The community structure of the geometric soft configuration model

Abstract

Network models serve as an approach to explain the properties of real networks. The geometric soft configuration model, also known as the S1/H2 model, can be used to generate synthetic networks that replicate many features of real complex networks —sparsity, a heterogeneous degree distribution, the small world property, a high level of clustering, and more— while randomizing others. In this work, a range of parameters of the S1/H2 model has been explored, satisfactorily manipulating the level of heterogeneity of the degree distribution with the parameter γ and the level of clustering with the parameter β, in order to probe the level of control that is possible to attain in the generation of random networks. Recent theoretical evidence supports that hyperbolic networks like this one possess topological community structure, up to being maximally modular in the thermodynamic limit, even if the model is not purposefully equipped with geometric communities. The community structure of the S1/H2 model was put under scrutiny using computational simulations, revealing that synthetic networks generated according to it could be consistently partitioned with a high modularity. The modularity of equally sized angular partitions of the generated random networks was evaluated, confirming that this model tends to maximal modularity in the limit of large network size and in a regime of high clustering. The Louvain method for community detection in the topology of complex networks using modularity maximization was employed as well, giving rise to no significantly better results in comparison with the initial approach. With the S1/H2 model, it was also explored how much of the community structure of real networks can be attributed to the effect of clustering in combination with their heterogeneous degree distribution —networks with these two features are called hierarchical—. The results suggest that the communities detected in some real networks are, in part or totally, a byproduct of their hierarchicity.