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

Hyperbolic network models offer a straightforward yet powerful method for understanding the small-world, scale-free, highly clustered, and modular characteristics typical of complex systems that are often called as real-world networks. These models involve randomly positioning nodes in a hyperbolic space and connecting them based on a probability that diminishes with distance. In this study, we examine the community structure within networks created by the Popularity Similarity Optimization model, a fundamental hyperbolic model, when the temperature parameter (responsible for tuning the clustering coefficient) is set to the limiting value of zero. We focus on link formation within communities and show their close relation with non-linear preferential attachment processes. Based on this, we derive analytical expressions that significantly improve previous estimates of the expected modularity for partitions formed by equally sized angular sectors in the 2d hyperbolic space. Our formulas can now predict average modularity, confirmed by numerical simulations, with high accuracy over a broader range of model parameters and community sizes relative to the whole network. These results advance our understanding of module formation in hyperbolic networks, highlighting the surprising emergence of communities despite the lack of explicit community formation steps in the model definition.