Networks as the fundamental constituents of the Universe

A first discrete measure of graph curvature was introduced by Forman [21, 22] using the topological constructs of CW (Closure-finite, Weak) cell complexes. In contrast to Regge calculus [59] and similar curvature measures used in the discretization of manifolds [8, 9], this curvature is not defined geometrically but only combinatorially. Its formulation draws upon an analogy with identities developed by Bochner [60] concerning the decomposition of the Riemannian-Laplace operator on the space of p-forms, defined for a manifold M. This decomposition yields a covariant derivative and a curvature correction known as the Bochner–Weitzenböck identity, whose discrete form is used to define the Forman–Ricci (FR) curvature.

Intuitively, a CW complex is a space made by gluing together simple building blocks of different dimensions (points, segments, disks, balls etc) along their boundaries. Formally, CW complexes (for a review see [61]) are constructed from p-cells (p denoting the dimension of the cells). A d-dimensional CW complex is made by gluing complexes along shared faces. Thus, a 1-complex, encompassing only nodes, i.e. 0-cells and links, i.e. 1-cells, is a graph. The original FR of an edge, thus, can depend at most on the degrees of the two vertices defining the edge. While this is a computationally simple construct, it is by far too coarse to capture the ‘geometry’ of networks.

The Forman curvature can by ‘augmented’ by considering graphs as 2-complexes, with graph cycles assumed to bound 2-cells [62]. These proposals, however focus typically only on triangles as 2-cells [63], which is still a too strong limitation when considering a discretization of Ricci curvature. Ricci curvature involves second derivatives of the metric, which is itself a measure of infinitesimal distances. The graph analogue of a metric is an edge, while the discrete graph equivalent of derivates up to second order involves up to the second-nearest neighbours of the two vertices defining this edge. Therefore, a more appropriate ‘enhancement’ of FR of graphs, treated as 2-complexes, should involve triangles, squares and pentagons [64]. This is essentially the discrete version of locality on graphs.

The boundary of a p-cell is the ensemble of -cells that ‘bound’ the cell. For a 1-cell, a graph edge, the boundary is the collection of the two vertices defining the edge. A general p-cell, α_p is a proper face of a -cell β if it is a member of the boundary set of β, and we write . A p-cell CW complex M over , is defined formally as a collection of cells , such that any two cells are joined along a common proper face, and all faces are contained in the cell complex.

A further important concept about cell complexes is the definition of the neighbours of a given p-cell [21, 22].

Definition 1. Let α₁ and α₂ be p-cells of a complex M. α₁,α₂ are neighbours if:

1. 1.  
   α₁ and α₂ share a -cell β such that and , or
2. 2.  
   α₁ and α₂ share a -cell γ such that and .

We can thus partition the set of neighbours of a cell into parallel and non-parallel ones. Two p-cells α₁,α₂ are parallel neighbours if one but not both of the conditions in definition 1 are true, in which case we write .

Using these concepts, one can define a series of maps , for each value of p,

where ε_q is a q-cell that is a parallel neighbour of α_p and q ≠ p. The symbol is intended to denote the number of such cells satisfying the condition in braces. Essentially this definition computes as the number of -cells that bound α_p, plus the number of -cells of which α_p is part of the boundary minus the number of parallel neighbours of α_p. For p = 1, defines the Forman curvature FR of edge e_ij. In this case, equation (1) is particularly simple; the vertices and edges constitute the 0- and 1-cells and the closed loops in the graph constitute the 2-cells. In equation (1) there is no arbitrary restriction to the length of cycles that are admissible as bounding a 2-cell. The so-called ‘augmented’ FR is obtained by restricting these cycles to triangles. The geometrically more appropriate ‘enhanced’ FR [64] is obtained by truncating the expansion to cycles of maximum length 5.

Another very important property of CW complexes is that of quasi-convexity [22].

Definition 2. A CW-complex M is quasi-convex if, for each pair of -cells α₁ and α₂ (viewed as topologically closed entities including their boundaries), if contains a p-cell β, then .

In the case of graphs, viewed as 2-complexes, this property reduces to the simple condition that any two cycles can share at most one edge (two edges share anyway always one link). Moreover, in view of the fact that only cycles of length up to 5 are relevant for local properties on the graph, we shall consider a reduced notion of quasi-convexity of graphs that imposes the condition of maximally one shared edge only for triangles, squares and pentagons. As we now show, this condition not only renders the computation of various combinatorial curvature measures tractable but it is a very physical condition representing a sort of ‘incompressibility’ of graphs; it was independently introduced in [28] and [29] where it was called ‘hard core condition’ or ‘condition of independent short cycles’.

For quasi-convex graphs, satisfying the independent short cycles condition, the prohibitive expression equation (1) reduces to the simple form

where k_i denotes the degree of vertex i and , and denote the number of triangles, squares and pentagons supported on edge (ij), respectively [64]. This further simplifies for k-regular quasi-convex graphs,

These expressions show that, when degrees do not vary too much, combinatorial curvature is mainly determined by the number of local loops based on an edge, triangles contributing the most and pentagons the least.

