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. We find 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.
Highlights
What is dark energy? Why is its density orders of magnitude smaller than one would expect from high-energy physics? Causal sets provide one of the simplest explanation attempts to date[58], but there are many other attempts[59,60,61,62,63,64,65], none commonly considered to be the final answer
We find that random geometric graphs only in manifolds with dark energy are navigable
If there is no dark energy, that is, in the dust-only spacetime, there is a finite fraction of paths for which geometric path finding fails, and that this fraction is constant—it does not depend on the cutoff time, i.e., the present cosmological time in the universe, if the average degree in the graph is kept constant
Summary
In network science and applied mathematics, random geometric graphs have attracted increasing attention over recent years[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44], since it was shown that if the space defining these graphs is not Euclidean but negatively curved, i.e., hyperbolic, these graphs provide a geometric explanation of many common structural and dynamical properties of many real networks, including scale-free degree distributions, strong clustering, community structure, and network growth dynamics[45,46,47] More interestingly, these graphs explain the optimality of many network functions related to finding paths in the network without global knowledge of the network structure[48, 49]. In spacetimes with dark energy, i.e., de Sitter spacetime and the spacetime of our universe, the fraction of unsuccessful paths quickly approaches zero as the cutoff time increases
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