Structural Complexity of One-Dimensional Random Geometric Graphs

by   Mihai-Alin Badiu, et al.

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.


page 1

page 2

page 3

page 4


On the Distribution of Random Geometric Graphs

Random geometric graphs (RGGs) are commonly used to model networked syst...

Another approach to non-repetitive colorings of graphs of bounded degree

We propose a new proof technique that aims to be applied to the same pro...

Compression and Symmetry of Small-World Graphs and Structures

For various purposes and, in particular, in the context of data compress...

Pathwidth vs cocircumference

The circumference of a graph G with at least one cycle is the length of ...

Bounds on the Wireless MapReduce NDT-Computation Tradeoff

We consider a full-duplex wireless Distributed Computing (DC) system und...

On the Number of Graphs with a Given Histogram

Let G be a large (simple, unlabeled) dense graph on n vertices. Suppose ...

Please sign up or login with your details

Forgot password? Click here to reset