Entropy of Random Geometric Graphs in High and Low Dimensions

Abstract

We use a multivariate central limit theorem (CLT) to study the distribution of random geometric graphs (RGGs) on the cube and torus in the high-dimensional limit with general node distributions. We find that the distribution of RGGs on the torus converges to the Erd\H os-R\'enyi (ER) ensemble when the nodes are uniformly distributed, but that the distribution for RGGs with non-uniformly distributed nodes on the torus, and for RGGs with any distribution of nodes with kurtosis greater than 1 on the cube is different. In these cases, the distribution has a lower maximum entropy than the ER ensemble, but is still symmetric. Soft RGGs in either geometry converge to the ER ensemble. An Edgeworth correction to the CLT is then developed to derive the $\mathcal{O}\left(d^{-\frac{1}{2}}\right)$ sub-leading term of the Shannon entropy of RGGs in dimension for both geometries. We also provide numerical approximations of maximum entropy in low-dimensional hard and soft RGGs, and calculate exactly the entropy of hard RGGs with 3 nodes in the one-dimensional cube and torus.