When clustering the nodes of a graph, a unique partition of the nodes is usually built, either the graph is undirected or directed. While this choice is pertinent for undirected graphs, it should be discussed for directed graphs because it implies that no difference is made between the clusters of source and target nodes. We examine this question in the context of probabilistic models with latent variables and compare the use of the  Stochastic Block Model (SBM) and of the Latent Block Model (LBM). We analyze and discuss this comparison  through simulated and real data sets and suggest some recommendation.