# Dirichlet Process Prior for Student's t Graph Variational Autoencoders
## 1 Introduction  
Variational autoencoders (VAEs), graph variational autoencoders (GVAEs) and their variants [1, 2, 3], the deep generative models that have emerged in recent years, integrate the auto-encoder and variational inference, not only keep the excellent expression of the neural network but solve the randomness of probabilistic latent variables in the generation process. At present, the development of VAEs in the probability models has two primary directions. The first is to design a more expressive variational posterior, the other one is to replace the standard normal distribution prior of VAEs with learning more intricate prior [4, 5, 6, 7, 8]. Although works in the first direction have made some signs of progress in various kinds of learning tasks, delving into learning more about whether the variational posterior is valid is still interesting. Reference [9] proved that even with a very strong posterior, VAEs always can not match the learned posterior with N(0, I) prior. So we focus on learning a prior with abundant semantics for GVAEs. The performance of GVAEs is mainly dependent on the validity of their data representation. Learning the representation of graph data, also known as network representation learning (NRL) [10, 11, 12], encodes nodes in the network and denotes them in a unified low-dimensional latent space. Hence, one way to improve GVAEs prior is to use more flexible probability distribution modeling latent variables. Gaussian distribution [13, 14, 15, 16] is the most widely used probability distribution of latent variables in VAE models [17, 18, 19]. This kind of phenomenon is because of the well-known \"central-limit theorem\" [20]. If the sample size is large enough, the distribution of the samples' mean is approximately Gaussian, irrelevant to the distribution of samples in the population. The mean value of the samples'mean distribution does not have to be zero, it depends on the value range of samples. More specifically, if there are a set of non-negative samples, their mean value is also non-negative. In addition, Gaussian distribution is the natural selection in the case of known mean and variance owing to its information entropy is the largest under this condition. However, it also has some disadvantages, especially its \"empirical rule\", which makes up about 99.7% of the numerical values distributed within three standard deviations centered on the mean. When the latent space dimension increases, its ability to capture more information will decline. Nevertheless, the choice of heavy-tailed Student's t (St) distribution [21; 22; 23; 24] can significantly remedy this defect. For the reason that it can adaptively model the networks according to the degree of freedom (dof) parameter. The dof parameter controls the curve shape changes of the St distribution. The smaller the dof, the flatter the St distribution curve. Conversely, the larger the dof, the curve is closer to the Gaussian distribution with the same mean and scale parameters. Because most of the real-world networks are not random, the degree distribution in different networks more or less has the power-law property [25; 26]. For instance, in the American airline network, nodes are airports, and the links between them are airlines. The nodes such as John F. Kennedy International Airport and Los Angeles International Airport are hubs with the power-law property. However, there are just several hubs in a network, and most nodes have a handful of links [27]. In addition, each latent vector is a node representation, so that we can fit the structure of real-world networks by applying the more general St distribution to model node representations. Concretely, ensuring the variational posterior of latent representation is close to a St distribution with zero mean and one scale, while dof remains unchanged [4]. They solve the problem that Gaussian prior never rejects outliers in the modeling. Furthermore, they deal with data with image characteristics, which is different from this paper.