Structure of de Sitter causets

De Sitter spacetime is the solution of Einstein's field equations for an empty universe with positive cosmological constant Λ. The (1+1)-dimensional de Sitter spacetime (the first ‘1’ stands for the space dimension; the second ‘1’—for time) can be visualized as a one-sheeted 2-dimensional hyperboloid embedded in a flat 3-dimensional Minkowski space (Fig. 2(a)). The length of horizontal circles in Fig. 2(a), corresponding to the volume of space at a moment of time, grows exponentially with time t. Since causet nodes are distributed uniformly over spacetime, their number also grows exponentially with time, while as we show below, their degree decays exponentially, resulting in a power-law degree distribution in the causet.

Figure 2 Mapping between the de Sitter universe and complex networks. Panel (a) shows the 1+1-dimensional de Sitter spacetime represented by the upper half of the outer one-sheeted hyperboloid in the 3-dimensional Minkowski space XY Z. The spacetime coordinates (θ, t), shown by the red arrows, cover the whole de Sitter spacetime. The spatial coordinate θ 0 of any spacetime event, e.g. point P, is its polar angle in the XY plane, while P's temporal coordinate t 0 is the length of the arc lying on the hyperboloid and connecting the point to the XY plane where t = 0. At any time t, the spatial slice of the spacetime is a circle. This 1-dimensional space expands exponentially with time. Dual to the outer hyperboloid is the inner hyperboloid—the hyperbolic 2-dimensional space, i.e. the hyperbolic plane, represented by the upper sheet of a two-sheeted hyperboloid. The mapping between the two hyperboloids is shown by the blue arrows. The green shapes show the past light cone of point P in the de Sitter spacetime and the projection of this light cone onto the hyperbolic plane under the mapping. Panel (b) depicts the cut of panel (a) by the YZ plane to further illustrate the mapping, shown also by the blue arrows. The mapping is the reflection between the two hyperboloids with respect to the cone shown by the dashed lines. Panel (c) projects the inner hyperboloid (the hyperbolic plane) with P's past light cone (the green shape) onto the XY plane. The red shape is the left half of the hyperbolic disc centered at P and having the radius equal to P's time t 0 , which in this representation is P's radial coordinate, i.e. the distance between P and the origin of the XY plane. The green and red shapes become indistinguishable at large times t 0 as shown in panels (d,e,f) where these shapes are drawn for t 0 = 5, 10, 15 using the exact expressions from Section II of Supplementary Notes. Assuming the average degree of , these t 0 times correspond to network sizes of approximately 40, 200 and 2000 nodes. Full size image

To obtain this result, we consider in Fig. 2(a) a patch of (1+1)-dimensional de Sitter spacetime between times t = 0 (the “big bang”) and t = t 0 > 0 (the “current” time) and sprinkle N nodes onto it with uniform density δ. In this spacetime the element of length ds (often called the metric because its expression contains the full information about the metric tensor) and volume dV (or area, since the spacetime is two-dimensional) are given by the following expressions (Supplementary Notes, Section II):

where is the angular (space) coordinate on the hyperboloid. In view of the last equation and uniform sprinkling, implying that the expected number of nodes dN in spacetime volume dV is dN = δ dV, the temporal node density ρ(t) at time is

where the last approximation holds for .

Since links between two nodes in the causet exist only if the nodes lie within each other's light cones, the expected degree of a node at time coordinate is proportional to the sum of the volumes of two light cones centered at the node: the past light cone cut below at t = 0 and the future light cone cut above at t = t 0 , similar to Fig. 1. Denoting these volumes by V p (t) and V f (t) and orienting causet links from the future to the past, i.e. from nodes with higher t to nodes with lower t, we can write , where and are the expected out- and in-degrees of the node. One way to compute V p (t) and V f (t) is to calculate the expressions for the light cone boundaries in the (t, θ) coordinates and then integrate the volume form dV within these boundaries. An easier way is to switch from cosmological time t to conformal time26 η(t) = arcsec cosh t. After this coordinate change, the metric becomes conformally flat, i.e. proportional to the metric ds2 = –dt2 + dx2 in the flat Minkowski space in Fig. 1,

so that the light cone boundaries are straight lines intersecting the coordinate (η, θ)-axes at 45°, as in Fig. 1 with (t, x) replaced by (η, θ). Therefore, the volumes can be easily calculated:

where approximations hold for and where we have used η(t) = arcsec cosh t ≈ π/2 – 2e–t. For large times , the past volume and consequently the out-degree are negligible compared to the future volume and in-degree, which decay exponentially with time t,

These results can be generalized to (d+1)-dimensional de Sitter spacetimes with any d and any curvature K = Λ/3 = 1/a2, where a, the inverse square root of curvature, is also known as the curvature radius of the de Sitter hyperboloid, or as its pseudoradius. Generalizing Eqs. (3,8), we can show that the temporal density of nodes and their expected in-degree in this case scale as and with α = β = d/a. In short, we have a combination of two exponentials, number of nodes ~ eαt born at time t and their degrees ~ e–βt. This combination yields a power-law distribution P(k) ~ k–γ of node degrees k in the causet, where exponent γ = 1 + α/β = 2.

