New results for the random nearest neighbor tree

TL;DR

This paper investigates the properties of the random nearest neighbor tree on a torus, providing new bounds on degrees, height, diameter, and establishing the local limit and recurrence of an infinite analog.

Contribution

It introduces new concentration inequalities, bounds on degrees, and characterizes the local limit of the random nearest neighbor tree in high dimensions.

Findings

Degree sequence decreases exponentially with k

Maximum degree is logarithmic in size

The height and diameter grow logarithmically with n

Abstract

In this paper, we study the online nearest neighbor random tree in dimension $d\in \mathbb N$ (called $d$-NN tree for short) defined as follows. We fix the torus $\mathbb T^d_n$ of dimension $d$ and area $n$ and equip it with the metric inherited from the Euclidean metric in $\mathbb R^d$. Then, embed consecutively $n$ vertices in $\mathbb T^d_n$ uniformly at random and independently, and let each vertex but the first one connect to its (already embedded) nearest neighbor. Call the resulting graph $G_n$. We show multiple results concerning the degree sequence of $G_n$. First, we prove that typically the number of vertices of degree at least $k\in \mathbb N$ in the $d$-NN tree decreases exponentially with $k$ and is tightly concentrated by a new Lipschitz-type concentration inequality that may be of independent interest. Second, we obtain that the maximum degree of $G_n$ is of logarithmic order. Third, we give explicit bounds for the number of leaves that are independent of the dimension and also give estimates for the number of paths of length two. Moreover, we show that typically the height of a uniformly chosen vertex in $G_n$ is $(1+o(1))\log n$ and the diameter of $\mathbb T^d_n$ is $(2e+o(1))\log n$, independently of the dimension. Finally, we define a natural infinite analog $G_{\infty}$ of $G_n$ and show that it corresponds to the local limit of the sequence of finite graphs $(G_n)_{n \ge 1}$. Moreover, we prove almost surely that $G_{\infty}$ is locally finite, that the simple random walk on $G_{\infty}$ is recurrent, and that $G_{\infty}$ is connected.