Structural Complexity of One-Dimensional Random Geometric Graphs
Mihai-Alin Badiu, Justin P. Coon · IEEE Transactions on Information Theory · 2022
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$\log _{2}(n!) = n \log _{2} n {-} n + O(\log _{2} n)$.