On the Distribution of the Number of Copies of Weakly Connected Digraphs in Random $k$NN Digraphs

TL;DR

This paper analyzes the asymptotic distribution of small subdigraphs in k-nearest neighbor digraphs constructed from random point processes, providing general theoretical results and specific corollaries.

Contribution

It introduces a general asymptotic framework for counting minuscule constructs in kNN digraphs derived from random point data.

Findings

Asymptotic behavior characterized for minuscule constructs in kNN digraphs

Results include distributions of vertices with fixed indegree

Findings cover shared and reflexive kNN pairs

Abstract

In a digraph with $n$ vertices, a minuscule construct is a subdigraph with $m<<n$ vertices. We study the number of copies of a minuscule constructs in $k$ nearest neighbor ($k$NN) digraph of the data from a random point process in $\mathbb{R}^d$. Based on the asymptotic theory for functionals of point sets under homogeneous Poisson process and binomial point process, we provide a general result for the asymptotic behavior of the number of minuscule constructs and as corollaries, we obtain asymptotic results for the number of vertices with fixed indegree, the number of shared $k$NN pairs and the number of reflexive $k$NN's in a $k$NN digraph.