A Probabilistic Characterization of Random Proximity Catch Digraphs and the Associated Tools

TL;DR

This paper provides a probabilistic framework and new tools for analyzing proximity catch digraphs (PCDs), which are graphs based on proximity regions, considering both uniform and non-uniform distributions of points.

Contribution

It introduces auxiliary tools and characterizations of PCDs based on their probabilistic behavior, including new proximity maps and analysis for various distributions.

Findings

Characterization of PCDs under uniform distributions

Extension to non-uniform distributions

Introduction of new proximity maps

Abstract

Proximity catch digraphs (PCDs) are based on proximity maps which yield proximity regions and are special types of proximity graphs. PCDs are based on the relative allocation of points from two or more classes in a region of interest and have applications in various fields. In this article, we provide auxiliary tools for and various characterizations of PCDs based on their probabilistic behavior. We consider the cases in which the vertices of the PCDs come from uniform and non-uniform distributions in the region of interest. We also provide some of the newly defined proximity maps as illustrative examples.