# Directed Graph Embeddings in Pseudo-Riemannian Manifolds

**Authors:** Aaron Sim, Maciej Wiatrak, Angus Brayne, Páidí Creed, Saee Paliwal  
**arXiv:** [2106.08678](https://arxiv.org/abs/2106.08678)  
**Date:** 2021-06-16  
**Summary page:** https://www.summarizepaper.com/en/arxiv-id/2106.08678v1/

## Key points

- The paper explores the use of embedding models to represent general directed graphs.
- The authors propose a novel approach that combines a pseudo-Riemannian metric structure, a non-trivial global topology, and a unique likelihood function.
- The method is effective for link prediction on synthetic and real directed graphs from natural language applications and biology.
- Lower-dimensional spacetimes can produce graph representations equal or better than higher-dimensional manifolds.
- Various datasets are considered, including protein-protein interaction networks, gene regulatory networks, and WordNet.
- Negative log-likelihood (NLL) loss based on probabilities is used with a negative sampling ratio of 4 throughout the experiments.
- The method shows promising capabilities for link prediction tasks in diverse domains.

## Summary

The paper "Directed Graph Embeddings in Pseudo-Riemannian Manifolds" explores the use of embedding models to represent general directed graphs. The authors propose a novel approach that combines three components: a pseudo-Riemannian metric structure, a non-trivial global topology, and a unique likelihood function that incorporates a preferred direction in the embedding space. They demonstrate the effectiveness of this method by applying it to the task of link prediction on synthetic and real directed graphs from natural language applications and biology. The authors compare different types of embedding spaces, including low-dimensional cylindrical Minkowski and anti-de Sitter spacetimes, with curved Riemannian manifolds of higher dimensions. Surprisingly, they find that these lower-dimensional spacetimes can produce graph representations that are equal or even better than those obtained from higher-dimensional manifolds. To evaluate their approach, the authors consider various datasets such as the Duplication Divergence Model, which simulates the growth and evolution of protein-protein interaction networks; DREAM5, which consists of gene regulatory networks across organisms; and WordNet, an acyclic hierarchical network of nouns. They aim to minimize negative log-likelihood (NLL) loss based on probabilities and use a negative sampling ratio of 4 throughout their experiments. Overall, this paper introduces a new method for representing directed graphs using pseudo-Riemannian manifolds. The results suggest that this approach has promising representational capabilities for link prediction tasks in diverse domains such as natural language processing and biology.