There are several variants of Ollivier combinatorial curvature (OR) [23–27]. Here we shall discuss only the simplest one, to be used to formulate GR on networks. Its basic idea is to reproduce in a discrete setting the geometric meaning of Ricci curvature.

While often introduced as a derivative expression of the metric, Ricci curvature on manifolds is a measure of how much infinitesimal spheres around a point contract (positive Ricci curvature) or expand (negative Ricci curvature) when they are parallel-transported along a geodesic with a given tangent vector at the point under consideration. The Ollivier curvature is simply a discrete version of the same measure. For two vertices i and j it compares the Wasserstein (or earth-mover) distance between the two uniform probability measures on the spheres around i and j to the graph distance and is defined as

The Wasserstein distance between two probability measures µ_i and µ_j on the graph is defined as

where the infimum has to be taken over all couplings (or transference plans) i.e. over all plans on how to transport a unit mass distributed according to µ_i on the unit sphere around i to the same mass distributed according to µ_j on the unit sphere around j,

While extremely intuitive from a geometric point of view, the OR seems even more cumbersome to compute in the generic case than the FR. This is why easier approximations have been developed [65–67]. Fortunately, however, these are not really necessary since the OR also simplifies substantially on quasi-convex graphs satisfying the independent short cycles condition [28]. In this case it takes the form [29],

where the subscript ‘+’ is defined as and , . For a k-regular quasi-convex graph this becomes

This shows that, when the degree distribution is rather peaked, also the OR of graphs is mostly determined by the number of short loops on an edge, where by short one means triangles, squares and pentagons. In this sense, the ‘geometry’ of graphs is a theory of loops. The quasi-convexity condition of independent short loops requires that such loops can touch along one edge but they cannot overlap more than this. It has thus a very physical interpretation as an incompressibility condition for loops (or graphs themselves), analogous to the hard-core condition for a gas of Bose particles [28]. As the hard-core condition prevents the infinite compressibility catastrophe of a Bose gas of particles, the corresponding hard-core requirement for loops also stabilizes a gas of loops on graphs, as we show below.

The incompressibility condition can be also formulated as an excluded sub-graph condition [29]. In figure 1 we list the subgraphs that must be absent for a graph to satisfy the independent short cycles condition and be thus quasi-convex.

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This leads immediately to another interpretation of quasi-convexity. On a Riemannian manifold, the metric component at a given point p defines a unique infinitesimal surface element on the tangent space in the plane spanned by the two directions i and j. The corresponding condition in the discrete graph setting requires that triangles, squares and pentagons, the fundamental local area elements, be uniquely defined by two edges. The excluded sub-graphs are exactly those for which this condition is not satisfied. In other words, quasi-convex graphs are those that satisfy the necessary condition for admitting a smooth continuum limit leading to a Riemannian manifold.

A general analytical comparison of the Forman and Ollivier curvatures is difficult because it is not simple to disentangle the contributions of the degrees and of the cycles. To make progress we will replace the degrees k_i with their average value , assuming that the degree distribution is sufficiently peaked. We then represent the so obtained OR curvature as a ‘mean field’ term, defined as the expression in equation (8) in which all brackets are simply summed up without taking into account the ‘+’ subscripts, plus a correction term. Essentially, the mean field term represents the dependence on the cycles alone, while the correction embodies deviations dependent on connectivity. This gives

where the correction to the mean field value is given by

This shows that the properly enhanced Forman curvature coincides (up to an overall factor) with the mean field Ollivier Ricci curvature,

This result is remarkable, given that the two discrete curvature constructions have completely different origins. It is also an indication that the combinatorial dependence on the number of cycles is unique for any local discrete curvature measure.

The two curvatures become identical (up to an overall factor) when the quantity I_ij defined by

satisfies , . When , we fall in the first case of equation (10) and the correction term vanishes. When , the second case of equation (10) is realized and the correction term vanishes because it coincides with I_ij. The condition , is realized for graphs with large connectivity and sparse cycles or, e.g. for all regular and semi-regular planar and hyperbolic tessellations involving triangles, squares and pentagons only. For these cases the appropriately enhanced Forman curvature and the Ollivier curvature coincide (up to a factor).

The case is particularly interesting. This is the case for tilings of D-dimensional space, for which the skeleton degree , so that at each vertex one can choose to ‘move’ forward or backwards along D possible directions and where each of the edges supports exactly k − 2 elementary faces, triangles, squares or pentagons . In this case the OR becomes

This demonstrates that, in this geometric framework, triangles contribute positive curvature, squares contribute zero curvature, and pentagons contribute negative curvature. This is intuitively clear: triangles act as elementary units where ‘geodesics’ combinatorially converge, squares where they remain parallel, and pentagons where they diverge.

