We introduce the variational graph autoencoder (VGAE), a framework for unsupervised learning on graph-structured data based on the variational auto-encoder (VAE) [2, 3] . This model makes use of latent variables and is capable of learning interpretable latent representations for undirected graphs (see Figure 1) . We demonstrate this model using a graph convolutional network (GCN) [4] encoder and a simple inner product decoder. Our model achieves competitive results on a link prediction task in citation networks. In contrast to most existing models for unsupervised learning on graph-structured data and link prediction [5, 6, 7, 8] , our model can naturally incorporate node features, which significantly improves predictive performance on a number of benchmark datasets. Definitions We are given an undirected, unweighted graph G = (V, E) with N = |V| nodes. We introduce an adjacency matrix A of G (we assume diagonal elements set to 1, i.e. every node is connected to itself) and its degree matrix D. We further introduce stochastic latent variables z i , summarized in an N × F matrix Z. Node features are summarized in an N × D matrix X. Inference model We take a simple inference model parameterized by a two-layer GCN: Here, µ = GCN µ (X, A) is the matrix of mean vectors µ i ; similarly log σ = GCN σ (X, A). The two-layer GCN is defined as GCN(X, A) =Ã ReLU Ã XW 0 W 1 , with weight matrices W i . GCN µ (X, A) and GCN σ (X, A) share first-layer parameters W 0 . ReLU(·) = max(0, ·) and 2 is the symmetrically normalized adjacency matrix. Generative model Our generative model is given by an inner product between latent variables: where A ij are the elements of A and σ(·) is the logistic sigmoid function. Learning We optimize the variational lower bound L w.r.t. the variational parameters W i :
Kipf and Welling REF introduced a variational graph autoencoder to learn a joint latent representation from an undirected graph and node features.
Variational Graph Auto-Encoders
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