Functional Central Limit Theorem for Subgraph Counting Processes

TL;DR

This paper establishes a functional central limit theorem for subgraph counting processes outside expanding regions, revealing how tail behaviors and expansion speed influence the limiting Gaussian processes.

Contribution

It introduces a novel functional CLT for subgraph counts in high-dimensional settings with different tail distributions and expansion regimes.

Findings

Different normalization regimes depending on tail behavior and expansion speed.

Limiting Gaussian processes vary with the presence of a weak core region.

Results applicable to high-dimensional geometric graph analysis.

Abstract

The objective of this study is to investigate the limiting behavior of a subgraph counting process. The subgraph counting process we consider counts the number of subgraphs having a specific shape that exist outside an expanding ball as the sample size increases. As underlying laws, we consider distributions with either a regularly varying tail or an exponentially decaying tail. In both cases, the nature of the resulting functional central limit theorem differs according to the speed at which the ball expands. More specifically, the normalizations in the central limit theorems and the properties of the limiting Gaussian processes are all determined by whether or not an expanding ball covers a region - called a weak core - in which the random points are highly densely scattered and form a giant geometric graph.