Stable and Fr\'echet limit theorem for subgraph functionals in the hyperbolic random geometric graph

TL;DR

This paper investigates the fluctuations of subgraph counts in hyperbolic random geometric graphs, revealing that a few central hubs dominate the structure and lead to stable and extremal limit behaviors.

Contribution

It establishes joint limit theorems for subgraph counts and maxima in hyperbolic graphs, highlighting the influence of hubs and deriving fluctuation results for clustering coefficients.

Findings

Limit processes involve stable Lévy and Fréchet processes.

Hubs near the center dominate subgraph counts and extremes.

Global clustering fluctuations are described by a bivariate Lévy process.

Abstract

We study the fluctuations of subgraph counts in hyperbolic random geometric graphs on the $d$-dimensional Poincar\'e ball 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, we consider two basic families of subgraphs: star shape counts and clique counts, and we analyze their global counts and maxima over the vertex set. Working in the parameter regime where a small number of vertices close to the center of the Poincar\'e ball carry very large degrees and act as hubs, we establish joint functional limit theorems 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 L\'evy process and an extremal Fr\'echet process, reflecting the fact that a small number of hubs dominates both the total number of local subgraphs and their extremes. As an application, we derive fluctuation results for the global clustering coefficient, showing that its asymptotic behavior is described by the ratio of the components of a bivariate L\'evy process with perfectly dependent stable jumps.