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

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 $$d>2$$ 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.

Model-independent embedding of directed networks into Euclidean and hyperbolic spaces

“…Our suggestion for the largest possible radial coordinate in the hyperbolic ball is r hyp;max ¼ C ζ Á lnðNÞ, where C is a constant. With this choice, the hyperbolic volume scales as V hyp d $ N CÁðdÀ1Þ with the number of nodes N, and at C = 2 we obtain the same volume as we would have in a network generated by the PSO model 15,22 . Based on that, the radial coordinate in the hyperbolic ball can be expressed as…”

“…In our hyperbolic embedding methods, we used the native representation of the hyperbolic space 14 , which is commonly used both in hyperbolic network models 15,21,22,26 and hyperbolic embeddings 16,17,43,47 . This representation visualises the d-dimensional hyperbolic space of curvature K = − ζ 2 < 0 in the Euclidean space as a d-dimensional ball of infinite radius (to which we refer as the native ball), in which the radial coordinate of a point (i.e., its Euclidean distance measured from the centre of the ball) is equal to the hyperbolic distance between the point and the centre of the ball, and the Euclidean angle formed by two hyperbolic lines is equal to its hyperbolic value.…”

“…Grasping the above properties all at once with a simple network model is a challenging task for which hyperbolic approaches offer an intuitive framework. The basic idea of hyperbolic network models is to place the nodes in the hyperbolic space and connect them with a probability decaying as the function of the hyperbolic distance [14][15][16][17][18][19][20][21][22] . Remarkably, the networks generated in this way are usually small-world, highly clustered and scale-free 14,15 , and according to recent results they can easily display a strong community structure as well 18,20,[23][24][25][26][27] .…”

Dimension matters when modeling network communities in hyperbolic spaces

“…The model has also been ex- * beatrice.desy.1@ulaval.ca tended to weighted [8], growing [5,9], bipartite [10], multilayer [11,12] or modular networks [13][14][15]. Now that hyperbolic networks of the lowest dimension have been shown to capture so many realistic properties, some attention has shifted to the study of higherdimensional models [16][17][18]. In these, there is still one radial coordinate for popularity, but there are D > 1 dimensions encoding similarity, or perhaps, similarities.…”

Maximally modular structure of growing hyperbolic networks

“…Furthermore, most real networks also display an intricate community structure [11][12][13] , corresponding to the presence of denser modules in the network topology, in a similar fashion to families and friendship circles in the society. Capturing the most essential properties of complex networks with the help of simple mathematical models has always been one of the key goals in this field, and a notable approach in this respect is given by hyperbolic models [14][15][16][17][18][19][20][21] , centred around the idea of placing the nodes in hyperbolic space and connecting node pairs with a probability depending on the hyperbolic distance.…”