To further test the remarkable equivalence (11) one can generate a variety of random graphs with varying connectivity and edge density and compute both curvatures numerically to compare them [64]. For this simulation we consider Erdös–Rényi random graphs (see, e.g. [31]) using a varying link probability p in the range 0.01 to 0.1, after which we manually enforce the independent short-cycle condition by removing edges that violate it. For a fixed number N = 100 of vertices this generates graphs with an average degree above the critical threshold for the emergence of a large connected subgraph. As p increases, the edge density of the graph increases, together with the density of short cycles. For each edge we compute I_ij in (12), from which we build an average value: we expect that the two measures of curvature become closer and closer (up to the factor of ) when this average of I_ij approaches zero.

In 2 we plot the average fractional difference between the two curvature measures, over all edges in randomly generated graphs against the average value of I_ij. For each link probability p we generate 10 graphs, to avoid the results being skewed by unusual graph configurations This simulation clearly shows that, here too, the two curvature measures differ when but, as I_ij increases and approaches zero they converge to the same value.

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We consider thus a statistical mechanics model defined as a sum on quasi-convex 2D-regular graphs (QCG) with N vertices. The partition function is

where g is the dimensionless coupling and the Hamiltonian is defined as

where we denote by the neighbour vertices j to vertex i in graph G, i.e. those connected to i by one edge (ij), 2D is the degree of the regular graph and is the Ollivier curvature (4) of edge (ij) (from now on we omit the superscript ‘OR’ since we will use only the Ollivier curvature). Note also that the constraints of quasi-convexity can alternatively be dynamically implemented by adding Kronecker-delta terms to the Hamiltonian [76].

It is important to stress that, contrary to the previously discussed case of RGGs, there is no embedding manifold assumed here. As we will show, however, at small coupling g the dominant graphs in the partition function (20) define themselves a manifold, specifically a surface, as 1-skeletons of 2-cell complexes. For this the constraint of quasi-convexity is a crucial ingredient; it is the free energy minimization under this constraint that leads to the emergence of manifolds.

Using the Ollivier curvature mean field plus correction representation (9) and (10), we can split the Hamiltonian into a global and a local contribution,

where , and are the total numbers of triangles, squares and pentagons on the graph and E₁ and E₂ are the ensembles of edges for which the respective summands are strictly positive.

The global term in (22) corresponds (up to a constant factor) to the Hamiltonian derived from the Forman curvature. It represents the Hamiltonian of a matrix model that depends solely on the total number of cycles in the graph. The energy is maximized for tree-like random regular graphs, which contain no cycles, and decreases as short cycles form. Among these, triangles contribute the most significant energy reduction, while pentagons contribute the least. The quasi-convexity condition, however, implies that a triangle can share an edge only with a pentagon, not with another triangle or a square, see figure 1. But it is energetically favourable to replace a triangle-pentagon couple with two squares, since . Therefore, triangles and pentagons can survive only as isolated defects but are excluded for an homogenous ground state and we can henceforth neglect them. The global Hamiltonian is then minimized by increasing the number of squares. The local Hamiltonian, however penalizes edges for which the number of squares is higher than (). The total expression (8) for the Ollivier curvature (with no triangles and pentagons) shows that these two effects cancel exactly when a configuration with squares per vertex and squares per edge is reached. Configurations with a higher number of squares (the maximum per edge is by regularity) are degenerate with this one. What happens is that, decreasing the coupling from the random configuration with sparse cycles, the number of squares at an edge increases until there are exactly squares based there. After this it pays energetically to increase the number of squares at another edge. This can continue until there are squares at each edge and globally. This is a hypercubic complex, which identifies D as the topological dimension of the ground state. For D = 2, this is a torus graph. The stability of the torus graph ground state has been discussed in detail and analysed numerically in [29]. To make numerical simulations less computationally intensive one can use bipartite graphs, for which there are no odd-length cycles. From here onward, we focus on the case D = 2 for which, as we now show, the ground state encodes the degrees of freedom of a 2D holographic screen, analogous to the framework proposed in Verlinde’s entropic gravity model [16, 17].

If only the global term is retained, the model undergoes a first-order phase transition in which the graph decomposes into isolated, weakly interacting hypercubic complexes [29, 77, 78]. If the full Hamiltonian is used, instead the model undergoes a continuous phase transition from a random graph phase to a geometric phase in which graphs become tessellations of smooth surfaces [28, 29, 34, 51, 52]. This phase transition is due to a condensation of square loops on the graph, while triangles and pentagons are suppressed and survive at best as isolated defects [29]. This is in agreement with a previous observation that the emergence of geometry in random networks is associated with clustering [79]. Random regular graphs are locally tree-like, with very sparse cycles, governed by a Poisson distribution with mean for cycles of length k [80]. The order parameter for the transition is the density of squares , which increases from (for N = 160 in the example below) in the random phase to reach the maximum value 1 for a torus graph. It is shown in figure 3 as a function of .

