Sub-tree counts on hyperbolic random geometric graphs

Abstract The hyperbolic random geometric graph was introduced by Krioukov et al. ( Phys. Rev. E 82 , 2010). Among many equivalent models for the space, we study d -dimensional Poincaré ball $d\ge 2$ ), with a general connectivity radius. While phase transitions are known expectation asymptotics of certain subgraph counts, very little is about second-order results. Two distinguishing characteristics graphs on space presence tree-like hierarchical structures and power-law behaviour degree distribution. We aim to reveal such in detail investigating sub-tree counts. show multiple variance resulting graph. In particular, counts exhibit an intricate dependence sequence tree under consideration. Additionally, unlike thermodynamic regime Euclidean graph, may different growth rates, which indicative behaviour. Finally, also prove normal approximation using Malliavin–Stein method Last Prob. Theory Relat. Fields 165 2016), along Palm calculus Poisson point processes.