Cosmological networks

Networks often represent systems that do not have a long history of studies in traditional fields of physics, albeit there are some notable exceptions such as energy landscapes and quantum gravity. Here we consider networks that naturally arise in cosmology. Nodes in these networks are stationary observers uniformly distributed in an expanding open FLRW universe with any scale factor, and two observers are connected if one can causally influence the other. We show that these networks are growing Lorentz-invariant graphs with power-law distributions of node degrees. These networks encode maximum information about the observable universe available to a given observer.

Here we add to this relatively short list of complex physical networks, a class of networks that naturally arise in cosmology. Specifically, we consider evolving networks of causal connections among stationary (comoving) observers, homogeneously distributed in any open Friedmann-Lemaître-Robertson-Walker (FLRW) spacetime [9]. These networks are purely classical. Nodes can represent a dust of classical particles, or galaxies, or indeed imaginary observers, scattered randomly throughout the space. The horizons of all the observers expand, and for any particular observer O at any given proper time τ , the network consists of all other observers within O's horizon, up to a certain cut-off time τ ν > 0 in the past, which can be interpreted as the Planck time, or the time of last scattering, or the red shift beyond which the observer cannot observe [9]. A directed link from observer B to observer A in this network exists if B is within A's retarded horizon, i.e., if B was within A's a) discretizations [2] of relativistic spacetimes [3]. Given a spacetime, one can construct a causal set corresponding to it by sprinkling points uniformly at random over the spacetime, linking all pairs of causally related points. Causal sets corresponding to asymptotically de Sitter spacetimes [4], such as our accelerating universe [5,6], are power-law graphs with strong clustering [7], similar to many complex networks [8][9][10]. More surprisingly, the growth dynamics of de Sitter causal sets [11,12] and complex networks [13] are asymptotically identical [7]. This similarity suggests a possibility to derive the fundamental laws of complex network dynamics, currently unknown, from an analogy to gravitation. Yet the similarity between causal sets and complex networks is not perfect. In particular, unlike causal sets, nodes in complex networks evolve in time, and links between existing nodes appear [8][9][10].
These observations motivate us here to consider evolving networks of causal connections among stationary observers, homogeneously distributed in any open Friedmann-Lemaître-Robertson-Walker (FLRW) spacetime [14]. The picture is purely classical. Nodes can be thought of as a dust of classical particles, or galaxies, or indeed imaginary observers, scattered randomly throughout the space. The universe expands, and so do the horizons of all the observers. For any particular observer O at any given time τ , the network consists of all other observers within O's horizon, up to a certain cut-off time τ ν in the past, which can be interpreted as the Planck time, or the time of last scattering, or the red shift beyond which the observer cannot observe [14]. A directed link from observer B to observer A in this network exists if B is within A's retarded horizon, i.e., if B was within A's horizon in time for A to signal B's existence to O by time τ , Fig. 1. That is, any directed path in this network represents a causal chain, or a sequence of events that could influence O at time τ . In what follows we show that this evolving network of maximum information about the universe that any observer a) b) c) These observations motivate us he evolving networks of causal connections ary observers, homogeneously distribut Friedmann-Lemaître-Robertson-Walker time [14]. The picture is purely classi be thought of as a dust of classical par ies, or indeed imaginary observers, scat throughout the space. The universe exp the horizons of all the observers. For an server O at any given time τ , the netw all other observers within O's horizon, cut-off time τ ν in the past, which can b the Planck time, or the time of last sc red shift beyond which the observer cann A directed link from observer B to obs network exists if B is within A's retard if B was within A's horizon in time for existence to O by time τ , Fig. 1. That path in this network represents a causa quence of events that could influence O what follows we show that this evolving n imum information about the universe th These observations motivat evolving networks of causal con ary observers, homogeneously Friedmann-Lemaître-Robertson time [14]. The picture is pur be thought of as a dust of cla ies, or indeed imaginary obser throughout the space.  [mb: please make all edges bl but then in b and c, overlay them on top of thick blue edges representing paths between X and O. label the starting node by X everywhere. and th make the plot readable in b&w. and then adjust t caption. -dk-] to A. The link between O and A is bi-directed bec they lie within each other horizons.
