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

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 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.

Network-based visualisation of frequent sequences

“…With this method, the visualisation of the network is enriched with nondirect connections, using multidimensional scaling (MDS) to define the relative position of the sequences on the network. The main principle of this approach is similar to hyperbolic embedding of networks [ 28 ], but uses different similarity measures.…”

Zoo guide to network embedding

“…Hyperbolic latent space contains geometric information associated with the original network; in the context of H 2 , a recent study showed that shortest paths in the hyperbolic latent space are aligned along geodesic curves connecting endpoint nodes, and this alignment is sufficiently strong to allow the identification of shortest path nodes even in the case of substantially incomplete networks [179]. Finally, note that the hyperbolic embedding in the case of directed networks has been addressed in [180]. • Embedding of hierarchical data.…”

Nearest-neighbor directed random hyperbolic graphs

“…(c) The left-hand side of Eqs. (32) for ξ min (red, top) and ξ max (blue, bottom) for m = 3, ν = 0.01. In the periphery, for ζ < r m (0) (lower dashed line),…”

“…These models, as well as somewhat similar Apollonian networks [25,26] and their generalizations [27][28][29][30], are the first equilibrium models that unify these three properties. However, up to now hyperbolic random graph models have been confined to undirected networks (except for two very recent papers [31,32], see Discussion).…”