Structural Complexity of One-Dimensional Random Geometric Graphs

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 $\log_2(n!) = n \log_2 n - n + O(\log_2 n)$.
Comment: 44 pages, 9 figures; revised some of results; significant additional results included