Examples include the E-PSO model that inherently accounts for the creation of internal links, i.e. Although this model has been shown to be capable of generating networks that are small-world, highly clustered and scale-free at the same time, several other variants of the original PSO model have been suggested in order to explain further features of real-world graphs. A mathematically equivalent model is given by the \(\mathbb \) of the resulting graphs can be adjusted as well. The random hyperbolic graph (RHG) 20, for instance, is a static network model where nodes are placed at random on the hyperbolic disk of constant curvature \(K=-\zeta ^2\), and the connection probability between any pair of nodes is a decreasing function of their hyperbolic distance. Besides these examples, a further notable approach is given by hyperbolic network models that are capable of simultaneously explaining many observed network characteristics in a natural manner by assuming that nodes are embedded into a negatively curved hidden metric space 20, 21, 22, 23, 24, 25. Along this line, a variety of different network models have been proposed so far, including the celebrated Barabási–Albert (BA) model with preferential attachement 12, the hidden variables formalism 13, 14, 15, 16, 17 or models based on the mechanism of triadic closure, which has been specifically designed for explaining the high clustering of social networks 18, 19. Incorporating all, or at least some of these universal properties into a unified modelling framework is, however, a non-trivial issue and still presents a theoretical challenge of high relevance. In the past decades, a vast number of related studies reported a few universal features that most of the real networks seem to have in common, such as sparsity 3, small-world property 4, 5, inhomogeneous degree distribution 6, 7, high clustering coefficient 8 or community structure 9, 10, 11. Network theory has become an essential and ubiquitous tool for modelling various types of complex systems ranging from the level of interactions within cells to the level of the Internet, economic networks, and the society 1, 2. 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. 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. With the motivation of better understanding hyperbolic random graphs, we hereby introduce the dPSO model, a generalisation of the PSO model to any arbitrary integer dimension \(d>2\). 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. Hyperbolic network models have gained considerable attention in recent years, mainly due to their capability of explaining many peculiar features of real-world networks.
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