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Effective graph analysis methods can reveal the intrinsic characteristics of graph data. However, graph is non-Euclidean data, which leads to high computation and space cost while applying traditional methods. Graph embedding is an efficient method for graph analysis. It converts original graph data into a low-dimensional space and retains key information to improve the performance of downstream tasks such as node classification, link prediction, and node clustering. Different from previous studies, this paper focuses on both static and dynamic graph embedding. Firstly, this paper proposes a universal taxonomy of static and dynamic methods, including matrix factorization based methods, random walk based methods, autoencoder based methods, graph neural networks (GNN) based methods and other embedding methods. Secondly, this paper analyzes the theoretical relevance of static and dynamic methods, and comprehensively summarizes the core strategy, downstream tasks and datasets. Finally, this paper proposes four potential research directions of graph embedding.

Graph convolutional autoencoders emerged as powerful graph representation learning methods with promising performance on link prediction. However, existing methods typically rely on graph convolutional network to encode adjacency matrix and attribute matrix. This strategy ignores high-order features such as second-order information. This paper proposes a new SeVGAE model based on second-order graph convolutional autoencoders to tackle this problem. Firstly, graph convolutional network and second-order graph convolutional network are used to generate the mean and variance of the variational autoencoders. Then, the embeddings are learned to represent the mixed probability distribution of the original graph. Finally, the topology is recovered by the inner product decoder. SeVGAE performs better than the baselines in the link prediction. AUC and AP are raised by 3.26% and 2.56% respectively compared to the original VGAE on Citeseer dataset. The results show that the second-order information can retain richer graph information in low-dimensional embeddings. Besides, the proposed method has obvious advantages in insufficient attribute and inaccurate topology. In the extreme case where both edges and attributes are missing up to 40%, SeVGAE still improves AUC and AP by 4.79% and 3.47% respectively compared to VGAE. [ABSTRACT FROM AUTHOR]