Ricci Curvature-Based Semi-Supervised Learning on an Attributed Network.

Wei Wu,Fucai Yu,Guangmin Hu

doi:10.3390/e23030292

Abstract

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.

Highlights

We found a geometrical quantity, Ricci curvature, which is better adapted to the polymerization process on the discrete manifolds. (ii) We propose a concrete implementation of the aggregation pattern mentioned above, the Ricci curvaturebased graph convolutional neural network (RCGCN), for network embedding
Our results show that RCGCN achieved the best performance on all five datasets, except that the classification accuracy on the CORA dataset was slightly behind that of GAT
We introduced RCGCN, a novel architecture that combines the neural network and Riemannian geometric methods

Summary

Introduction

The conventional approach, which uses a set of edges to represent a network, has proven that it is difficult to process complicated network data. As an efficient tool for data mining, is designed to convert the information within the network to a continuous low-dimensional vector representation. We review the related notion of curvature on the manifold. We introduce Ollivier’s coarse Ricci curvature, which generalizes Ricci curvature on the manifold to metric space by Wasserstein distance. The Riemannian curvature tensor or Riemannian tensor is the standard way of expressing the curvature of a Riemannian manifold. As we can see in Equation (2), we can transform the Riemannian curvature tensor to a (0,4)-tensor with the help of the Riemannian metric g.

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