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Functional central limit theorem for subgraph counts in a dynamic random connection model

Rajat Subhra Hazra, Nikolai Kriukov, Michel Mandjes, Moritz Otto

TL;DR

This work studies a dynamic variant of the random connection model where vertices switch between active and inactive states over time. It proves a functional central limit theorem for a multivariate vector of subgraph counts by developing a dynamic cumulant approach that extends Poisson-U-statistic techniques to the temporal setting. The authors derive explicit mean and covariance structures in both dense and sparse regimes and use a delta method to obtain functional limits for ratio-type processes, including the clustering coefficient. The results provide a rigorous Gaussian approximation for time-evolving spatial networks and offer tools for inference on temporal subgraph patterns. Overall, the paper establishes a robust framework connecting cumulant methods, diagram formulas, and functional limit theorems in dynamic random geometric-type graphs.

Abstract

We prove a functional central limit theorem for subgraph counts in a dynamic version of the random connection model. To establish tightness, we develop a dynamic extension of the cumulant method.

Functional central limit theorem for subgraph counts in a dynamic random connection model

TL;DR

This work studies a dynamic variant of the random connection model where vertices switch between active and inactive states over time. It proves a functional central limit theorem for a multivariate vector of subgraph counts by developing a dynamic cumulant approach that extends Poisson-U-statistic techniques to the temporal setting. The authors derive explicit mean and covariance structures in both dense and sparse regimes and use a delta method to obtain functional limits for ratio-type processes, including the clustering coefficient. The results provide a rigorous Gaussian approximation for time-evolving spatial networks and offer tools for inference on temporal subgraph patterns. Overall, the paper establishes a robust framework connecting cumulant methods, diagram formulas, and functional limit theorems in dynamic random geometric-type graphs.

Abstract

We prove a functional central limit theorem for subgraph counts in a dynamic version of the random connection model. To establish tightness, we develop a dynamic extension of the cumulant method.

Paper Structure

This paper contains 13 sections, 7 theorems, 69 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Model and main results
  3. Model description
  4. Main result
  5. Preliminaries
  6. Cumulants
  7. Partitions and diagram formulae
  8. Two auxiliary results
  9. Parameter specifications
  10. Expectation and covariance structure
  11. Finite-dimensional convergence
  12. Tightness
  13. Subgraph ratio process

Key Result

Theorem 1

If $\nu_n$ satisfies eq:nu, then $\boldsymbol{\Gamma}_n^*(\cdot)\to\boldsymbol{\Gamma}(\cdot)$ as $n\to\infty$ (in distribution in $\mathbb{D}([0,T],\mathbb R^{m})$).

Figures (1)

  • Figure 1: The dynamic RCM at different times. Nodes alternate between active (full dot) and inactive (empty dot). Potential edges (dotted line) appear (solid line) when both adjacent nodes are active.

Theorems & Definitions (13)

  • Theorem 1
  • Remark 2
  • Proposition 3: Subgraph ratio process
  • Example 4: Clustering coefficient process
  • Lemma 5
  • proof
  • Lemma 6
  • Lemma 7
  • proof
  • Lemma 8
  • ...and 3 more