Digraph Representation Learning: A Magnetic Adaptive Propagation based Approach

TL;DR

The paper tackles limitations of MagDGs in digraph learning caused by hand-crafted and coarse propagation. It introduces MAP, a plug-and-play, weight-free complex-domain propagation strategy that encodes topology- and feature-driven uncertainty into adaptive edge directions, and MAP++, a learnable extension enabling edge-wise propagation and node-wise aggregation for deeper receptive fields. Theoretical analysis links MAP to graph attribute synchronization and angular synchronization, providing robustness guarantees and connections to existing synchronization-based methods. Empirically, MAP improves a range of MagDG baselines across 12 datasets, while MAP++ achieves state-of-the-art results on multiple tasks with scalable training, including large-scale graphs. The work demonstrates that tailoring complex-domain propagation to node context yields significant performance gains and offers a practical path toward scalable, directed-graph representation learning.

Abstract

The -parameterized magnetic Laplacian serves as the foundation of directed graph (digraph) convolution, enabling this kind of digraph neural network (MagDG) to encode node features and structural insights by complex-domain message passing. As a generalization of undirected methods, MagDG shows superior capability in modeling intricate web-scale topology. Despite the great success achieved by existing MagDGs, limitations still exist: (1) Hand-crafted : The performance of MagDGs depends on selecting an appropriate -parameter to construct suitable graph propagation equations in the complex domain. This parameter tuning, driven by downstream tasks, limits model flexibility and significantly increases manual effort. (2) Coarse Message Passing: Most approaches treat all nodes with the same complex-domain propagation and aggregation rules, neglecting their unique digraph contexts. This oversight results in sub-optimal performance. To address the above issues, we propose two key techniques: (1) MAP is crafted to be a plug-and-play complex-domain propagation optimization strategy in the context of digraph learning, enabling seamless integration into any MagDG to improve predictions while enjoying high running efficiency. (2) MAP++ is a new digraph learning framework, further incorporating a learnable mechanism to achieve adaptively edge-wise propagation and node-wise aggregation in the complex domain for better performance. Extensive experiments on 12 datasets demonstrate that MAP enjoys flexibility for it can be incorporated with any MagDG, and scalability as it can deal with web-scale digraphs. MAP++ achieves SOTA predictive performance on 4 different downstream tasks.
Paper Structure (28 sections, 3 theorems, 16 equations, 6 figures, 6 tables, 2 algorithms)

This paper contains 28 sections, 3 theorems, 16 equations, 6 figures, 6 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations and Problem Formulation
  4. Directed Graph Neural Networks
  5. Empirical Investigation
  6. Magnetic Adaptive Propagation
  7. Topology-related Uncertainty Encoding
  8. Feature-related Correlation Encoding
  9. MAP Framework
  10. MAP++ Framework
  11. Theoretical Analysis
  12. Experiments
  13. Performance Comparison
  14. Ablation Study
  15. Efficiency Comparison
  16. ...and 13 more sections

Key Result

Theorem 1

The adaptive phase matrix $\boldsymbol\Theta^{(q^\star)}$ encoding by MAP is skew-symmetric, and $\mathbf{H}$ is Hermitian, where $\mathbf{H}=\exp\left(i\boldsymbol\Theta^{(q^\star)}\right)$.

Figures (6)

  • Figure 1: (Left a,b,c) The illustration of different semantic contexts for two current nodes within two-hop neighbors, where colors denote label classes. (Right, d,e) The empirical study on directed CoraML (3.3k Nodes at Upper tables) and arXiv (169k Nodes at Lower tables) with different scales.
  • Figure 2: Node-C Performance with $q$ guidance.
  • Figure 3: The visualization of MAP++ in arXiv.
  • Figure 4: Convergence improvement brought by MAP.
  • Figure 5: Running efficiency performance.
  • ...and 1 more figures

Theorems & Definitions (5)

  • Definition 1
  • Definition 2
  • Theorem 1
  • Theorem 2
  • Theorem 3