Department of Physics, Northeastern University, 360 Huntington Ave., Boston, MA, 02115, United States.
Department of Computing and Mathematical Sciences, California Institute of Technology, 1200 E. California Blvd., Pasadena, CA, 91125, United States.
Sci Rep. 2017 Aug 18;7(1):8699. doi: 10.1038/s41598-017-08872-4.
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.
在双曲空间中的随机几何图解释了许多真实网络的常见结构和动态特性，但它们无法预测真实网络中观察到的幂律度分布指数的正确值。在这方面，渐近德西特时空中的随机几何图，例如我们加速宇宙的黎曼时空中的图，更具吸引力，因为它们的预测与真实网络中的观测结果更一致。双曲图的另一个重要性质是它们的可导航性，而德西特图是否像双曲图一样可导航尚不清楚。在这里，我们研究了对应于仅充满暗能量（德西特时空中的时空）、仅充满物质以及充满暗能量和物质混合物的三个黎曼流形中的随机几何图的可导航性。我们发现这些图仅在具有暗能量的流形中是可导航的。该结果意味着，就可导航性而言，渐近德西特时空中的随机几何图与随机双曲图一样好。它还在存在暗能量和时空离散因果结构的可导航性之间建立了联系，为宇宙学中暗能量问题提供了一种不同的方法的基础。
