Directed Networks in Hyperbolic Space

This paper demonstrates several approaches to modeling directional networks that have a small diameter, a large clustering coefficient, and a power-law distribution of incoming degrees of nodes. An example of real networks with such properties are networks of free associations of Russian and English languages, i.e. networks in which the nodes are words, and the role of connections is played by associations between them, measured experimentally by interviewing native speakers. Many experimentally observed undirected networks have similar topological properties, and about 10 years ago it was discovered that geometric graphs in hyperbolic space are well suited for their modeling. Moreover until now only undirected graphs have been considered in the literature, while networks of free associations are one of many practically important examples in which the direction of connections is a fundamental property. This motivates the question of the possibility of generalizing the results obtained for undirected geometric graphs in hyperbolic space to a directed case. In this paper, we consider the graph of nearest neighbours in a hyperbolic space, i.e. directed graph in which each vertex is connected to m nearest neighbours (according to the hyperbolic metric). In addition to the properties inherited from undirected networks in hyperbolic space, the new model networks have a fundamentally different shell-core structure.