Structural Complexity of One-Dimensional Random Geometric Graphs

TL;DR

This paper analyzes the structural complexity of one-dimensional random geometric graphs, providing bounds on the number of possible unlabeled graph structures and their entropy, depending on the connection range and node distribution.

Contribution

It derives universal bounds on the structural entropy of 1D random geometric graphs for various connection ranges and node distributions, including asymptotic behaviors.

Findings

Number of structures is $ heta(a^{2n})$ with $a=a(r)$ for fixed $r$

Normalized entropy asymptotically reaches 2 bits per node when $r_n=O(1/n)$

Entropy bounds decrease linearly with $r$ when $r$ is bounded away from zero and one

Abstract

We study the richness of the ensemble of graphical structures (i.e., unlabeled graphs) of the one-dimensional random geometric graph model defined by $n$ nodes randomly scattered in $[0,1]$ that connect if they are within the connection range $r\in[0,1]$. We provide bounds on the number of possible structures which give universal upper bounds on the structural entropy that hold for any $n$, $r$ and distribution of the node locations. For fixed $r$, the number of structures is $\Theta(a^{2n})$ with $a=a(r)=2 \cos{\left(\frac{\pi}{\lceil 1/r \rceil+2}\right)}$, and therefore the structural entropy is upper bounded by $2n\log_2 a(r) + O(1)$. For large $n$, we derive bounds on the structural entropy normalized by $n$, and evaluate them for independent and uniformly distributed node locations. When the connection range $r_n$ is $O(1/n)$, the obtained upper bound is given in terms of a function that increases with $n r_n$ and asymptotically attains $2$ bits per node. If the connection range is bounded away from zero and one, the upper and lower bounds decrease linearly with $r$, as $2(1-r)$ and $(1-r)\log_2 e$, respectively. When $r_n$ is vanishing but dominates $1/n$ (e.g., $r_n \propto \ln n / n$), the normalized entropy is between $\log_2 e \approx 1.44$ and $2$ bits per node. We also give a simple encoding scheme for random structures that requires $2$ bits per node. The upper bounds in this paper easily extend to the entropy of the labeled random graph model, since this is given by the structural entropy plus a term that accounts for all the permutations of node labels that are possible for a given structure, which is no larger than $\log_2(n!) = n \log_2 n - n + O(\log_2 n)$.