Ollie Baker

Entropy of Random Geometric Graphs in High and Low Dimensions

We study the effect of dimension on the entropy of random spatial networks in the d-cube and d-torus. First, in low dimensions we provide the first exact calculations of the entropy of hard random geometric graphs, and give numerical simulations which hint at the behaviour in higher dimensions. To investigate this, we study the entropy in the high-dimensional limit using a multivariate central limit theorem. First, we show that random geometric graphs with probabilistic connection functions always converge to the Erdos-Renyi ensemble in high dimensions. We then give conditions for when hard random geometric graphs exhibit this behaviour. Finally, we use these results along with an Edgeworth correction to the central limit theorem to derive the scaling of entropy in dimension.