We consider the problem of modeling network interactions and identifying latent groups of network nodes. This problem is challenging due to the facts i) that the network nodes are interdependent instead of independent, ii) that the network data are very noisy (e.g., missing edges), and iii) that the network interactions are often sparse. To address these challenges, we propose a Sparse Matrix-variate t process Blockmodel (SMTB). In particular, we generalize a matrix-variate t distribution to a t process on matrices with nonlinear covariance functions. Due to this generalization, our model can estimate latent memberships for individual network nodes. This separates our model from previous t distribution based relational models. Also, we introduce sparse prior distributions on the latent membership parameters to select group assignments for individual nodes. To learn the model efficiently from data, we develop a variational method. When compared with several state-of-the-art models, including the predictive matrix-variate t models and mixed membership stochastic blockmodels, our model achieved improved prediction accuracy on real world network datasets.