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 characterize the number of possible structures and obtain a universal upper bound on the structural entropy of $2n - \frac{3}{2} \log_2 n - \frac{1}{2} \log_2 \pi$, which holds for any $n$, $r$ and distribution of the node locations. For large $n$, we derive bounds on the structural entropy normalized by $n$, 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 bound decreases with $r$ as $2(1-r)$. 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 \emph{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 $\log_2(n!) \sim n \log_2 n$.