Abstract
Prediction and control of the dynamics of complex networks is a central problem in network science. Structural and dynamical similarities of different real networks suggest that some universal laws might accurately describe the dynamics of these networks, albeit the nature and common origin of such laws remain elusive. Here we show that the causal network representing the largescale structure of spacetime in our accelerating universe is a powerlaw graph with strong clustering, similar to many complex networks such as the Internet, social, or biological networks. We prove that this structural similarity is a consequence of the asymptotic equivalence between the largescale growth dynamics of complex networks and causal networks. This equivalence suggests that unexpectedly similar laws govern the dynamics of complex networks and spacetime in the universe, with implications to network science and cosmology.
Introduction
Physics explains complex phenomena in nature by reducing them to an interplay of simple fundamental laws. This very successful tradition seems to experience certain difficulties in application to complex systems in general and to complex networks in particular, where it remains unclear if there exist some unique universal laws explaining a variety of structural and dynamical similarities found in many different real networks^{1,2,3,4,5,6,7}. One could potentially remedy this situation by identifying a wellunderstood physical system whose largescale dynamics would be asymptotically identical to the dynamics of complex networks. One could then try to use the extensively studied dynamical laws of that physical system to predict and possibly control the dynamics of networks. At the first glance, this program seems to be quite difficult to execute, as there are no indications where to start. Yet we show here that there exists a very simple but completely unexpected connection between networks and cosmology.
In cosmology, de Sitter spacetime plays a central role as the exact solution of Einstein's equations for an empty universe, to which our universe asymptotically converges. Here we show that graphs encoding the largescale causal structure of de Sitter spacetime and our universe have structure common to many complex networks^{8,9,10} and that the largescale growth dynamics of these causal graphs and complex networks are asymptotically the same. To show this, we describe the causal graphs first.
The finite speed of light c is a fundamental constant of our physical world, responsible for the nontrivial causal structure of the universe^{11}. If in some coordinate system the spatial distance x between two spacetime events (points in space and time) is larger than ct, where t is the time difference between them, then these two events cannot be causally related since no signal can propagate faster than c (Fig. 1(a)). Causality is fundamental not only in physics, but also in fields as disparate as distributed systems^{12,13} and philosophy^{14}.
The main physical motivation for quantum gravity is that at the Planck scale (l_{P} ~ 10^{–35} meters and t_{P} ~ 10^{–43} seconds), one expects spacetime not to be continuous but to have a discrete structure^{15}, similar to ordinary matter, which is not continuous at atomic scales but instead is composed of discrete atoms. The mathematical fact that the structure of a relativistic spacetime is almost fully determined by its causal structure alone^{16,17,18} motivates the causal set approach to quantum gravity^{19}. This approach postulates that spacetime at the Planck scale is a discrete causal set, or causet. A causet is a set of elements (Planckscale “atoms” of spacetime) endowed with causal relationships among them. A causet is thus a network in which nodes are spacetime quanta and links are causal relationships between them. To make contact with General Relativity, one expects the theory to give rise to causal sets which are constructed by a Poisson process, i.e. by sprinkling points into spacetime uniformly at random and then connecting each pair of points if they lie within each other's light cones (Fig. 1(b)). According to the theorem in Ref. 20, causets constructed by Poisson sprinkling are relativistically invariant, as opposed to regular lattices, for example. Therefore we will use Poisson sprinkling here to construct causets corresponding to spacetimes. An important goal in causal set quantum gravity (not discussed here) is to identify fundamental physical laws of causet growth consistent with Poisson sprinkling onto realistic spacetimes in the classical limit^{21,22}.
In 1998 the expansion of our universe was found to be accelerating^{23,24}. Positive vacuum energy, or dark energy, corresponding to a positive cosmological constant Λ in the Einstein equations (Supplementary Notes, Section II), is currently the most plausible explanation for this acceleration, even though the origin and nature of dark energy is one of the deepest mysteries in contemporary science^{25}. Positive Λ implies that the universe is asymptotically (at late times) described by de Sitter spacetime^{26,27}. We first consider the structure of causets sprinkled onto de Sitter spacetime, then quantify how different this structure is for the real universe and finally prove the asymptotic equivalence between the growth dynamics of de Sitter causets and complex networks.
