Navigability of the Universe

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. Introduction Random geometric graphs1–3 formalize the notion of “discretization” of a continuous geometric space or manifold. Nodes in these graphs are points, sprinkled randomly at constant sprinkling density, over the manifold, thus representing “atoms” of space, while links encode geometry—two nodes are connected if they happen to lie close in the space. These graphs are also a central object in algebraic topology since their clique complexes4 are Rips complexes5, 6 whose topology is known to converge to the manifold topology under very mild assumptions7. In network science and applied mathematics, random geometric graphs have attracted increasing attention over recent years8–41, since it was shown that if the space defining these graphs is not Euclidean but negatively curved, i.e., hyperbolic, then 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 dynamics42–44. Yet more interestingly, these graphs also explain the optimality of many network functions related to finding paths in the network without global knowledge of the network structure45, 46. Random hyperbolic graphs appear to be optimal, that is, maximally efficient, with respect to the greedy path finding strategy that uses only spatial geometry to navigate through a complex network structure by moving at each step from a current node to its neighbor closest to the destination in the space37, 42. The efficiency of this process is called network navigability47. High navigability of random hyperbolic graphs led to practically viable applications, including the design of efficient routing in the future Internet48, 49, and demonstration that the spatiostructural organization of the human brain is nearly as needed for optimal information routing between different parts of the brain50. Yet if random hyperbolic graphs are truly geometric, meaning that if the sprinkling density is indeed constant with respect to the hyperbolic volume form, then the exponent γ of the distribution P(k)∼ k−γ of node degrees k in the resulting graphs is exactly γ = 342. In contrast, in random geometric graphs in de Sitter spacetime, which is asymptotically the spacetime of our accelerating universe, or indeed in the spacetime representing the exact large-scale Lorentzian geometry of our universe, this exponent asymptotically approaches γ = 251, as in many real networks52. Yet it remains unclear if these random Lorentzian graphs are as navigable as random hyperbolic graphs. In physics, random geometric graphs in Lorentzian spacetimes are known as causal sets, a central object in the causal set approach to quantum gravity53. A seemingly unrelated big, if not the biggest unsolved problem in cosmology is the dark energy puzzle54. 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 date55, but there are many other attempts56–62, none commonly considered to be the final answer. Here we study the navigability of random geometric graphs in three Lorentzian manifolds. One manifold is de Sitter 1 ar X iv :1 70 3. 09 05 7v 1 [ gr -q c] 1 6 M ar 2 01 7 spacetime, corresponding to a universe filled with dark energy only, and no matter. Another manifold is the other extreme, a universe filled only with dust matter, and no dark energy. The third manifold is a universe like ours, containing both matter and dark energy. This last manifold interpolates between the other two. At early times and small graph sizes, it is matter-dominated and “looks” like the dust-only spacetime. At later times and large graph sizes, it is dark-energy-dominated and “looks” increasingly more like de Sitter spacetime. We find that random geometric graphs only in manifolds with dark energy are navigable. Specifically, 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. In contrast, 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. For network science this finding implies that in terms of navigability, random geometric graphs in Lorentzian spacetimes with dark energy are as good as random hyperbolic graphs. For physics, this finding 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. Results Lorentzian Manifolds While Riemannian manifolds are manifolds with positive-definite metric tensors gi j defining geodesic distances ds by ds2 = ∑i, j=1 gi j dxi dx j, where d is the manifold dimension, Lorentzian manifolds are manifolds whose metric tensors gμν , μ,ν = {0,1, . . . ,d}, have signature (−++ . . .+), meaning that if diagonalized by a proper choice of the coordinate system, these tensors have one negative entry on the diagonal, while all other entries are positive. In general relativity, Lorentzian manifolds represent relativistic spacetimes, which are solutions of Einstein’s equations. Typically, the dimension of a Lorentzian manifold is denoted by d+1, with the “+1” referring to the temporal (zeroth) dimension, while the other d dimensions are spatial. In this paper we consider only (3+1)-dimensional Lorentzian manifolds, that is, manifolds of dimension equal to the dimension of our universe63. The Lorentzian metric structure naturally defines spacetime’s causal structure: timelike intervals with ∆s2 < 0 connect pairs of causally related events, i.e., timelike-separated points on a manifold. Einstein’s equations are a set of ten coupled non-linear partial differential equations: Rμν − 1 2 Rgμν +Λgμν = 8πTμν , (1) where we use the natural units G = c = 1. The Ricci curvature tensor Rμν and Ricci scalar R measure the manifold curvature, the cosmological constant Λ is proportional to the dark energy density in the spacetime, and the stress-energy tensor Tμν represents the matter content. Spacetimes which are homogeneous and isotropic are called Friedmann-Robertson-Walker (FRW) spacetimes, which have a metric of the form ds2 =−dt2 +a(t)2dΣ2. The time-dependent function a(t) in front of the spatial metric dΣ is called the scale factor. This function characterizes the expansion of the volume form in a spatial hypersurface with respect to time; it alone tells whether there is a “Big Bang” at t = 0, i.e., whether a(0) = 0. The scale factor is derived explicitly as a solution to the 00-component (μ = ν = 0) of (1), known as the first Friedmann equation: ( ȧ a )2 = Λ 3 − K a 2 + c a3g . (2) The variable g represents the type of matter in the spacetime: in this work we use the values g = {0,1} to indicate no matter and dust matter, respectively. The spatial curvature of the spacetime is captured by K: K = {+1,0,−1} implies positive, zero, or negative spatial curvature, respectively. Motivated by the observation that our universe is nearly flat64, in this work we use K = 0, which significantly simplifies the calculations below. In the flat case, the spatial metric, hereafter using dimensionless spherical coordinates, becomes dΣ2 = dr2 + r2 dθ 2 + r2 sin2 θ dφ 2. Finally, c is a constant proportional to the density of matter in the universe. The total energy density in the universe is known to come from four sources: the matter (dark and baryonic) density ρM , the dark energy density ρΛ, the radiation energy density ρR, and the curvature K. The densities may be rescaled by a critical density: Ω≡ ρ/ρc, where ρc ≡ 3H2 0/8π; H0 ≡ ȧ0/a0 is the Hubble constant and a0 ≡ a(t0), i.e., the scale factor at the present time. Similarly, the curvature density parameter may be written as ΩK ≡ −K/(a0H0) so that we obtain the state equation ΩM +ΩΛ +ΩR +ΩK = 1. This allows us to rewrite (2) in the integral form65 H0t = ∫ a/a0 0 dx x √ ΩΛ +ΩKx−2 +ΩMx−3 +ΩRx−4 . (3)