The Total Acquisition Number of Random Geometric Graphs

TL;DR

This paper determines the asymptotic behavior of the total acquisition number in random geometric graphs, revealing how it scales with the number of vertices and the connection radius.

Contribution

It provides a precise asymptotic characterization of the total acquisition number for all ranges of the connection radius in random geometric graphs.

Findings

For large r, the total acquisition number is approximately proportional to n / (r log r)^2.

When r < 1, the total acquisition number is proportional to n.

When r log r > sqrt(n), the total acquisition number is approximately constant.

Abstract

Let $G$ be a graph in which each vertex initially has weight 1. In each step, the weight from a vertex $u$ to a neighbouring vertex $v$ can be moved, provided that the weight on $v$ is at least as large as the weight on $u$. The total acquisition number of $G$, denoted by $a_t(G)$, is the minimum cardinality of the set of vertices with positive weight at the end of the process. In this paper, we investigate random geometric graphs $G(n,r)$ with $n$ vertices distributed u.a.r. in $[0,\sqrt{n}]^2$ and two vertices being adjacent if and only if their distance is at most $r$. We show that asymptotically almost surely $a_t(G(n,r)) = \Theta( n / (r \lg r)^2)$ for the whole range of $r=r_n \ge 1$ such that $r \lg r \le \sqrt{n}$. By monotonicity, asymptotically almost surely $a_t(G(n,r)) = \Theta(n)$ if $r < 1$, and $a_t(G(n,r)) = \Theta(1)$ if $r \lg r > \sqrt{n}$.