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The red dots are obtained by starting with a torus graph and gradually increasing the coupling g; the blue dots, instead, correspond to decreasing the coupling from the random phase. Note the absence of hysteresis, as expected for a continuous phase transition. As could be expected, the Monte Carlo algorithm does not always manage to reach an exact ground state with deep in the geometric phase when starting from a random graph configuration. Of course, several other diagnostics for a second order transition have been tested [29], with positive results.

Any finite graph has a genus, which is defined as the smallest genus of a surface on which it can be embedded as a 2-cell complex without any edge crossings [81]. Random graphs of N vertices can be embedded without edge crossings on a Riemann surface of genus N [82]. Therefore, decreasing the coupling from its critical value amounts to decreasing the genus of the emerged manifold until the ground state torus graph at g = 0 is reached. In figure 4 we show the example of a two-torus state at a small but finite value of g in the geometric phase.

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As always in the case of continuous phase transitions, there in an emergent scale. This is the correlation length , which vanishes in the ground state at g = 0 and diverges at the critical point. In the present case, let us consider the connected correlation function

between the number of squares and based on vertices i and j at fixed graph distance d, where denotes the variance (over the vertices) of the number of squares. The typical behaviour of this correlation in the geometric phase, as a function of d (at fixed g) is shown in figure 5.

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At extremely small distances the correlation is high since, given a square based at vertex i, there are always at least two vertices at distance 1 and one vertex at distance 2 that also support a square. At intermediate distances the correlation turns negative, because there is no correlation between squares on these distances but we are in the geometric phase, in which the average square number is large. On these intermediate distances the graph behaves like a random graph. Correlations then suddenly pick up at the distance , before decaying again to zero on even larger distances. In this case, the approach to zero is not due to absence of correlations but, rather, to each vertex supporting the same number of squares . This is the geometric manifold phase on large scales. As usual, the correlation length separates the ordered phase (here the geometric phase), realized at large scales, from the remnants of the disordered phase (here the random phase) at small scales. The graphs in the geometric phase behave thus as geometric manifolds on distances larger than and as random graphs below this scale. To make contact with manifolds, we shall henceforth endow the links of the graph with a uniform length scale and identify the correlation length with the Planck length, . With this definition, the Planck length vanishes at g = 0 whereas at it equals the graph diameter.

Note that, strictly speaking, the continuous phase transition in this model of emergent geometry is fully characterized not just by the mean number of squares, but by the entire distribution of squares across the ensemble. In the random phase, this distribution is a Poisson distribution, with variance equal to its mean, as previously noted. In the geometric phase, the distribution becomes sharply peaked around its mean, approaching a delta function in the limit of zero coupling. Both phases are governed by a finite correlation length, which sets the scale for fluctuations and keeps the variance finite. At the critical point, however, the distribution becomes scale-free, and the variance diverges. This signals the emergence of ensemble fluctuations that include clusters of 4-cycles and random links on all length scales.

2D quantum gravity is often described as both trivial and pathological. It is considered trivial because it lacks local degrees of freedom, only the topological genus of the manifold plays a role. At the same time, it is deemed pathological because large quantum fluctuations cause the geometry to crumple into fractal structures with a Hausdorff dimension . This behaviour is observed in both the continuum approach, such as Liouville theory, and in discrete formulations like matrix models or EDTs (see, e.g. [83, 84] for reviews). In contrast, the model studied here belongs to a different universality class. It features two distinct phases, one of which does not admit a manifold interpretation: this ensures the existence of local degrees of freedom at scales below the Planck length. Meanwhile, in the geometric phase, the quasi-convex graph structure gives rise to smooth manifolds with a Hausdorff dimension .

In this context, the Hausdorff dimension characterizes how the graph’s diameter scales with the number N of vertices

This scaling relation captures the fractal nature of the emergent geometry in the large-scale limit. In the random phase, with logarithmic distance scaling, the Hausdorff dimension is infinite. Figure 6 shows the behaviour of as a function of graph size N deep within the geometric phase. For graph sizes up to N = 1000, the Hausdorff dimension increases to values between 2.1 and 2.2. However, for significantly larger graphs, this growth is not only halted, it gradually reverses, with approaching 2 (the gap in data between N = 1000 and N = 2000 reflects the high computational cost: simulations at each value of N require months of CPU time). These findings support the conclusion that sufficiently large graphs in the geometric phase effectively approximate a smooth 2D manifold, unlike all previously studied models. This 2D manifold at large scales plays the role of a holographic screen analogous to what proposed in the framework of Verlinde’s model [16, 17], with some notable differences to be discussed below.

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Regular graphs are ultrasparse and, as such, their combinatorial curvature is not sufficient to sample the continuum Ricci curvature on a manifold on which they are embedded, as we have shown above. There is, however, another way to map the combinatorial curvature onto a geometric curvature, as we now show.

