Representing networks in a low dimensional latent space is a crucial task with many interesting application in graph learning problems, such as link prediction and node classification. A widely applied network representation learning paradigm is based on the combination of random walks with the traditional Skip-Gram approach, modeling center-context node relationships. In this paper, we emphasize on exponential family distributions to capture rich interaction patterns between nodes in random walk sequences. We introduce the generic exponential family graph embedding (EFGE) model, that generalizes random walk-based network representation learning techniques to exponential family conditional distributions. Our experimental evaluation demonstrates that the proposed technique outperforms well-known baseline methods in two downstream machine learning tasks.