Distribution of the Relative Density of Central Similarity Proximity Catch Digraphs Based on One Dimensional Uniform Data

TL;DR

This paper analyzes the distribution of a graph invariant called relative density in central similarity proximity catch digraphs based on one-dimensional uniform data, providing asymptotic results and parameter optimization.

Contribution

It establishes the asymptotic normality of the relative density of central similarity PCDs and explores their distribution across parameter ranges for one-dimensional uniform data.

Findings

Relative density is a U-statistic.

Asymptotic normality is proven under mild conditions.

Optimal parameters for fastest convergence are identified.

Abstract

We consider the distribution of a graph invariant of central similarity proximity catch digraphs (PCDs) based on one dimensional data. The central similarity PCDs are also a special type of parameterized random digraph family defined with two parameters, a centrality parameter and an expansion parameter, and for one dimensional data, central similarity PCDs can also be viewed as a type of interval catch digraphs. The graph invariant we consider is the relative density of central similarity PCDs. We prove that relative density of central similarity PCDs is a U-statistic and obtain the asymptotic normality under mild regularity conditions using the central limit theory of U-statistics. For one dimensional uniform data, we provide the asymptotic distribution of the relative density of the central similarity PCDs for the entire ranges of centrality and expansion parameters. Consequently, we determine the optimal parameter values at which the rate of convergence (to normality) is fastest. We also provide the connection with class cover catch digraphs and the extension of central similarity PCDs to higher dimensions.