From the perspective of the hyperbolic plane, the h zon of the observer O is a disk of radius χ h , whereas horizon of observer A is a disk of radius χ h − χ cent at A who is located at radial coordinate χ. This dis tangent to O's horizon as illustrated in Fig. 2(b). The erage in-degree of observer A is thus given by the num of points within a disk of radius χ h − χ: On the other hand, since the points are distributed formly according to the hyperbolic metric, the densit points located at radial coordinate χ is given by di bution Therefore, for large networks (χ h 1) the average degree scales as k in ∼ πδχ h ≈ πδ ln (N/πδ) whe the in-degree distribution is discretizations [2] of relativistic spacetimes [3]. G a spacetime, one can construct a causal set corresp ing to it by sprinkling points uniformly at random the spacetime, linking all pairs of causally related po Causal sets corresponding to asymptotically de S spacetimes [4], such as our accelerating universe [5 are power-law graphs with strong clustering [7], sim to many complex networks [8][9][10]. More surprisingly growth dynamics of de Sitter causal sets [11,12] and plex networks [13] are asymptotically identical [7]. similarity suggests a possibility to derive the fundam laws of complex network dynamics, currently unkn from an analogy to gravitation. Yet the similarity tween causal sets and complex networks is not per In particular, unlike causal sets, nodes in complex works evolve in time, and links between existing n appear [8][9][10].
These observations motivate us here to con evolving networks of causal connections among sta ary observers, homogeneously distributed in any Friedmann-Lemaître-Robertson-Walker (FLRW) sp time [14]. The picture is purely classical. Nodes be thought of as a dust of classical particles, or g ies, or indeed imaginary observers, scattered rand throughout the space. The universe expands, and s the horizons of all the observers. For any particula server O at any given time τ , the network consis all other observers within O's horizon, up to a ce cut-off time τ ν in the past, which can be interprete the Planck time, or the time of last scattering, or red shift beyond which the observer cannot observe A directed link from observer B to observer A in network exists if B is within A's retarded horizon, if B was within A's horizon in time for A to signa existence to O by time τ , Fig. 1. That is, any dire path in this network represents a causal chain, or quence of events that could influence O at time τ what follows we show that this evolving network of m imum information about the universe that any obs b)

a) b) c)
These observations motivate evolving networks of causal con ary observers, homogeneously d Friedmann-Lemaître-Robertsontime [14]. The picture is pure be thought of as a dust of class ies, or indeed imaginary observ throughout the space. [mb: please make but then in b and c, overlay them on blue edges representing paths between label the starting node by X everywh make the plot readable in b&w. and t caption. -dk-] to A. The link between O and A is bithey lie within each other horizons.
From the perspective of the hyperboli zon of the observer O is a disk of radius horizon of observer A is a disk of radius at A who is located at radial coordinat tangent to O's horizon as illustrated in F erage in-degree of observer A is thus giv of points within a disk of radius On the other hand, since the points are formly according to the hyperbolic metr points located at radial coordinate χ i bution Therefore, for large networks (χ h 1 degree scales as k in ∼ πδχ h ≈ πδ ln the in-degree distribution is [2] of relativistic spac a spacetime, one can construct a caus ing to it by sprinkling points uniform the spacetime, linking all pairs of caus Causal sets corresponding to asymp spacetimes [4], such as our accelerati are power-law graphs with strong clu to many complex networks [8][9][10]. Mo growth dynamics of de Sitter causal set plex networks [13] are asymptotically similarity suggests a possibility to deriv laws of complex network dynamics, c from an analogy to gravitation. Yet tween causal sets and complex netwo In particular, unlike causal sets, node works evolve in time, and links betw appear [8][9][10]. These observations motivate us evolving networks of causal connectio ary observers, homogeneously distrib Friedmann-Lemaître-Robertson-Walke time [14]. The picture is purely cla be thought of as a dust of classical p ies, or indeed imaginary observers, sc throughout the space. The universe e the horizons of all the observers.   horizon in time for A to signal B's existence to O by time τ . That is, assuming there are some physical processes running at each observer, directed paths between observers X and O in this network represent causal relations between X and O, albeit these relations are indirect if the path is longer than one hop, Fig. 1. In what follows we show that this evolving network of maximum information about the universe that any observer can collect by her proper time τ , is a growing power-law graph in any open homogeneous and isotropic (FLRW) spacetime.

