Title: Navigability of Random Geometric Graphs in the Universe and Other Spacetimes,
Abstract: Random geometric graphs in hyperbolic spaces explain many common structural
and dynamical properties of real networks, yet they fail to predict the correct
values of the exponents of power-law degree distributions observed in real
networks. In that respect, random geometric graphs in asymptotically de Sitter
spacetimes, such as the Lorentzian spacetime of our accelerating universe, are
more attractive as their predictions are more consistent with observations in
real networks. Yet another important property of hyperbolic graphs is their
navigability, and it remains unclear if de Sitter graphs are as navigable as
hyperbolic ones. Here we study the navigability of random geometric graphs in
three Lorentzian manifolds corresponding to universes filled only with dark
energy (de Sitter spacetime), only with matter, and with a mixture of dark
energy and matter as in our universe. We find that these graphs are navigable
only in the manifolds with dark energy. This result implies that, in terms of
navigability, random geometric graphs in asymptotically de Sitter spacetimes
are as good as random hyperbolic graphs. It also establishes a connection
between the presence of dark energy and navigability of the discretized causal
structure of spacetime, which provides a basis for a different approach to the
dark energy problem in cosmology.