The Hyperbolic Geometric Block Model and Networks with Latent and Explicit Geometries

In hyperbolic geometric networks the vertices are embedded in a latent metric space and the edge probability depends on the hyperbolic distance between the nodes. These models allows to produce networks with high clustering and scale-free degree distribution, where the coordinates of the vertices abstract their centrality and similarity. Based on the principles of hyperbolic models, in this paper we introduce the Hyperbolic Geometric Block Model, which yields highly clustered, scale-free networks while preserving the desired group mixing structure. We additionally study a parametric network model whose edge probability depends on both the distance in an explicit euclidean space and the distance in a latent geometric space. Through extensive simulations on a stylized city of 10K inhabitants, we provide experimental evidence of the robustness of the HGBM model and of the possibility to combine a latent and an explicit geometry to produce data-driven social networks that exhibit many of the main features observed in empirical networks.