We will henceforth consider the limit , in which the emerging manifolds are the covering spaces of genus g Riemann surfaces, i.e. infinite surfaces of negative curvature, so-called Cartan–Hadamard surfaces (for a review see [85, 86]). To simplify things, we will initially forget about the small scale randomness and, also, we will consider first configurations in which each vertex supports the same number of squares. These correspond to a quantized constant negative curvature, as we now show.

In the geometric phase, graphs become the 1-skeleton of a 2-cell embedding, for which the graph cycles are homeomorphic to open disks on a surface. The 2-cell embedding is called a ‘map’. A combinatorial map can be geometrized by assigning the same fixed geodesic length to each edge so that it becomes a tessellation of the surface, in which the combinatorial cycles become regular polygonal faces of the tessellation [87–89].

Each edge of the map supports exactly two regular polygons, each vertex supports exactly four polygons. Let k_i, , be the number of edges of the 4 polygons surrounding a vertex and let us call the vector the vertex type. For simplicity, we shall consider homogeneous maps, for which the edge type is the same for all vertices, so that the geometrization gives rise to semi-regular tessellations. For such maps, the tessellated surface is a constant curvature surface given by the angle sum parameter

If α < 2 the surface is spherical, i.e. of positive constant curvature, if α = 2 it is the flat Euclidean plane and, finally, if α > 2 it is hyperbolic, i.e. of negative constant curvature [87–89].

For every hyperbolic tessellation there is exactly one geodesic length so that the sum of interior angles at each vertex sums exactly to 2π. To see this, let us use the cosine rule for a hyperbolic n-gon,

to write the total interior angle at a vertex of a tessellation as

where R is the radius of curvature of the Poincaré disk on which the tessellation is constructed. First of all let us note that this is a monotonically decreasing function of . Second, for the total interior angle vanishes. In the opposite limit , the solution of (27) requires α = 2, i.e. the geometry becomes Euclidean. Therefore, given a specific semi-regular hyperbolic tessellation, there is exactly one possible parameter that solves (27). If the length unit is held fixed, the hyperbolic radius R of the Poincaré disk varies and, with it, the curvature . Otherwise, we can describe varying curvature at a fixed radius by letting the length unit change. The tessellation realized by the graph minimizing the free energy at every coupling g determines the value of the curvature K(g) in units of .

The geometric phase is realized by the condensation of squares and corresponds, thus to vertex types with at least one and all other or for all vertices. At each coupling, one of the four possible vertex types of the semi-regular tessellation, with one, two, three or four squares will characterize the minimum of the free energy, the remaining larger cycles determined by the detailed balance between the combinatorial energy and entropy. When the coupling is decreased from the critical value, the number of squares corresponding to the free-energy minimum will typically increase. For sufficiently large steps, one of the k_i decreases from a higher value to 4. In order to maintain the total interior angle sum as 2π, as in (27), the quantity has to decrease, which means the absolute value of the curvature of the tessellated surface typically decreases, until it vanishes for four squares around each vertex. In figure 7. we show one example of such a tessellation with three squares and one hexagon per vertex, corresponding to a constant negative-curvature surface at a small but non-zero value of the coupling g.

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Note that one cannot take the length scale to zero without the curvature also vanishing. In the infrared, the model is pushed to the completely ordered zero-coupling state corresponding to a flat geometry. At small, but non-vanishing coupling g the surfaces emerging from the network gravity model are characterized by two connected scales, a fundamental length scale and a radius of curvature R. At zero coupling the former vanishes and the latter diverges, at the critical coupling the former diverges and the latter vanishes.

Via this construction, the purely combinatorial Ollivier curvature is mapped to geometric curvature. The value of this curvature is determined by the statistical balance between the combinatorial energy dependent only on square cycles and the entropy representing the possible choices of residual larger cycles. Of course, in the general case, the tessellation will not be semi-regular, corresponding to surfaces with possibly varying negative curvature, so-called Cartan–Hadamard surfaces [85, 86]. And finally, on scales smaller than , there are random microscopic degrees of freedom.

To proceed we shall consider a large-scale hydrodynamic approach consisting of point defects in Brownian motion on a constant negative-curvature surface. It turns out that curvature has a profound effect on Brownian motion, with momentous consequences: in the case of negative curvature it is anomalous and becomes asymptotically ballistic.

Let us consider a constant negative curvature surface as the upper sheet of a two-sheeted hyperboloid embedded in 3D Minkwoski space with metric as

The metric is d, with with and the curvature.

For (with the diffusion coefficient), the Brownian motion b(t) on such a manifold can be expressed as

where is the Brownian bridge, the Brownian motion b(t) constrained to come back to the origin x = o for t = T. The quantity is called the infinite Brownian loop [50]. If for , as in the typical case, the infinite Brownian loop is the Brownian motion itself. Not so, however, if the Brownian motion is ballistic, for . This is exactly the case in negative curvature.