Results
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 onesheeted 2dimensional hyperboloid embedded in a flat 3dimensional 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 powerlaw degree distribution in the causet.
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 twodimensional) 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 indegrees 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 time^{26} η(t) = arcsec cosh t. After this coordinate change, the metric becomes conformally flat, i.e. proportional to the metric ds^{2} = –dt^{2} + dx^{2} 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 outdegree are negligible compared to the future volume and indegree, 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/a^{2}, 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 indegree in this case scale as and with α = β = d/a. In short, we have a combination of two exponentials, number of nodes ~ e^{αt} born at time t and their degrees ~ e^{–βt}. This combination yields a powerlaw 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 largescale 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 Sitter^{26,27}, 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 ds^{2} = –dt^{2} + R^{2}(t)dΩ^{2}, 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) ~ sinh^{2/3} (3τ/2). In both cases, R(t) ~ e^{τ} 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 a^{4} = 1/K^{2} and sprinkling density (one causet element per unit Planck 4volume) is astronomically huge in the universe, δa^{4} ~ 10^{244}, compared to in sparse causets with a small average degree. Collectively these two differences result in that the present universe causet is also a powerlaw graph, but with a different exponent γ = 3/4 (Fig. 3(a)).
However, the γ = 2 scaling currently emerges (Fig. 3(b)) as a part of a cosmic coincidence known as the “why now?” puzzle^{28,29,30,31}. 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 matterdominated era with τ < 1, the degree distribution is a power law with exponent 3/4 up to a soft cutoff that grows with time. Above this soft cutoff, the distribution decays sharply. Once we reach times τ ~ 1, e.g. today, we enter the darkenergydominated era. The part of the distribution with exponent 3/4 freezes, while the soft cutoff transforms into a crossover to another power law with exponent 2, whose cutoff grows exponentially with time. The crossover point is located at k_{cr} ~ δa^{4}. Nodes of small degrees k < k_{cr} obey the γ = 3/4 part of the distribution, while highdegree 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 presentday universe is that it is a graph with a powerlaw degree distribution, which currently transitions from the past matterdominated era (τ < 1) with exponent γ = 3/4 to the future darkenergydominated 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 2^{8,9,10}. In Fig. 4(a) we show a few paradigmatic examples of largescale 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. Degreedependent 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 magnitude^{8,9,10}.
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 tradeoffs between popularity and similarity, shown to accurately describe the largescale structure and dynamics of some complex networks, such as the Internet, social trust network, etc^{32}. 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 tradeoff 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 twodimensional 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 distance^{33} 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 powerlaw 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 largescale structure and growth dynamics of different real networks with a remarkable accuracy^{32}. 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 onesheeted hyperboloid representing the half of de Sitter spacetime with t > 0, to the upper sheet of the dual twosheeted 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(df)) 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.
Discussion
Geometrically, this equivalence is due to a simple duality between the two hyperboloids in Fig. 2. The inner hyperboloid represents the popularity×similarity hyperbolic geometry of complex networks; the outer hyperboloid is the de Sitter spacetime, which is the solution to Einstein's equations for a universe with positive vacuum energy. In that sense, Einstein's equations provide an adequate baseline description for the structure and dynamics of complex networks, which can be used for predicting network dynamics at the large scale. De Sitter spacetime is homogeneous and isotropic, as is the hyperbolic space, but if we take a real network, e.g. the Internet and map it to this homogeneous space, then after the mapping, the node density in the space is nonuniform^{35}. In real networks, the space thus appears homogeneous only at the largest scale, while at smaller scales there are inhomogeneities and anisotropies, similar to the real universe, in which matter introduces spacetime inhomogeneities at smaller scales and leads to nontrivial coupled dynamics of matter density and spacetime curvature, described by the same Einstein equations. In view of this analogy, equations similar to Einstein's equations may also apply to complex networks at smaller scales. If so, these equations can be used to predict and possibly control the finegrained dynamics of links and nodes in networks.
