Navigability of Random Geometric Graphs in the Universe and Other Spacetimes

William Cunningham, Konstantin Zuev, Dmitri Krioukov
Nature Scientific Reports
7: 8699
August 18, 2017

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.

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