Brownian motion on a negative curvature surface decomposes in two independent radial and angular Brownian motions [44–46] (for a review see [47–49]). For , the angular component converges to a limiting angle , while the radial component becomes ballistic,

where is the inertial velocity of the defect and is the diffusion coefficient, which depends on its size. Asymptotically, the manifold coordinate z is dynamically soldered to the absolute time and Brownian motion acquires a deterministic component describing inertia in negative curvature. The exponential volume growth due to negative curvature creates a surface of last scattering: after the typical time scale background fluctuations decouple and the defect moves on radially with the velocity v it had after the last scattering [51, 52].

Let us focus first on this deterministic component. The asymptotic inertial trajectories (31) at different angles diverge because the negative curvature implies an exponential volume growth when z increases. Up to an exponentially small correction, these trajectories are the same as the comoving geodesics on a 2D Lorentzian manifold of positive curvature , realized by the embedding

with the metric

This is 2D de Sitter space-time (for a review see [90]), in which comoving geodesics diverge dynamically because of the Hubble expansion with parameter J (see [90]). Asymptotically, inertially diffusing observers in Riemannian negative curvature are equivalent, up to an exponentially small correction, to comoving observers in Lorentzian positive curvature [51, 52]. One can use both descriptions; however, with positive curvature, the universal time t becomes the coordinate time of a Lorentzian manifold. This is a form of Wick equivalence arising between the two signs of curvature.

In general, diffusion processes probe the intrinsic geometry of a manifold via the return probability kernel . The quantity is called the spectral dimension and measures the effective number of dimensions available to a random walker on the scale reached by time t, essentially on small scales for t → 0 and on very large scales for . The spectral dimension is of paramount importance, since it is the dimension perceived by local physical processes.

Unfortunately, this does not work as usual in the present framework, because of the dominant ballistic component of Brownian motion, which is much too fast to probe the local geometry. The Laplacian on a constant negative curvature manifold has a spectral gap

representing the bottom of the spectrum of the positive operator , which is associated with asymptotic ballistic behaviour. As a consequence, the return probabilities K(t) are dominated by an exponential behaviour at large t [55, 91],

which gives a spectral dimension linearly diverging in time.

The large-scale local geometry, however, is probed by the sub-dominant random component of Brownian motion, the infinite Brownian loop [50], whose return probabilities are given by (35) without the dominant exponential factor and give, therefore the spectral dimension function

The spectral dimension on scales smaller than is D and coincides with the topological dimension (D = 2 in our case), while the large-scale spectral dimension at distances larger than becomes 3, independently of D. In the mathematical literature this is called the ‘pseudo-dimension’, or ‘dimension at infinity’ of the constant negative curvature manifold [50]. The large-scale spectral dimension 3 is not confined to a constant negative curvature manifold though, but is valid for any manifold with strictly non-positive curvature and is a consequence of the central limit theorem in negative curvature [54]. This scale-dependent spectral dimension reproduces similar results in causal dynamical triangulations [8, 9] and Horava–Lifshitz gravity [3], with the difference that, here, there is a fundamental scale separating the two regimes.

When we take into account the sub-dominant random component of negative-curvature Brownian motion two things happen: first, the universal time t does not coincide anymore with one coordinate, secondly, an additional space dimension emerges spontaneously, a phenomenon we shall denote by hyperbolic holography. Let us consider the radial heat kernel on negative curvature, for distances , such that the angular distribution is uniform,

where the radial Laplacian is given by

and we have introduced, for simplicity of presentation, dimensionless quantities and . The kernel admits the representation [50]

where f₀ is the symmetric ground state of the Laplacian corresponding to the spectral gap [55]

for . The residual kernel describes the infinite Brownian loop random process, obtained by subtracting out the dominant ballistic component. It encodes the random dynamics of the defects as seen by an observer comoving with the inertial flow and satisfies the modified heat equation

Using (40) and , this equation becomes

where is the radial Laplacian in a 3D Euclidean space. Therefore

where we have reintroduced the original variables. This is the isotropic heat kernel on a 3D Euclidean manifold with distance norm z ‘inherited’ from the hyperbolic distance on the holographic screen [50]. In the Riemannian negative-curvature picture, the observer is comoving with the inertial flow of ballistic diffusion; in the Lorentzian positive-curvature picture, the observer is comoving with the Hubble flow on the holographic screen. If this Lorentzian picture is used for the dominant flow, the comoving random process must be treated as time-symmetric in the forward and backward directions, which gives immediately the Schrödinger evolution [92]

where we have used the previously introduced time measured in seconds. This is the 3D Schrödinger propagator for a particle of mass , velocity and therefore Compton wavelength . This is the definition of the ratio , more about this in the next section. At large scales, the local random process, as seen by geodesically free-falling observers on a de Sitter holographic screen, is quantum mechanics in an effective 3D Euclidean space with distances given by the hyperbolic distance on the holographic screen.

