Intra-community link formation and modularity in ultracold growing hyperbolic networks

Abstract

Hyperbolic network models, centered around the idea of placing nodes at random in a hyperbolic space and drawing links according to a probability that decreases as a function of the distance, provide a simple, yet also very capable framework for grasping the small-world, scale-free, highly clustered and modular nature of complex systems that are often referred to as real-world networks. In the present work we study the community structure of networks generated by the Popularity Similarity Optimization model (corresponding to one of the fundamental, widely known hyperbolic models) when the temperature parameter (responsible for tuning the clustering coefficient) is set to the limiting value of zero. By focusing on the intra-community link formation we derive analytical expressions for the expected modularity of a partitioning consisting of equally sized angular sectors in the native disk representation of the 2d hyperbolic space. Our formulas improve earlier results to a great extent, being able to estimate the average modularity (measured by numerical simulations) with high precision in a considerably larger range both in terms of the model parameters and also the relative size of the communities with respect to the entire network. These findings enhance our comprehension of how modules form in hyperbolic networks. The existence of these modules is somewhat unexpected, given the absence of explicit community formation steps in the model definition.