# Interpretable Variational Graph Autoencoder with Noninformative Prior
## 1 Introduction  
Many complex systems in real life, such as social networks (e.g., Facebook) and coauthor networks, can be abstracted into graph-structured data for analysis. The analysis results can be leveraged to guide practical applications, such as link prediction and personalized recommendation. The variational graph autoencoder (VGAE) proposed by [PERSON] and [PERSON] [1], which extended the variational autoencoder (VAE) [2] from the field of Euclidean structure data (e.g., image) to that of non-Euclidean data (e.g., graph-structured data), is one of the commonly utilized methods for studying graph-structured data. It could capture the distribution of the samples through a two-layer graph convolutional network (GCN) [3] and reconstruct the graph structure through a decoder. In recent years, the variations of VAE and VGAE [4; 5; 6; 7] have emerged in an endless stream, further improving the performance of the variational (graph) autoencoder on traditional machine learning tasks such as link prediction. Although VGAE has been widely applied and promoted, there are two urgent challenges to be solved:  
How to set a reasonable prior without additional expert knowledge. VGAE and its variants assumed that the prior distribution of latent variables obeyed a simple standard normal distribution. Under this assumption, the Kullback-Leibler (KL) term in the objective function usually caused the posterior distribution to overfit the shape and parameters of the prior and encouraged samples to focus on the origin. Therefore, the phenomenonof Posterior Collapse [8] will occur. Two types of methods are proposed to handle the issue. The first is to improve the loss function of the variational (graph) autoencoder [9; 10; 11; 12]. However, most of the methods were designed for Euclidean structure data based on the variational autoencoder. These methods are difficult to extend to graph-structured data since they are not arranged neatly, making it difficult to define the same convolution operation as image data. The second is to assume the latent variables follow a more complex prior or posterior distribution [13; 14; 15; 16; 17]. Unfortunately, these methods applied a more complex distribution as the prior for latent variables. In general, a more complex prior means more expert experience is involved. Thus, it is an arduous task to determine a suitable prior when we lack prior knowledge. How to improve the interpretability of VGAE. Graph neural networks (GNNs) have achieved great success in processing graph-structured data in recent years. However, this type of methods is often a black box, which makes the learned low-dimensional representations unexplainable. For example, we cannot explain the meanings of the neural networks' parameters when we want to convince people of the prediction results in the neural networks. Existing methods improved the interpretability of GNNs by analyzing and explaining the prediction results of the algorithms [18] or constructing an interpretable model [19]. However, the above methods made the node embeddings still black boxes because they did not point out the meanings of embeddings themselves, i.e., they could not explain the meaning of each dimension of the learned representations. To cope with the above challenges, we propose a noninformative prior [20] based interpretable variational graph autoencoder called NPIVGAE, which utilizes the noninformative prior distribution as the prior of the latent variables. Such a prior is more reasonable, especially when we lack prior knowledge. It no longer forces the posterior distribution to tend to the standard normal distribution, so that almost all the posterior distribution parameters are learned from the sample data. In addition, we leverage a new perspective to understand the node representations to improve the interpretability of the model. Specifically, each dimension of the latent variables is regarded as the soft assignment probability that the node belongs to each community (also called block in the stochastic blockmodel).