Structure of the universe and complex networks

The large-scale causet structure of the universe in the standard model differs from the structure of sparse de Sitter causets in many ways, two of which are particularly important. First, the universe is not empty but contains matter. Therefore it is only asymptotically de Sitter26,27, meaning that only at large times , or rescaled times , space in the universe expands asymptotically the same way as in de Sitter spacetime. In a homogeneous and isotropic universe, the metric is ds2 = –dt2 + R2(t)dΩ2, where dΩ is the spatial part of the metric and function R(t) is called the scale factor. In de Sitter spacetime, the scale factor is R(t) ~ cosh τ, while in a flat universe containing only matter and dark energy, R(t) ~ sinh2/3 (3τ/2). In both cases, R(t) ~ eτ at large times , but at early times the scaling is different. In particular, at τ → 0 the universe scale factor goes to zero, resulting in a real big bang. The second difference is even more important: the product between the square of inverse curvature a4 = 1/K2 and sprinkling density (one causet element per unit Planck 4-volume) is astronomically huge in the universe, δa4 ~ 10244, compared to in sparse causets with a small average degree. Collectively these two differences result in that the present universe causet is also a power-law graph, but with a different exponent γ = 3/4 (Fig. 3(a)).

Figure 3 Degree distribution in the universe. Panel (a) shows the rescaled distribution Q(κ, τ 0 ) = δa4P(k, t 0 ) of rescaled degrees κ = k/(δa4) in the universe causet at the present rescaled time τ 0 = t 0 /a = 0.85, where δ is the constant node density in spacetime and . As shown in Section III of Supplementary Notes, the rescaled degree distribution does not depend on either δ or a, so we set them to δ = 104 and a = 1 for convenience. The size N of simulated causets can be also set to any value without affecting the degree distribution and this value is N = 106 nodes in the figure. The degree distribution in this simulated causet is juxtaposed against the numeric evaluation of the analytical solution for Q(κ, τ 0 ) shown by the blue dashed line. The inset shows this analytic solution for the whole range of node degrees in the universe, where δ ~ 10173 and a ~ 5 × 1017. Panel (b) shows the same solution for different values of the present rescaled time τ, tracing the evolution of the degree distribution in the universe in its past and future. All further details are in Section I of Supplementary Methods. Full size image

However, the γ = 2 scaling currently emerges (Fig. 3(b)) as a part of a cosmic coincidence known as the “why now?” puzzle28,29,30,31. The matter and dark energy densities happen to be of the same order of magnitude in the universe today. This coincidence implies that the current rescaled time τ 0 ≡ t 0 /a is approximately 1. Figure 3(b) traces the evolution of the degree distribution in the universe in its past and future. In the matter-dominated era with τ < 1, the degree distribution is a power law with exponent 3/4 up to a soft cut-off that grows with time. Above this soft cut-off, the distribution decays sharply. Once we reach times τ ~ 1, e.g. today, we enter the dark-energy-dominated era. The part of the distribution with exponent 3/4 freezes, while the soft cut-off transforms into a crossover to another power law with exponent 2, whose cut-off grows exponentially with time. The crossover point is located at k cr ~ δa4. Nodes of small degrees k < k cr obey the γ = 3/4 part of the distribution, while high-degree nodes, k > k cr , lie in its γ = 2 regime. At the future infinity τ → ∞, the distribution becomes a perfect double power law with exponents 3/4 and 2.

In short, the main structural property of the causet in the present-day universe is that it is a graph with a power-law degree distribution, which currently transitions from the past matter-dominated era (τ < 1) with exponent γ = 3/4 to the future dark-energy-dominated era (τ > 1) with γ = 2. In many (but not all) complex networks the degree distribution is also a power law with γ close to 28,9,10. In Fig. 4(a) we show a few paradigmatic examples of large-scale technological, social and biological networks for which reliable data are available and juxtapose these networks against a de Sitter causet. In all the shown networks, the exponent γ ≈ 2. This does not mean however that the networks are the same in all other respects. Degree-dependent clustering, for example (Fig. 4(b)), is different in different networks, although average clustering is strong in all the networks. Strong clustering is another structural property often observed in complex networks: average clustering in random graphs of similar size and average degree is lower by orders of magnitude8,9,10.

