Structural Complexity of One-Dimensional Random Geometric Graphs - Badiu, Mihai-Alin; Coon, Justin P. (IEEE transactions on information theory, 2023)
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 Θ (a^{2n}) with a=a(r)=2 \cos {\left ({\frac {π }{\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 ≈ 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) .