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This paper considers the problem of embedding directed graphs in Euclideanspace while retaining directional information. We model a directed graph as afinite set of observations from a diffusion on a manifold endowed with a vectorfield. This is the first generative model of its kind for directed graphs. Weintroduce a graph embedding algorithm that estimates all three features of thismodel: the low-dimensional embedding of the manifold, the data density and thevector field. In the process, we also obtain new theoretical results on thelimits of "Laplacian type" matrices derived from directed graphs. Theapplication of our method to both artificially constructed and real datahighlights its strengths.

Estimating Vector Fields on Manifolds and the Embedding of Directed Graphs