Figure 4 Degree distribution and clustering in complex networks and de Sitter spacetime. The Internet is the network representing economic relations between autonomous systems, extracted from CAIDA's Internet topology measurements38. The network size is N = 23752 nodes, average degree and average clustering . Trust is the social network of trust relations between people extracted from the Pretty Good Privacy (PGP) data39; N = 23797, , . Brain is the functional network of the human brain obtained from the fMRI measurements in Ref. 40; N = 23713, , . De Sitter is a causal set in the 1 + 1-dimensional de Sitter spacetime; N = 23739, , . Panel (a) shows the degree distribution P(k), i.e. the number of nodes N(k) of degree k divided by the total number of nodes N in the networks, P(k) = N(k)/N, so that . Panel (b) shows average clustering of degree-k nodes c(k), i.e. the number of triangular subgraphs containing nodes of degree k, divided by N(k)k(k – 1)/2, so that . All further details are in Section I of Supplementary Methods. Full size image

Dynamics of de Sitter causets and complex networks

Is there a connection revealing a mechanism responsible for the emergence of this structural similarity? Remarkably, the answer is yes. This mechanism is the optimization of trade-offs between popularity and similarity, shown to accurately describe the large-scale structure and dynamics of some complex networks, such as the Internet, social trust network, etc32. The following model of growing networks, with all the parameters set to their default values, formalizes this optimization in Ref. 32. New nodes n in a modeled network are born one at a time, n = 1, 2, 3, …, so that n can be called a network time. Each new node is placed uniformly at random on circle . That is, the angular coordinates θ n for new nodes n are drawn from the uniform distribution on [0, 2π]. Circle models a similarity space. The closer the two nodes on , the more similar they are. All other things equal, the older the node, the more popular it is, the higher its degree. Therefore birth time n of node n models its popularity. Upon its birth, new node n 0 optimizes between popularity and similarity by establishing its fixed number m of connections to m existing nodes n < n 0 that have the minimal values of the product nΔθ, where is the angular distance between nodes n and n 0 . One dimension of this trade-off optimization strategy is to connect to nodes with smaller birth times n (more popular nodes); the other dimension is to connect to nodes at smaller angular distances Δθ (more similar nodes). After placing each node n at radial coordinate r n = ln n, all nodes are located on a two-dimensional plane at polar coordinates (r n , θ n ). For each new node n 0 , the set of nodes minimizing nΔθ is identical to the set of nodes minimizing , where x is equal to the hyperbolic distance33 between nodes n 0 and n if r, r 0 and Δθ are sufficiently large. One can compute the expected distance from node n 0 to its mth closest node and find that this distance is equal to , where approximations hold for large n 0 . In other words, new node n 0 is born at a random location on the edge of an expanding hyperbolic disc of radius r 0 = ln n 0 and connects to asymptotically all the existing nodes lying within hyperbolic distance r 0 from itself. The connectivity perimeter of new node n 0 at time n 0 is thus the hyperbolic disc of radius r 0 centered at node n 0 . The resulting connection condition x < r 0 , satisfied by nodes n to which new node n 0 connects, can be rewritten as

This model yields growing networks with power-law degree distribution P(k) ~ k–γ and γ = 2. The networks in the model also have strongest possible clustering, i.e. the largest possible number of triangular subgraphs, for graphs with this degree distribution. The model and its extensions describe the large-scale structure and growth dynamics of different real networks with a remarkable accuracy32. We next show that the described network growth dynamics is asymptotically identical to the growth dynamics of de Sitter causets.

To show this, consider a new spacetime quantum P that has just been born at current time t = t 0 in Fig. 2. That is, assume that the whole de Sitter spacetime is sprinkled by nodes with a uniform density, but only nodes between t = 0 and t = t 0 are considered to be “alive.” We can then model causet growth as moving forward the current time boundary t = t 0 one causet element P at a time. By the causet definition, upon its birth, P connects to all nodes in its past light cone shown by green. As illustrated in Fig. 2, we then map the upper half of the outer one-sheeted hyperboloid representing the half of de Sitter spacetime with t > 0, to the upper sheet of the dual two-sheeted inner hyperboloid, which is the standard hyperboloid representation of the hyperbolic space 34. This mapping sends a point with coordinates (t, θ) in to the point with coordinate (r, θ) in , where r = t. Since in the conformal time coordinates the light cone boundaries are straight lines intersecting the (η, θ)-axes at 45°, the coordinates (t, θ) of all points in P's past light cone satisfy inequality . If , then we can neglect the second term in the last expression and the coordinates (t n , θ n ) of existing causet nodes n to which new node P connects upon its birth are given by

which is identical to Eq. (9) since r n = t n . In Section II of Supplementary Notes we fill in further details of this proof, extend it to any dimension and curvature and show that the considered mapping between de Sitter spacetime and hyperbolic space is relativistically invariant.

In short, past light cones of new nodes, shown by green in Fig. 2, are asymptotically equal (Fig. 2(d-f)) to the corresponding hyperbolic discs, shown by red. The green light cone bounds the set of nodes to which node P connects as a new causet element. The red hyperbolic disc bounds the set of nodes to which P connects as a new node in the hyperbolic network model that accurately describes the growth of real networks. Since these two sets are asymptotically the same, we conclude that not only the structure, but also the growth dynamics of complex networks and de Sitter causets are asymptotically identical.