Estimating Vector Fields on Manifolds and the Embedding of Directed Graphs

TL;DR

This paper introduces a novel generative model for embedding directed graphs in Euclidean space by capturing manifold structure, data density, and vector fields, supported by new theoretical insights and practical algorithms.

Contribution

It presents the first generative model for directed graphs that estimates manifold embeddings, density, and vector fields simultaneously.

Findings

Effective embedding of directed graphs preserving directional information

Theoretical limits of Laplacian matrices from directed graphs established

Successful application to artificial and real datasets

Abstract

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.