A1582
Title: Stable and Frechet limit theorem for subgraph functionals in the hyperbolic random geometric graph
Authors: Takashi Owada - Purdue University (United States) [presenting]
Christian Hirsch - Aarhus University (Denmark)
Ruiting Tong - Purdue University (United States)
Abstract: Fluctuations of subgraph counts in hyperbolic random geometric graphs on the d-dimensional Poincare ball are studied in the heterogeneous, heavy-tailed degree regime. In a hyperbolic random geometric graph whose vertices are given by a Poisson point process on a growing hyperbolic ball, two basic families of subgraphs are considered: star shape counts and clique counts, with their global counts and maxima over the vertex set analyzed. Working in the parameter regime where a small number of vertices close to the center of the Poincare ball carry very large degrees and act as hubs, joint functional limit theorems are established for suitably normalized star shape and clique count processes together with the associated maxima processes. The limits are given by a two-dimensional dependent process whose components are a stable Levy process and an extremal Frechet process, reflecting the fact that a small number of hubs dominates both the total number of local subgraphs and their extremes. As an application, fluctuation results for the global clustering coefficient are derived, showing that its asymptotic behavior is described by the ratio of the components of a bivariate Levy process with perfectly dependent stable jumps.
