Estimating Vector Fields on Manifolds and the Embedding of Directed Graphs
This provides a novel generative model for directed graphs, addressing a domain-specific problem in graph embedding and manifold learning.
The paper tackles the problem of embedding directed graphs in Euclidean space while preserving directional information by modeling them as observations from a diffusion on a manifold with a vector field, introducing an algorithm that estimates the embedding, data density, and vector field, and applying it to artificial and real data.
This paper considers the problem of embedding directed graphs in Euclidean space while retaining directional information. We model a directed graph as a finite set of observations from a diffusion on a manifold endowed with a vector field. This is the first generative model of its kind for directed graphs. We introduce a graph embedding algorithm that estimates all three features of this model: the low-dimensional embedding of the manifold, the data density and the vector field. In the process, we also obtain new theoretical results on the limits of "Laplacian type" matrices derived from directed graphs. The application of our method to both artificially constructed and real data highlights its strengths.