Model-independent embedding of directed networks into Euclidean and hyperbolic spaces

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 present 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 measure of proximity based on the shortest path length, 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 of any number of dimensions with the aim of reproducing a distance matrix measured on the given (un)directed network. According to various commonly used quality scores, our methods are capable of producing high-quality embeddings for several real networks. Numerous results support that networks representing the structure of complex systems can be accompanied by hidden metric spaces. The present paper introduces a framework for embedding directed networks into hyperbolic spaces using dimension reduction techniques and a universal conversion method between Euclidean and hyperbolic node coordinates.