[Paper Review] Graph Convolutional Neural Networks based on Quantum Vertex Saliency

Lu Bai, Yuhang Jiao+4 more|arXiv (Cornell University)|Sep 4, 2018

Advanced Graph Neural Networks56 references3 citations

TL;DR

This paper proposes Quantum Spatial Graph Convolutional Neural Networks (QSGCNN), a novel GCN model that uses transitive vertex alignment and quantum vertex saliency to transform arbitrary-sized graphs into fixed-sized grid structures. By leveraging continuous-time quantum walks for multi-scale feature learning and preserving spatial order, QSGCNN enables end-to-end graph classification with improved structural representation and reduced information loss compared to existing methods.

ABSTRACT

This paper proposes a new Quantum Spatial Graph Convolutional Neural Network (QSGCNN) model that can directly learn a classification function for graphs of arbitrary sizes. Unlike state-of-the-art Graph Convolutional Neural Network (GCNN) models, the proposed QSGCNN model incorporates the process of identifying transitive aligned vertices between graphs, and transforms arbitrary sized graphs into fixed-sized aligned vertex grid structures. In order to learn representative graph characteristics, a new quantum spatial graph convolution is proposed and employed to extract multi-scale vertex features, in terms of quantum information propagation between grid vertices of each graph. Since the quantum spatial convolution preserves the grid structures of the input vertices (i.e., the convolution layer does not change the original spatial sequence of vertices), the proposed QSGCNN model allows to directly employ the traditional convolutional neural network architecture to further learn from the global graph topology, providing an end-to-end deep learning architecture that integrates the graph representation and learning in the quantum spatial graph convolution layer and the traditional convolutional layer for graph classifications. We demonstrate the effectiveness of the proposed QSGCNN model in relation to existing state-of-the-art methods. The proposed QSGCNN model addresses the shortcomings of information loss and imprecise information representation arising in existing GCN models associated with the use of SortPooling or SumPooling layers. Experiments on benchmark graph classification datasets demonstrate the effectiveness of the proposed QSGCNN model.

Motivation & Objective

• Address the challenge of representing arbitrary-sized graphs as fixed-size numeric features for deep learning.
• Overcome information loss and imprecise structural representation in existing GCN models that rely on pooling layers like SumPooling or SortPooling.
• Bridge the gap between spatial graph convolution and traditional CNNs by preserving vertex spatial order through grid transformation.
• Enhance graph representation by incorporating quantum information propagation via continuous-time quantum walks on aligned vertex grids.
• Develop an end-to-end deep learning framework that integrates quantum spatial convolution with standard CNNs for graph classification.

Proposed method

• Apply transitive vertex alignment to map vertices across different graphs into a fixed-sized grid structure, ensuring consistent spatial ordering.
• Propose a quantum spatial graph convolution layer that models vertex feature propagation using the average mixing matrix of continuous-time quantum walks.
• Use the quantum walk's spectral properties to better distinguish graph structures compared to classical random walks or Laplacian-based methods.
• Preserve the original spatial sequence of vertices during convolution, enabling direct integration with standard CNN architectures.
• Combine the quantum spatial convolution layer with traditional convolutional layers to learn hierarchical global graph topology.
• Formulate the graph convolution operation as a quantum analog of the Weisfeiler-Lehman algorithm, reducing the tottering problem common in classical GCNs.

Experimental results

Research questions

• RQ1 Can transitive vertex alignment effectively map graphs of arbitrary sizes into a fixed-size grid while preserving structural relationships?
• RQ2 Can quantum information propagation via continuous-time quantum walks improve the representation of multi-scale graph features compared to classical methods?
• RQ3 Does preserving vertex spatial order in the convolution process enable better integration with standard CNNs for graph classification?
• RQ4 To what extent does the quantum spatial convolution reduce the tottering problem observed in classical graph neural networks?
• RQ5 Can the proposed QSGCNN model outperform state-of-the-art GCNs and graph kernels on benchmark graph classification datasets?

Key findings

• The QSGCNN model achieves superior performance on benchmark graph classification datasets by preserving full graph topology through quantum spatial convolution.
• Unlike models using SumPooling or SortPooling, QSGCNN avoids information loss by maintaining local vertex features and their spatial order.
• The quantum walk-based convolution mechanism better distinguishes complex graph structures due to its spectral properties, outperforming classical random walk and Laplacian-based approaches.
• The model overcomes the tottering problem inherent in classical Weisfeiler-Lehman-based GCNs by using a quantum analog of the algorithm.
• Experiments demonstrate that QSGCNN consistently outperforms state-of-the-art models including DGCNN, PSGCNN, GCCNN, ECC, and AWE on standard graph classification benchmarks.
• The proposed method enables end-to-end learning without relying on permutation-invariant aggregation, allowing direct use of traditional CNNs on graph-structured data.