Random geometric graphs as models of complex networks
The zero clustering (infinite temperature) limit of random hyperbolic graphs is the hypersoft configuration model (HSCM), which is an exchangeable and projective maximum-entropy model of random graphs with a given power-law degree distribution. Another limit (infinite curvature or power-law exponent) of random hyperbolic graphs is (soft) random geometric graphs on $\mathbb{R}^1$ or $\mathbb{S}^1$, which are known to be equivalent to exchangeable and projective maximum-entropy random graphs with expected clustering of every node fixed to a constant. Yet sparse random hyperbolic graphs outside of any of these two limits are not projective for any combination of finite values of curvature (power-law exponent) and temperature (clustering). On the other hand, random graphs in any Lorentzian manifold are manifestly exchangeable and projective, have non-zero clustering, and if the manifold is de Sitter, they also have a power-law degree distribution. We will discuss exchangeably, projectivity, Lorentz invariance, and similarity and differences between random hyperbolic and dual de Sitter graphs.