The modularity of random graphs on the hyperbolic plane

TL;DR

This paper studies the modularity of a hyperbolic random graph model, showing that under certain conditions, the modularity converges to 1, indicating highly modular network structure.

Contribution

It provides the first analysis of modularity in hyperbolic random graphs, demonstrating convergence to 1 in specific parameter regimes.

Findings

Modularity converges to 1 in probability for the model.

The model exhibits typical complex network features such as power-law degree distribution.

Networks are highly modular under certain parameters.

Abstract

Modularity is a quantity which has been introduced in the context of complex networks in order to quantify how close a network is to an ideal modular network in which the nodes form small interconnected communities that are joined together with relatively few edges. In this paper, we consider this quantity on a recent probabilistic model of complex networks introduced by Krioukov et al. (Phys. Rev. E 2010). This model views a complex network as an expression of hidden hierarchies, encapsulated by an underlying hyperbolic space. For certain parameters, this model was proved to have typical features that are observed in complex networks such as power law degree distribution, bounded average degree, clustering coefficient that is asymptotically bounded away from zero, and ultra-small typical distances. In the present work, we investigate its modularity and we show that, in this regime, it converges to 1 in probability.