The inherent community structure of hyperbolic networks

Abstract

A remarkable approach for grasping the relevant statistical features of real networks with the help of random graphs is offered by hyperbolic models, centred around the idea of placing nodes in a low-dimensional hyperbolic space, and connecting node pairs with a probability depending on the hyperbolic distance. It is widely appreciated that these models can generate random graphs that are small-world, highly clustered and scale-free at the same time; thus, reproducing the most fundamental common features of real networks. In the present work, we focus on a less well-known property of the popularity-similarity optimisation model and the $${\mathbb {S}}^1/{\mathbb {H}}^2$$ S 1 / H 2 model from this model family, namely that the networks generated by these approaches also contain communities for a wide range of the parameters, which was certainly not an intention at the design of the models. We extracted the communities from the studied networks using well-established community finding methods such as Louvain, Infomap and label propagation. The observed high modularity values indicate that the community structure can become very pronounced under certain conditions. In addition, the modules found by the different algorithms show good consistency, implying that these are indeed relevant and apparent structural units. Since the appearance of communities is rather common in networks representing real systems as well, this feature of hyperbolic models makes them even more suitable for describing real networks than thought before.

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

“…As it has already been reported in Ref. 25 , the original, two-dimensional PSO model 21 is capable of generating networks with strong communities for a wide range of the parameter settings, despite the fact that it does not include any explicitly built-in community structure generating mechanism. Based on the high modularity values measured on d PSO networks (Figs.…”

“…As it was detailed in Ref. 25 in the case of the two-dimensional PSO model, the emergence of a strong community structure can be achieved under two conditions: the existence of inner nodes that are distant from each other enough to provide well-separated attractive centres for the different angular regions, and the localisation of the connections. The distance between the inner community cores can be increased by accelerating their outward drift that simulates the popularity fading via decreasing the popularity fading parameter .…”

“…The automated extraction of communities based solely on the network topology is a challenging problem, with an immense number of different solutions proposed in the literature 9 – 11 . An interesting related feature of two-dimensional hyperbolic networks is that they also contain communities for the major part of the parameter space, in spite of the lack of any explicit built-in community formation mechanism in the graph generation algorithms 25 . Motivated by this and the overall importance of communities in network science, here we examine also the community structure of d -dimensional PSO networks.…”

“…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 – 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 . 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 – 25 .…”

“…Or, the nonuniform popularity-similarity optimisation (nPSO) model 24 , 29 that allows the generation of networks with an adjustable community structure by assuming a heterogeneous angular node distribution with multiple peaks is also worth mentioning. Nevertheless, it has also been revealed quite recently that both the RHG and the PSO models can generate networks that possess strong community structure despite lacking any explicitly built-in community generating mechanisms 25 , 30 – 34 .…”