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Stable and Fréchet limit theorem for subgraph functionals in the hyperbolic random geometric graph

Christian Hirsch, Takashi Owada, Ruiting Tong

TL;DR

This work analyzes fluctuations of subgraph counts in hyperbolic random geometric graphs on the $d$-dimensional Poincaré ball in a heavy-tailed degree regime driven by hub vertices near the origin. It develops a unified framework based on moment analysis, point-process convergence to Poisson random measures, and continuous-mapping arguments to prove joint functional limit theorems for star-shape and clique counts, with scaling induced by stable Lévy processes and extremal Fréchet processes. A key feature is the hub-driven dependence that yields non-Gaussian fluctuations and perfectly dependent jumps, which also govern the global clustering coefficient via a ratio limit of stable processes. The results establish a robust picture of non-Gaussian fluctuations in scale-free spatial networks and lay groundwork for extending functional limit theory to broader subgraph statistics within hyperbolic and GIRG/SIRG models.

Abstract

We study the fluctuations of subgraph counts in hyperbolic random geometric graphs on the $d$-dimensional Poincaré 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 Poincaré 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 Lévy process and an extremal Fréchet 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 Lévy process with perfectly dependent stable jumps.

Stable and Fréchet limit theorem for subgraph functionals in the hyperbolic random geometric graph

TL;DR

This work analyzes fluctuations of subgraph counts in hyperbolic random geometric graphs on the -dimensional Poincaré ball in a heavy-tailed degree regime driven by hub vertices near the origin. It develops a unified framework based on moment analysis, point-process convergence to Poisson random measures, and continuous-mapping arguments to prove joint functional limit theorems for star-shape and clique counts, with scaling induced by stable Lévy processes and extremal Fréchet processes. A key feature is the hub-driven dependence that yields non-Gaussian fluctuations and perfectly dependent jumps, which also govern the global clustering coefficient via a ratio limit of stable processes. The results establish a robust picture of non-Gaussian fluctuations in scale-free spatial networks and lay groundwork for extending functional limit theory to broader subgraph statistics within hyperbolic and GIRG/SIRG models.

Abstract

We study the fluctuations of subgraph counts in hyperbolic random geometric graphs on the -dimensional Poincaré 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 Poincaré 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 Lévy process and an extremal Fréchet 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 Lévy process with perfectly dependent stable jumps.
Paper Structure (16 sections, 23 theorems, 283 equations, 3 figures)

This paper contains 16 sections, 23 theorems, 283 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Star shape counts
  3. Moments
  4. Point process convergence
  5. Proof of Theorem \ref{['t:joint.sum.max.star']}
  6. Clique counts
  7. Moments
  8. Point process convergence
  9. Proof of Theorem \ref{['t:joint.sum.max.clique']}
  10. Limit theorem for the global clustering coefficient
  11. Appendix
  12. Fundamental properties of hyperbolic random geometric graphs
  13. Proof of \ref{['e:PPD_n']}
  14. Proof of \ref{['e:main(i).cond4']}
  15. Negligibility of $B_n(u)$, $C_{n,\ell}(u)$, $D_n(u)$, and $E_n(u)$
  16. ...and 1 more sections

Key Result

Theorem 2.1

$(i)$ If $\frac{1}{2} (k-1)<\alpha< k-1$, then as $n\to\infty$, in the space $D([0,1], {\mathbb R} \times [0,\infty))$ of right-continuous functions from $[0,1]$ to ${\mathbb R}\times [0,\infty)$ with left limits. The weak limit in e:weak.conv.join.sum.max.star is defined by the hybrid characteristic–distribution function where and $\mathsf{m}_{2\alpha/(k-1)}$ is a Radon measure on $(0,\infty]$

Figures (3)

  • Figure 1: Simulations of $\mathsf{HG}(R_n; \alpha)$ with $d=2$, $R_n = 2 \log 1000 = 13.82$, and different values of $\alpha$. This figure is taken from owada:yogeshwaran:2022.
  • Figure 2: A $4$-clique on the vertices $\{X_1, X_2, X_3, p\}$ with $m=4$ and $d=2$. The asymptotic negligibility of $A_{n,2}^{(2)}(u)$ (resp. $A_{n,3}^{(2)}(u)$) corresponds to the negligibility of the edge $X_2\to p$ (resp. $X_3\to p$). Moreover, \ref{['e:rela.angle.main']} gives the probability of the configuration with edges $X_1 \to p$, $X_1\to X_i$ for $i=2,3$, and $X_2 \to X_3$.
  • Figure 3: The point $a$ lies in $\mathcal{A}_d$ and $b$ lies in $H^{d-1}$. We set $\Theta_i = \overrightarrow{oa}$, $S_i = \overrightarrow{ob}$, and $T_i = \overrightarrow{{\bf 0}b}$.

Theorems & Definitions (46)

  • Theorem 2.1
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof : Proof of \ref{['e:PPmu_n']}
  • Proposition 2.6
  • proof : Proof of Proposition \ref{['p:main.mu.version.star']}
  • ...and 36 more