Exact Recovery of Two-Latent Variable Stochastic Block Model with Side Information

The two-latent variable stochastic block model is a new graph synthetic model making a connection between the conventional stochastic block model and real-world networks. In this model, each node contains two latent variables such that at least one of these two latent variables is unknown. Still, this model lonely is not able to model a real-world network. Side information is another component that sometimes exists beside a real-world network. In this paper, we will investigate the asymptotic behavior of the two-latent variable stochastic block model in the presence of side information. Two different types of side information are considered in this paper: noisy labels and partially revealed labels side information. For each case, the necessary and sufficient conditions for the exact recovery of the desired latent variable are obtained via semidefinite programming optimization. It is shown that these conditions are tight and create a phase transition for the exact recovery.