Wu Wei, Hu Guangmin, Yu Fucai
School of Information and Communication Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China.
Entropy (Basel). 2021 Feb 27;23(3):292. doi: 10.3390/e23030292.
In recent years, on the basis of drawing lessons from traditional neural network models, people have been paying more and more attention to the design of neural network architectures for processing graph structure data, which are called graph neural networks (GNN). GCN, namely, graph convolution networks, are neural network models in GNN. GCN extends the convolution operation from traditional data (such as images) to graph data, and it is essentially a feature extractor, which aggregates the features of neighborhood nodes into those of target nodes. In the process of aggregating features, GCN uses the Laplacian matrix to assign different importance to the nodes in the neighborhood of the target nodes. Since graph-structured data are inherently non-Euclidean, we seek to use a non-Euclidean mathematical tool, namely, Riemannian geometry, to analyze graphs (networks). In this paper, we present a novel model for semi-supervised learning called the Ricci curvature-based graph convolutional neural network, i.e., RCGCN. The aggregation pattern of RCGCN is inspired by that of GCN. We regard the network as a discrete manifold, and then use Ricci curvature to assign different importance to the nodes within the neighborhood of the target nodes. Ricci curvature is related to the optimal transport distance, which can well reflect the geometric structure of the underlying space of the network. The node importance given by Ricci curvature can better reflect the relationships between the target node and the nodes in the neighborhood. The proposed model scales linearly with the number of edges in the network. Experiments demonstrated that RCGCN achieves a significant performance gain over baseline methods on benchmark datasets.
近年来，在借鉴传统神经网络模型的基础上，人们越来越关注用于处理图结构数据的神经网络架构设计，这类架构被称为图神经网络（GNN）。GCN，即图卷积网络，是GNN中的神经网络模型。GCN将卷积运算从传统数据（如图像）扩展到图数据，它本质上是一个特征提取器，将邻域节点的特征聚合到目标节点的特征中。在聚合特征的过程中，GCN使用拉普拉斯矩阵为目标节点邻域中的节点赋予不同的重要性。由于图结构数据本质上是非欧几里得的，我们寻求使用一种非欧几里得数学工具，即黎曼几何，来分析图（网络）。在本文中，我们提出了一种用于半监督学习的新型模型，称为基于里奇曲率的图卷积神经网络，即RCGCN。RCGCN的聚合模式受到GCN的启发。我们将网络视为离散流形，然后使用里奇曲率为目标节点邻域内的节点赋予不同的重要性。里奇曲率与最优传输距离相关，它能够很好地反映网络底层空间的几何结构。由里奇曲率给出的节点重要性能够更好地反映目标节点与邻域中节点之间的关系。所提出的模型与网络中的边数呈线性比例关系。实验表明，在基准数据集上，RCGCN相对于基线方法取得了显著的性能提升。
