Generalised popularity-similarity optimisation model for growing hyperbolic networks beyond two dimensions

Sámuel G. Balogh,Gergely Palla,Bianka Kovács

doi:10.1038/s41598-021-04379-1

Sámuel G. Balogh, Gergely Palla + Show 1 more

Open Access

https://doi.org/10.1038/s41598-021-04379-1

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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 dPSO 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.

Highlights

Hyperbolic network models have gained considerable attention in recent years, mainly due to their capability of explaining many peculiar features of real-world networks
By converting the hidden variables into radial coordinates we arrive to the hyperbolic H2 model[27] that is equivalent to the random hyperbolic graph (RHG) model; the RHG is often referred to as the S1/H2 model
We show analytically that the degree distribution of dPSO networks can be written as P(k) ∼ k−γ in the large k regime, where the degree decay exponent γ is directly related to the dimension d and the popularity fading parameter β as γ = 1 + (d−11)β

Summary

Introduction

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. Incorporating all, or at least some of these universal properties into a unified modelling framework is, a non-trivial issue and still presents a theoretical challenge of high relevance Along this line, a variety of different network models have been proposed so far, including the celebrated Barabási–Albert (BA) model with preferential a ttachement[12], the hidden variables formalism[13,14,15,16,17] or models based on the mechanism of triadic closure, which has been designed for explaining the high clustering of social networks[18,19]. The degree of the nodes is determined by the radial coordinate, and owing to an outward shift of the nodes (referred to as the popularity fading, controlled by a parameter β ), the degree distribution takes the scaling form of P(k) ∼ k−γ with a tuneable decay exponent γ

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