The metric in an open FLRW spacetime is given by
where τ > 0 and χ > 0 are the cosmic time and "radial" coordinates, dΩ 2 d−1 is the metric on the unit (d − 1)dimensional sphere, and R(τ ) is the scale factor of the universe given by the Friedmann equations [9]. The "edge of the universe" corresponds to particles receding from O at the speed of light. This edge is thus a circle of radius R edge = τ centered at O. Observer O observes not all particles within this edge, since particles are "lit" not at τ = 0 but at τν > 0. These events lie on the invariant hyperboloid t 2 = τ 2 ν + x 2 + y 2 shown in blue. The horizon of any given observer is then induced by the intersection of her past light cone with this hyperboloid, and defines the maximum speed of a particle within the horizon. In particular, the radius of O's horizon in the (x, y) plane is R horizon = τ [1 − (τν /τ )]/[1 + (τν /τ )] (τν = 1.5 and τ = 5 in the figure). The thick red arrows show the world-lines of stationary observers O and also A who is at rest at radial coordinate χ = const. The retarded horizon of observer A at proper time τχ is induced by the intersection of A's past light cone with the blue hyperboloid. Projected into the (x, y) plane, this retarded horizon encompasses all the observers that can causally influence O indirectly via A, Fig. 1. Observer O has incoming connections from observers A, B, and C since they all lie within O's horizon. Observer A has incoming connections from O and B, but not from C who is outside A's horizon. is the hyperbolic d-dimensional space of constant curvature K = −1/R(τ ). To simplify the calculations, we assume that R(τ ) = τ , meaning that we are considering the Milne universe-a completely empty universe without any matter or dark energy [10]. The results presented henceforth do not depend on a particular form of scale factor R(τ ). We discuss this important point at the end. In (2 + 1) dimensions (the generalization to (d + 1) with d > 2 is straightforward), the change of coordinates (τ, χ, θ) to x = τ sinh χ cos θ y = τ sinh χ sin θ t = τ cosh χ (2) transforms the metric in Eq. (1) into the Minkowski metric However, this transformation does not map the original spacetime in Eq. (1) to the whole Minkowski spacetime, but only to the future light cone of the event t = x = y = 0. Indeed, the radial Minkowski coordinate r = x 2 + y 2 of an event at coordinates (τ, χ, θ) is r = t tanh χ. This means that a stationary observerthat is, an observer at rest in the co-moving coordinates (χ, θ) in H 2 -is receding from the origin x = y = 0 at constant speed v = tanh χ ≤ 1. Consistent with homogeneity and isotropy of the universe, we assume that stationary observers are also homogeneously and isotropically distributed throughout space with constant density δ. These observers are therefore points distributed in the hyperbolic space H 2 according to a Poisson point process with point density δ. In the Milne cosmology, an infinite number of such observers are thus initially at the origin of coordinates (the big bang), and then they all start moving in all directions within a bubble -in the considered case this bubble is a disk in R 2 -that expands at the speed of light (see the (x, y) plane in Fig. 2). Because the distribution of observers is uniform in H 2 , any stationary observer will "see" all other observers receding from her with the Lorentz-invariant density of speeds v Without loss of generality or breaking Lorentz invariance, in what follows we focus on the stationary observer O at rest at coordinate χ = 0, and therefore also at rest at x = y = 0. According to Eq. (2), O's proper time τ is equal to the time coordinate t in the Minkowski spacetime. First, we determine the horizon of O at any given proper time τ . This horizon is the radius of the part of the universe that O can observe, up to the past cut-off time τ ν , which can be any positive number, 0 < τ ν < τ . This radius is determined by the intersection of O's past light cone with the hyperboloid at time τ ν , Fig. 2. At time τ > τ ν , the farthest particle that O can observe is moving at a speed such that light emitted at proper time τ ν reaches O at this time τ , yielding the following simple expression for the hyperbolic radius of O's horizon: The number of other observers that O can observe is then the number of points within a hyperbolic disk of radius χ h , growing asymptotically linearly with time τ : Any two observers A and B in O's horizon are connected by a directed link from B to A if B lies within the retarded horizon of A. If A's radial coordinate is χ, then the retarded horizon of A is defined as its horizon at time τ χ = τ e −χ . According to Eq. (5), τ χ is such that if A emits light at her proper time τ χ , then this light reaches O at time τ . This means that if A has some physical state (possibly causally influenced by B) at time τ χ , then this state can causally influence O by time τ . Figure 2 shows observer A lying within the horizon of observer O. Observer B is connected to A because B lies within A's retarded horizon at time τ χ , the latest time in A's history that can influence O at time τ . Observer C is outside of this horizon and therefore is not connected to A. The link between O and A is bi-directed because they lie within each other horizons.
