Normal approximation for subgraph counts in age-dependent random connection models

TL;DR

This work establishes normal approximation (central limit theorems) for subgraph counts in the age-dependent random connection model (ADRCM) under a light-tailed regime where moments of order $(2+\varepsilon)$ are finite. Using a combination of Malliavin-Stein techniques for Poisson functionals and association-based CLTs, the authors obtain a multivariate CLT for clique counts across multiple dimensions and a CLT for rooted subtree counts, with explicit variance and covariance asymptotics that scale linearly in the system size. The analysis hinges on delicate moment bounds for first- and second-order difference operators and a careful treatment of neighborhood sizes on the torus, yielding tight rates and positivity of limiting variances. Together with a companion paper addressing stable limits in the infinite-variance regime, these results provide a comprehensive picture of the Gaussian–stable phase transition for subgraph statistics in ADRCMs and support statistical inference for clustering-like quantities such as the clustering coefficient in spatially embedded networks.

Abstract

We study normal approximation of subgraph counts in a model of spatial scale-free random networks known as the age-dependent random connection model. In the light-tailed regime where only moments of order $(2 + \varepsilon)$ are finite, we study the asymptotic normality of both clique and subtree counts. For clique counts, we establish a multivariate quantitative normal approximation result through the Malliavin-Stein method. In the more delicate case of subtree counts, we obtain distributional convergence based on a central limit theorem for sequences of associated random variables.

Figures (1)

• Figure 1: Example of directed wedge trees of two leaves (i.e., $\ell=2$), where two vertices are connected to a root with a lower mark. The quantity $\mathcal{T}_{n,\mathsf{T}}$ counts only the "visualized" rooted wedges drawn by solid lines. Note that each such rooted wedge admits two isomorphisms, and therefore we have $\mathcal{T}_{n,\mathsf{T}} = 14$.

Theorems & Definitions (34)

• Theorem 1
• Theorem 2
• Remark 3: Higher dimensions and non-compactly supported profile functions
• Proposition 4: Covariance asymptotics
• Proposition 5: Bounds for $\Gamma_i$
• Remark 6: Kolmogorov distance
• Lemma 7
• Lemma 8: Size of up- and down-neighborhoods
• Lemma 9: Auxiliary expectation computations
• Lemma 10: Auxiliary covariance computations I