Model-independent methods for embedding directed networks into Euclidean and hyperbolic spaces

TL;DR

This paper introduces a versatile framework for embedding directed networks into Euclidean and hyperbolic spaces, using dimension reduction and conversion techniques to improve accuracy and applicability across various real-world networks.

Contribution

It presents a novel approach combining proximity measures, Euclidean-hyperbolic conversion, and direct hyperbolic embedding for directed networks, filling a gap in existing methods.

Findings

High-quality embeddings achieved for multiple real networks

Effective graph reconstruction and routing performance

Versatile methods applicable to directed and undirected networks

Abstract

The arrangement of network nodes in hyperbolic spaces has become a widely studied problem, motivated by numerous results suggesting the existence of hidden metric spaces behind the structure of complex networks. Although several methods have already been developed for the hyperbolic embedding of undirected networks, approaches able to deal with directed networks are still in their infancy. Here, we propose a framework based on the dimension reduction of proximity matrices reflecting the network topology, coupled with a general conversion method transforming Euclidean node coordinates into hyperbolic ones even for directed networks. While proposing a new measure of proximity, we also incorporate an earlier Euclidean embedding method in our pipeline, demonstrating the widespread applicability of our Euclidean-hyperbolic conversion. Besides, we introduce a dimension reduction technique that maps the nodes directly into the hyperbolic space with the aim of reproducing a distance matrix measured on the given (un)directed network. According to mapping accuracy, graph reconstruction performance and greedy routing score, our methods are capable of producing high-quality embeddings for several real networks.