Structural Complexity of One-Dimensional Random Geometric Graphs
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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∈[0,1]. We characterize the number of possible structures and obtain a universal upper bound on the structural entropy of 2n - 3/2log_2 n - 1/2log_2 π, 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 ∝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.
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