Our results may also have important implications for cosmology. In particular, de Sitter causal sets have exactly the same graph structure that maximizes network navigability^{36}. Translated to asymptotically sparse causal sets, does this property imply that the expanding portion of de Sitter spacetime (t > 0) is the spacetime that maximizes the probability that two random Planckscale events have an ancestor in their common past? If it does, then this uniqueness of de Sitter spacetime may lead to a different perspective on the cosmic coincidence problem, as well as on dark energy, possibly casting the latter as a phenomenon emerging from certain optimization principles encoded in the causal network structure.
The degree distributions in some complex networks deviate from clean power laws, the exponents of these power laws vary a lot across different real networks and so do clustering, correlation and many other structural properties of these networks^{8,9,10}. Therefore it may seem unlikely that de Sitter causets can model the full spectrum of structural diversity observed in complex networks. Focusing on the trust network in Fig. 4 for instance, we have already observed that its degreedependent clustering is quite different from the one in de Sitter causets. Yet, given that these causets are asymptotically identical to growing hyperbolic networks, this observation appears as a paradox, because the hyperbolic networks were shown to accurately match not only clustering peculiarities, but also a long list of other important structural properties of the same trust network, as well as of other networks^{32}. The explanation of this paradox lies in the fact that the hyperbolic network model has parameters to tune the degree distribution exponent, clustering strength, node fitness and other network properties, while in de Sitter causets, only the number of nodes and average degree can be controlled. Do the hyperbolic network parameters have their duals in the de Sitter settings, what are the physical meanings of these dual parameters and do they lead to similar modeling versatility—all these questions are open.
We conclude with the observation that the node density in growing hyperbolic networks with the default parameters corresponding to de Sitter causets, is not uniform in the hyperbolic space^{32}. This observation means that these networks are not random geometric graphs^{37} and that their structure does not exactly reflect the geometry of the underlying hyperbolic space. Informally, a random geometric graph is a coarse, discrete representation of a smooth geometric space. Our finding that asymptotically the same networks have a uniform node density in de Sitter spacetimes dual to hyperbolic spaces, strongly suggests that real networks are random geometric graphs that grow in spacetimes similar to the asymptotically de Sitter spacetime of our accelerating universe.
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Acknowledgements
We thank M.Á. Serrano, M. Norman, Z. Toroczkai, A.L. Barabási, J. Garriga, A. Vilenkin, R. Sorkin, G. Gibbons, F. Papadopoulos, G. Bianconi, A. Petersen, L. Braunstein, F. Bonahon, kc claffy, M. Fomenkov, C. Carlson, L. Klushin, S. Paston, V.D. Lyakhovsky and A. Krioukov for useful discussions and suggestions. Special thanks to D. Chialvo for sharing his brain data with us. We also thank M. Norman for providing computing time at SDSC through the director's discretionary fund and to J. Cheng and B. Huffaker for their help with Fig. 1. This work was supported by DARPA grant No. HR00111210012; NSF grants No. CNS0964236 and CNS1039646; Cisco Systems; Foundational Questions Institute grant No. FQXiRFP31018; George W. and Carol A. Lattimer Campus Professorship at UCSD; MICINN project No. FIS201021781C0202; Generalitat de Catalunya grant No. 2009SGR838; and by the ICREA Academia prize, funded by the Generalitat de Catalunya.
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D.K., D.R., D.M. and M.B. designed research; D.K., M.K. and M.B. performed research; D.K., M.K., R.S. and M.B. analyzed data and performed simulations; D.K., D.R. and M.B. wrote the manuscript; all authors discussed the results and reviewed the manuscript.
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Krioukov, D., Kitsak, M., Sinkovits, R. et al. Network Cosmology. Sci Rep 2, 793 (2012). https://doi.org/10.1038/srep00793
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DOI: https://doi.org/10.1038/srep00793
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