This paper considers the problem of embedding directed graphs in Euclidean space while retaining directional information. We model the observed graph as a sample from a manifold endowed with a vector field, and we design an algorithm that separates and recovers the features of this process: the geometry of the manifold, the data density and the vector field. The algorithm is motivated by our analysis of Laplacian-type operators and their continuous limit as generators of diffusions on a manifold. We illustrate the recovery algorithm on both artificially constructed and real data.