Limit theorems under heavy-tailed scenario in the age dependent random connection models

TL;DR

This paper establishes limit theorems for subgraph counts in age-dependent random connection models, showing convergence to stable laws under heavy-tailed conditions and providing new insights into clique and sub-tree distributions.

Contribution

It introduces new limit theorems for subgraph counts in age-dependent models, including stable convergence and expectation asymptotics for cliques.

Findings

Subgraph counts converge to stable distributions under heavy tails.

Weak convergence of point processes to Poisson processes is established.

Expectation asymptotics for clique counts are derived.

Abstract

This paper considers limit theorems associated with subgraph counts in the age-dependent random connection model. First, we identify regimes where the count of sub-trees converges weakly to a stable random variable under suitable assumptions on the shape of trees. The proof relies on an intermediate result on weak convergence of associated point processes towards a Poisson point process. Additionally, we prove the same type of results for the clique counts. Here, a crucial ingredient includes the expectation asymptotics for clique counts, which itself is a result of independent interest.