Maximally modular structure of growing hyperbolic networks

摘要

Hyperbolic network models provide a particularly successful approach to explain many peculiar features of real complex networks including, for instance, the small-world and scale-free properties, or the relatively high clustering coefficient. Here we show that for the popularity-similarity optimisation (PSO) model from this family, the generated networks become also extremely modular in the thermodynamic limit, despite lacking any explicitly built-in community formation mechanism in the model definition. In particular, our analytical calculations indicate that the modularity in PSO networks can get arbitrarily close to its maximal value of 1 as the network size is increased. We also derive the convergence rate, which turns out to be dependent on the popularity fading parameter controlling the degree decay exponent of the generated networks. During the past decade, hyperbolic network models have received considerable attention due to their ability to capture many peculiar features of real complex networks, including for instance, the small-world and scale-free properties, or the high clustering coefficient. Here we show that for the popularity-similarity optimisation (PSO) model from this family, the generated networks do not only display the above properties but become also extremely modular in the thermodynamic limit, even though there is no explicitly built-in community formation mechanism in the model definition.