Large deviation result for the empirical locality measure of typed random geometric graphs

TL;DR

This paper establishes a large deviation principle for the empirical locality measure in typed random geometric graphs, providing insights into the probabilities of rare configurations and extending to classical Erdős-Rényi graphs.

Contribution

It introduces a large deviation principle for the empirical locality measure in typed random geometric graphs, linking it to pair and type measures, and extends results to Erdős-Rényi graphs.

Findings

Large deviation principle for empirical locality measure

Derived LDP for degree distribution and isolated nodes

Extended results to classical Erdős-Rényi graphs

Abstract

In this article for a finite typed random geometric graph we define the empirical locality distribution, which records the number of nodes of a given type linked to a given number of nodes of each type. We find large deviation principle (LDP) for the empirical locality measure given the empirical pair measure and the empirical type measure of the typed random geometric graphs. From this LDP, we derive large deviation principles for the degree measure and the proportion of detached nodes in the classical Erdos-Renyi graph defined on [0, 1]^d. This graphs have been suggested by (Canning and Penman, 2003) as a possible extension to the randomly typed random graphs.