Growing hyperbolic networks beyond two dimensions: the generalised popularity-similarity optimisation model

TL;DR

This paper extends the popular two-dimensional hyperbolic network model to higher dimensions, revealing how network properties are influenced by the hyperbolic space's dimension and enabling broader applications.

Contribution

It introduces the $d$PSO model, a novel generalisation of the PSO model to arbitrary dimensions, enhancing understanding of hyperbolic random graphs.

Findings

Network properties vary non-trivially with dimension

The model generalises existing 2D hyperbolic embeddings

Provides a foundation for higher-dimensional hyperbolic techniques

Abstract

Hyperbolic network models have gained considerable attention in recent years, mainly due to their capability of explaining many peculiar features of real-world networks. One of the most widely known models of this type is the popularity-similarity optimisation (PSO) model, working in the native disk representation of the two-dimensional hyperbolic space and generating networks with small-world property, scale-free degree distribution, high clustering and strong community structure at the same time. With the motivation of better understanding hyperbolic random graphs, we hereby introduce the $d$PSO model, a generalisation of the PSO model to any arbitrary integer dimension $d>2$. The analysis of the obtained networks shows that their major structural properties can be affected by the dimension of the underlying hyperbolic space in a non-trivial way. Our extended framework is not only interesting from a theoretical point of view but can also serve as a starting point for the generalisation of already existing two-dimensional hyperbolic embedding techniques.