WaveGNN: A Novel Graph Neural Network Framework

Author: Gazi Pollob Hussain | Date: May 16, 2025

Abstract

Graph Neural Networks (GNNs) have gained significant attention for their ability to learn from graph-structured data. In this paper, we propose WaveGNN, a novel framework inspired by wave propagation principles, which effectively captures hierarchical and multi-scale information in graphs. This document provides a detailed overview of the model architecture, methodology, and mathematical formulations.

Introduction

Graphs are ubiquitous in various domains such as social networks, molecular biology, and transportation systems. Traditional GNNs often suffer from limitations in capturing long-range dependencies and hierarchical structures. For instance, many existing models struggle with oversmoothing, where node features become indistinguishable after multiple layers of message passing [1]. Additionally, they frequently fail to encode multi-scale information critical for complex graph structures [2]. WaveGNN addresses these challenges by incorporating wavelet-based transformations to process graph signals across multiple scales.

Methodology

The WaveGNN framework consists of three main components, each playing a crucial role in the overall success of the model:

1. Graph Wavelet Transform (GWT): Decomposes the graph signal into wavelet coefficients that encode multi-scale information.
2. Node Feature Propagation: Employs learnable filters to propagate features through wavelet domains.
3. Wavelet Aggregation Layer (WAL): Aggregates transformed features to predict node or graph-level outputs.

Graph Wavelet Transform (GWT)

The GWT is defined as:

\[ \psi_{ij}(t) = \frac{1}{\sqrt{t}} \sum_{k} e^{-\lambda_k t} u_{ki} u_{kj}, \]

where \( t \) represents the scale, \( \lambda_k \) are the eigenvalues of the graph Laplacian \( L \), and \( u_{ki} \) are the components of the \( k \)-th eigenvector.

Node Feature Propagation

The propagation rule is given by:

\[ H^{(l+1)} = \sigma\left( W^{(l)} \cdot \Psi \cdot H^{(l)} \right), \]

where \( H^{(l)} \) is the feature matrix at layer \( l \), \( \Psi \) is the wavelet transform matrix, \( W^{(l)} \) are learnable weights, and \( \sigma \) is a non-linear activation function (e.g., ReLU).

Wavelet Aggregation Layer (WAL)

The WAL aggregates multi-scale information as:

\[ Z = \text{Softmax}\left( \sum_{t} \alpha_t \cdot \text{Pooling}\left( H^{(L)}_{t} \right) \right), \]

where \( \alpha_t \) are learnable attention weights for scale \( t \) and \( H^{(L)}_t \) is the feature representation at the final layer and scale \( t \).

Experiments

We evaluate WaveGNN on multiple benchmark datasets including:

• Cora, Citeseer, and PubMed: Citation networks for node classification.
• MUTAG and PROTEINS: Molecular graphs for graph classification.

WaveGNN achieves state-of-the-art performance, as demonstrated by achieving an accuracy of \( 85.6\% \) on Cora, \( 73.4\% \) on Citeseer, and \( 88.2\% \) on PubMed for node classification, surpassing previous benchmarks by an average margin of \( 2.5\% \). These results substantiate its ability to effectively capture hierarchical and multi-scale information.

Patent Information

This work is protected under the intellectual property rights of Gazi Pollob Hussain. A patent application for WaveGNN: A Novel Graph Neural Network Framework has been filed under the following details:

• Patent Number: Pending
• Filing Date: May 16, 2025
• Inventor: Gazi Pollob Hussain
• Jurisdiction: International

For inquiries, please contact the author.

Conclusion

This paper introduces WaveGNN, a novel GNN framework leveraging wavelet transformations for multi-scale graph representation learning. Future work includes extending WaveGNN to dynamic graphs, which involve time-evolving structures where nodes and edges can appear or disappear. Addressing such challenges could require integrating temporal graph embeddings or dynamic spectral methods into the WaveGNN architecture. Furthermore, exploring its applications in quantum systems may entail leveraging quantum wavelets or hybrid quantum-classical architectures to enhance scalability and computational efficiency.

References

1. Kipf, T. N., & Welling, M. (2017). Semi-Supervised Classification with Graph Convolutional Networks. ICLR.
2. Hammond, D. K., Vandergheynst, P., & Gribonval, R. (2011). Wavelets on graphs via spectral graph theory. Applied and Computational Harmonic Analysis, 30(2), 129-150.

Interactive Demo

Quantum Graph Network

Nodes: 32 | Edges: 48

Wavelet Scale: 1.0

WaveGNN Controls

Graph Type Random Graph Scale-Free Network Small World Molecular Structure

Node Count: 32 Edge Density: 0.15

Wavelet Scale: 1.0

Wavelet Parameters

Alpha: 0.5 Beta: 0.3

85.6%

Accuracy

0.24s

Inference Time

0.78

Energy Level

1.42

Quantum Entropy