Quantum mechanics arises because the physical evolution parameter t, identified with coordinate time at large scales, must be treated as Lorentzian, when using the comoving interpretation of the dominant inertial flow. When constructing the Lorentz-invariant, but non-reparametrization-invariant action in terms of t, the fluctuations at small scales are sampled with the phase factor exp(iS). The Wick rotation is automatically enforced by the dominant comoving flow. The holographic promotion of the screen distance to an effective distance in a 3D Euclidean space immediately then implies quantum behaviour in this emergent space. The appearance of quantum behaviour, therefore goes hand-in-hand with the Lorentzian signature of space-time and vanishes with it below a scale fixed by the radius of curvature of the holographic screen.

In the previous section we have derived the emergence of an effective 3D Euclidean space ‘seen’ at large distances by quantum particles in a comoving frame of a de Sitter holographic screen. In this section we consider the holographic screen and the emergent space as embedded in a common large-scale bulk Universe manifold. To this end let us consider the isometry groups of the two components. The isometry group of a constant-curvature 2D holographic screen is , where the + indicates the orthochronous subgroup if the curvature is negative, in which case an arrow of time is built in. The isometry group of the emergent 3D Euclidean space is SO(3). The construction of an embedding Universe must be such that these two isometry groups share a common SO(2), corresponding to the angular variable of the holographic screen, and that 3D spatial distances scale with the same exponential factor as on the holographic screen, see (43). These requirements are satisfied by an isometry group SO(1,4) [93, 94], which corresponds, in the effective Lorentzian picture, to a (1+3)-dimensional de Sitter spacetime. The effective curvature experienced in this ‘bulk’ de Sitter Universe need not coincide with that of the holographic screen, since rescalings of time and of the speed of light can be absorbed into the definition of the de Sitter radius. Consequently, the non-relativistic quantum mechanics (44) can be ‘lifted’ to relativistic quantum field theory in an effective bulk de Sitter space.

Let us now consider a random phase bubble embedded in a constant curvature graph at low, but finite coupling. The interior vertices of the bubble support no squares (apart accidental ones). The boundary vertices of the bubble also support typically less squares than the surrounding graph, since at least one edge has to connect them to the random graph inside. Moreover, the remaining long cycles based on edges of the boundary vertices are much longer than in the surrounding geometric phase, since they have to cycle along many vertices in the interior random graph before coming back. These energy and entropy effects conspire to generate the higher free energy of the random phase bubble with respect to the geometric surrounding. But also, each vertex on the boundary has a lower curvature since less squares and longer cycles imply a larger excess angle at the vertex, so that the curvature is more negative than in the geometric phase, as explained above. Since the number of boundary vertices depends on the bubble size, this lower negative curvature is associated with the higher free energy, but this relation is statistical in nature since many microstates of slightly different curvature, due to long cycles entropy, characterize the ensemble with a given free energy. The ensemble equation is thus

in units where the free energy F is measured in Joule with respect to the surrounding geometric matrix with curvature . is Newton’s gravitational constant and n(g) is a numerical factor depending on the original coupling g.

In the Lorentzian picture the curvature changes sign. Let us consider also the large-scale continuum approximation in the bulk Universe, where the random bubble becomes a point-like excitation at rest in the comoving frame. In this case we must consider density distributions by dividing both sides by . This gives

where f is the free energy density, measured here with respect to the bulk curvature , and we have used the standard notation R(x) for the continuum Ricci curvature in (1+3) dimensions. Since both the free energy and the curvature of these random phase defects are higher than the surrounding manifold on which they live it is natural to identify them with matter particles, although they are made of the same microscopic degrees of freedom as the surrounding space.

In a generic frame, the energy–momentum tensor of a dust of such particles moving along geodesics must be defined by a local (maximum two derivatives), symmetric geometric tensor which is covariantly conserved and has equation (46) as its trace. The unique such geometric tensor is the Einstein tensor augmented by a cosmological constant term [95], which gives

These are Einstein’s equations of GR (with a cosmological term), with the difference that T_µν must be interpreted as a (density) thermodynamic potential and the equations hold as statistical ensemble averages. In this model these are not dynamical equations but, rather, macroscopic constitutive equations of matter as defined by the microscopic network degrees of freedom.

Matter, in this model is simply the random phase of the microscopic network. Matter particles have two sizes: a gravitational size where their associated excess curvature is concentrated and a quantum size, the Compton wavelength, set by the screen radius of curvature. In the present model, at large scales, it is ‘geometry that defines matter’, not the other way around, as usual.

In this model, there is an obvious possible solution to the cosmological constant problem (for a review see, e.g. [96]): the zero-point energy of quantum fields curves only the holographic screen, only matter and geometric fluctuations curve the bulk Universe. Note that the free energy of the model is, by construction, a very small positive quantity for a Universe empty of matter, with only geometric fluctuations on the scale of the cutoff .

As discussed above, matter is modelled as the random phase of the network. At finite coupling , matter particles are identified with the residual bubbles of random phase embedded in the dominant geometric matrix, representing the space manifold. Their typical size, the Planck scale, is stabilized by the balance between energy and entropy at the given coupling.