From the perspective of the hyperbolic plane, the horizon of the observer O is a disk of radius χ h , whereas the horizon of observer A is a disk of radius χ h − χ centered at A who is located at radial coordinate χ. This disk is tangent to O's horizon as illustrated in Fig. 3. The average in-degree of observer A is thus given by the number of points within a disk of radius χ h − χ: On the other hand, since observers are distributed uniformly according to the hyperbolic metric, the density of them located at radial coordinate χ is given by distribution Therefore, for large networks (χ h 1) the average indegree scales as k in ∼ πδχ h ≈ πδ ln (N/πδ) whereas the in-degree distribution is that is, a scale-free network with exponent γ = −2, similar to many real complex networks. The average out-degree of a node located at (χ, 0) is given by the number of points within a domain in H 2 defined as the locus of points (χ , θ) such that their hyperbolic distances to the point (χ, 0), x, are smaller than the radius of their retarded horizons χ h − χ , that is, where Θ(·) is the Heaviside step function. In the limit χ h 1, the integration yields where K(·) is the complete elliptic integral of the first kind. In the regime 1 < χ < χ h , the average out degree is well approximated bȳ This result, combined with Eq. (8), implies that the outdegree distribution scales as with logarithmic corrections. We notice however that observers near (but not exactly at) the edge of the horizon have out-degrees approximately equal to χ h . Therefore, the out-degree distribution is asymptotically a power law with a lower cut-off that grows as χ h with time. New connections in this network appear not only between new and existing nodes, but also between pairs of already existing nodes, not previously connected. This effect is a simple consequence of the continuous expansion of the horizons of all observers. The resulting network dynamics is illustrated in Fig. 4, where three snapshots of a growing network are taken. The horizon of the central observer O (the blue dashed circle) grows over time, discovering an exponentially increasing number of new observers. Gray connections indicate purely directed causal relations between observers, that is, one is aware of the other. As time goes on, directed connections are reciprocated (connections in red), meaning that an increasing number of pairs of observers are getting mutually aware of each other.
Up to this point, we have assumed that all observers entering the horizon of another observer are detected. If we assume that the probability of connection between observers decays exponentially with the hyperbolic distance x between them, p(x) ∝ e −βx , where parameter β ∈ [0, 1), then the average in-degree of an observer at coordinate χ is As a consequence, the in-degree distribution scales asymptotically as a power law P (k in ) ∼ k −γ in with exponent which can take any value between 2 and ∞, as shown in Fig. 5. Finally we emphasize that the very same network construction can be extended beyond the Milne universe to any open FLRW universe with any scale factor R(τ ). The same picture shown in Fig. 3 would apply there. The only difference is the rate at which new nodes join the network, defined by the radius of the observer's horizon as a function of time. In general, this radius is generalizing Eq. (5).
In summary, the physical network of (indirect) causal relations between observers uniformly distributed in any open FLRW universe is a Lorentz-invariant scale-free graph with strong clustering, Fig. 5. This network is the network of maximal information about the universe that any particular observer can collect by a certain time. More precisely, paths in this network are all possible communication channels between observers. Perhaps coincidentally, in the perfect case without information loss (β = 0), this network has the same statistical properties (γ = 2 and strong clustering) as the maximally navigable networks [11], i.e., networks that are most conductive with respect to targeted information signaling. The crucial requirement for this coincidence is that the universe must be open, Eq. (1). Bubble universes are open in most inflationary cosmologies [12], and the current measurements of our universe do not preclude that it is open either, although it is definitely close to being flat [13].
Here we have considered an idealized case of massless points distributed uniformly in the space. It remains unclear how the picture would change if points have masses, perhaps distributed according to some heterogeneous distributions similar to the distribution of the masses of galaxies in the universe [14]. and if the spatial distribution of points deviates from uniform, as it does for galaxies [15] and for complex networks embedded in hyperbolic spaces [16].