Entropy of Random Geometric Graphs in High and Low Dimensions (2503.11418v2)

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-Reny\' i (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, 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 $\bigO{d^{-\frac{1}{2}}}$ scaling of 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.