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Hyperbolic network models have gained considerable attention in recent years,mainly due to their capability of explaining many peculiar features ofreal-world networks. One of the most widely known models of this type is thepopularity-similarity optimisation (PSO) model, working in the native diskrepresentation of the two-dimensional hyperbolic space and generating networkswith small-world property, scale-free degree distribution, high clustering andstrong community structure at the same time. With the motivation of betterunderstanding hyperbolic random graphs, we hereby introduce the $d$PSO model, ageneralisation of the PSO model to any arbitrary integer dimension $d>2$. Theanalysis of the obtained networks shows that their major structural propertiescan be affected by the dimension of the underlying hyperbolic space in anon-trivial way. Our extended framework is not only interesting from atheoretical point of view but can also serve as a starting point for thegeneralisation of already existing two-dimensional hyperbolic embeddingtechniques.

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