Clustering of points randomly distributed in n-dimensional space.

Area of Science:

• * Computational geometry
• * Statistical physics
• * Spatial analysis

Background:

• * Understanding spatial patterns is crucial in various scientific fields.
• * Random point distributions are fundamental models in statistical physics and data analysis.
• * Cluster formation in spatial data influences network properties and information diffusion.

Purpose of the Study:

• * To derive and present the size distribution of clusters in n-dimensional space.
• * To analyze cluster formation based on nearest and next nearest neighbor connections.
• * To provide a mathematical framework for analyzing randomly distributed point sets.

Main Methods:

• * Consideration of point distributions in n-dimensional Euclidean space.
• * Definition of clusters based on connectivity to nearest and/or next nearest neighbors.
• * Mathematical derivation and presentation of cluster size distributions.

Main Results:

• * The study presents the size distribution of clusters formed under specified connectivity rules.
• * Findings are applicable to n-dimensional spaces, offering generalized insights.
• * The derived distributions characterize the typical sizes of emergent clusters.

Conclusions:

• * The size distribution of clusters in random spatial point sets is mathematically characterized.
• * The findings offer a foundational understanding of percolation and connectivity in high-dimensional spaces.
• * This work provides tools for analyzing clustered data in diverse scientific applications.