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Rethinking the Graph Polynomial Filter via Positive and Negative Coupling Analysis

Haodong Wen, Bodong Du, Ruixun Liu, Deyu Meng, Xiangyong Cao

TL;DR

This work argues that incorporating graph information into basis construction can enhance understanding of polynomial basis, and further facilitate simplified polynomial filter design, and proposes a simple GNN based on the new basis.

Abstract

Recently, the optimization of polynomial filters within Spectral Graph Neural Networks (GNNs) has emerged as a prominent research focus. Existing spectral GNNs mainly emphasize polynomial properties in filter design, introducing computational overhead and neglecting the integration of crucial graph structure information. We argue that incorporating graph information into basis construction can enhance understanding of polynomial basis, and further facilitate simplified polynomial filter design. Motivated by this, we first propose a Positive and Negative Coupling Analysis (PNCA) framework, where the concepts of positive and negative activation are defined and their respective and mixed effects are analysed. Then, we explore PNCA from the message propagation perspective, revealing the subtle information hidden in the activation process. Subsequently, PNCA is used to analyze the mainstream polynomial filters, and a novel simple basis that decouples the positive and negative activation and fully utilizes graph structure information is designed. Finally, a simple GNN (called GSCNet) is proposed based on the new basis. Experimental results on the benchmark datasets for node classification verify that our GSCNet obtains better or comparable results compared with existing state-of-the-art GNNs while demanding relatively less computational time.

Rethinking the Graph Polynomial Filter via Positive and Negative Coupling Analysis

TL;DR

This work argues that incorporating graph information into basis construction can enhance understanding of polynomial basis, and further facilitate simplified polynomial filter design, and proposes a simple GNN based on the new basis.

Abstract

Recently, the optimization of polynomial filters within Spectral Graph Neural Networks (GNNs) has emerged as a prominent research focus. Existing spectral GNNs mainly emphasize polynomial properties in filter design, introducing computational overhead and neglecting the integration of crucial graph structure information. We argue that incorporating graph information into basis construction can enhance understanding of polynomial basis, and further facilitate simplified polynomial filter design. Motivated by this, we first propose a Positive and Negative Coupling Analysis (PNCA) framework, where the concepts of positive and negative activation are defined and their respective and mixed effects are analysed. Then, we explore PNCA from the message propagation perspective, revealing the subtle information hidden in the activation process. Subsequently, PNCA is used to analyze the mainstream polynomial filters, and a novel simple basis that decouples the positive and negative activation and fully utilizes graph structure information is designed. Finally, a simple GNN (called GSCNet) is proposed based on the new basis. Experimental results on the benchmark datasets for node classification verify that our GSCNet obtains better or comparable results compared with existing state-of-the-art GNNs while demanding relatively less computational time.
Paper Structure (24 sections, 4 theorems, 12 equations, 9 figures, 6 tables)

This paper contains 24 sections, 4 theorems, 12 equations, 9 figures, 6 tables.

Table of Contents

  1. Introdution
  2. Notations and Preliminaries
  3. Positive and Negative Coupling Analysis
  4. Basic concept
  5. Effect of Positive and Negative Activation
  6. Explain PNCA via Message Propagation
  7. Proposed Model
  8. Analysis on popular basis in GNNs via PNCA
  9. GSCNet
  10. Experiment
  11. Node classification on real-world datasets
  12. Node classification on Large-scale datasets
  13. Discussions
  14. Conclusion
  15. Proof
  16. ...and 9 more sections

Key Result

Theorem 3.2

The node activation is permutation invariant, i.e., given any permutation matrix $\bm{P}\in\{0,1\}^{n\times n}$ and node $t$ with its K-order neighbors $\textbf{x}_{t}=(\bm{x}_t,\bm{x}_{s}), s\in N_k{(t)}$, where $f(\textbf{x}_t)=\alpha _{t} \bm{x}_t + \sum_{k=1}^K\sum_{s \in N_k{(t)}} \alpha_s \bm{x}_s.$

Figures (9)

  • Figure 1: Effects of positive activation, negative activation and mixed activation on the feature distribution of the homophily and heterophily graphs. Under the setting of CSBM deshpande2018contextual, we generate an initial graph as shown on the left, then we further generate homophily and heterophily graphs with two different classes in the middle. For the homophily graph, we first tried positive activation, and representation distributions of different classes become apart. After using the negative activation, the two classes are more distinctly separated. The results are the same for the heterophily graph.
  • Figure 2: Effects of positive activation with degree $K_1$ and negative activation degree $K_2$ in our GSCNet. We evaluate each activation on both homophily graph (marked with orange) and heterophily graph (marked with green). The accuracy change with the degree parameters $K_1$ and $K_2$ are shown on the right.
  • Figure 3: Comparison of different methods for alleviating the over-smoothing issue.
  • Figure 4: Accuracy of different degree parameters ($K_1$ and $K_2$) for our GSCNet.
  • Figure 5: Performance comparison of different activations.
  • ...and 4 more figures

Theorems & Definitions (10)

  • Definition 3.1
  • Theorem 3.2
  • Definition 3.3
  • Definition 3.4
  • Definition 3.5
  • Lemma 3.6
  • Theorem 3.7
  • Theorem 3.8
  • Definition 1.1
  • Definition 1.2