Voronoi Cell Patterns: theoretical model and applications

TL;DR

This paper introduces a simple fragmentation model to analyze the statistical distribution of Voronoi cell sizes in 1D and 2D, applicable to various systems like island nucleation, car parking, administrative divisions, and metro station patterns.

Contribution

The paper presents a unified fragmentation model for Voronoi patterns that incorporates probabilistic distributions and applies it to diverse real-world systems.

Findings

Model accurately describes Voronoi cell size distributions

Applicable to systems like island nucleation and metro patterns

Provides insights into physical properties influencing patterns

Abstract

We use a simple fragmentation model to describe the statistical behavior of the Voronoi cell patterns generated by a set of points in 1D and in 2D. In particular, we are interested in the distribution of sizes of these Voronoi cells. Our model is completely defined by two probability distributions in 1D and again in 2D, the probability to add a new point inside an existing cell and the probability that this new point is at a particular position relative to the preexisting point inside this cell. In 1D the first distribution depends on a single parameter while the second distribution is defined through a fragmentation kernel; in 2D both distributions depend on a single parameter. The fragmentation kernel and the control parameters are closely related to the physical properties of the specific system under study. We use our model to describe the Voronoi cell patterns of several systems. Specifically, we study the island nucleation with irreversible attachment, the 1D car parking problem, the formation of second-level administrative divisions, and the pattern formed by the Paris M\'etro stations.