These bubbles diffuse due to the underlying network fluctuations and, in the continuum approximation at large scales, their dominant dynamics in the Lorentzian picture is described by a thermodynamic energy momentum tensor defined by the Einstein equations statistically in terms of network curvature. As a consequence, matter particles will tend to aggregate under the influence of gravitational forces. This is balanced by repulsive effects, among which quantum ones, such as the uncertainty relation and statistical pressure for fermions, but also thermal and radiation pressure and other interactions. However, if many particles do manage to approach each other on distances of the Planck length order, their random networks can merge to form a larger bubble of random network. And if the resulting bubble becomes larger than the radius of curvature scale, we have a single macroscopic 3D quantum object. This is a black hole.

Actually the hole is not really black. Let us consider the typical times τ_G and τ_R necessary for information to traverse a geometric and a random region of the same size L. In the geometric phase , where is the number of network vertices making up the region and D = 2 on the screen or D = 3 in the emergent space. In the random phase, instead, . To achieve the same dimension L, we must choose . Therefore, to cover the same distance L in the random phase, information must take exponentially more steps than in the geometric phase. If every edge is traversed with the same speed, this entails that the time needed to traverse a region of size L of random phase is exponentially larger than the time it takes for the same distance on the geometric matrix,

The random phase acts as a retardant medium for information propagation due to the disorder associated with the lack of a coherent geometric matrix. From the point of view of an observer in the geometric matrix, clocks in the random phase tick exponentially slower. Albeit exponentially large, however, the time delay associated with traversing a large random phase bubble is still finite. After a very long time, information will come out of the random phase region; only in the continuum limit and with finite does the hole develop an event horizon and become really black, since information has to make infinite steps to come out. However, in this limit no random phase survives. True black holes are thus only a continuum idealization which does not exist at finite Universe curvature.

Note that k-regular () random graphs are excellent expanders: the number of vertices within a ball of radius r scales like the number of vertices at distance r, where r denotes the level of hops along the tree leaves. Asymptotically for , the ‘surface to volume ratio’ of the graph is , 2/3 for 4-regular graphs. This means that there is essentially no distinction between volume and surface since 2/3 of the volume is taken up by the boundary surface. Since the boundary is well-defined from the embedding in the geometric matrix, the black hole is essentially only a very thin surface layer. All the vertices of the random bubble are concentrated there. This is the origin of the holographic principle: the number of degrees of freedom of a black hole scales with the area of the (would be) event horizon. The holographic principle for black holes is thus an immediate consequence of the expander property of regular random graphs.

The same expander property is shared by hyperbolic space. Since the geometric matrix of the screen is a negative-curvature surface, it is natural to visualize the black hole also as embedded in a 2D hyperbolic space, for which the surface to volume ratio is , the inverse of the radius of curvature. To do so we must assign to the graph edges a fixed length unit and then equate to , which gives a curvature radius . Finally, if we want to compare this to the surrounding geometric matrix, we must rewrite the logarithmic distances in this hyperbolic bubble as where is the number of vertices in it and . For large bubbles, the ‘inside’ effective edge length is exponentially smaller than the corresponding one in the geometric phase outside, which implies that the black hole can be seen as a hyperbolic bubble of exponentially large curvature with all degrees of freedom on the surface.

The expander property of the random phase implies that black holes are essentially large circles of radius R on the holographic screen. Their free energy, measured in Joule, can thus be expressed as

where u and s are numerical constants determining internal energy and entropy, respectively. We identify the internal energy as the black hole mass (times velocity squared),

which, for u = 1 is the Schwarzschild value. Seen from afar, these black holes appear as 2D surfaces, since their scale is much larger than the screen radius of curvature. If we want to interpret correspondingly the entropy as distributed on this surface, we must multiply it by a factor and divide simultaneously the dimensionless ‘temperature’ g by the same factor. This gives the physical temperature and entropy seen in the bulk,

For g = 1 and these are the Hawking temperature [97] and the Bekenstein entropy [98]. The exact values of these numerical factors depend, of course, on the microscopic statistical mechanics of the underlying network. However, the qualitative facts that black holes are characterized by an entropy proportional to their perceived horizon area and by a temperature inversely proportional to their mass are natural, but non-trivial predictions of the network model.

Bubbles of random phase much larger than the Planck length are not stable since they are not minima of the network free energy. When such a large bubble forms it will stabilize by shrinking to its equilibrium size by emitting both energy and entropy. This is a simple explanation of the Hawking radiation of black holes. They are simply statistically unstable and they emit energy and entropy to return to their equilibrium size.

Finally, since information is always stored in the fundamental network, there is no information paradox. When information falls into the black hole it is simply stored provisionally in another network phase and then released back when the hole stabilizes to its equilibrium size.

Dark matter is one of the major puzzles of contemporary cosmology (for a review see, e.g. [99]). The mass-energy of the Universe is made only by 5% of ordinary baryonic matter, the remainder being 27% dark matter, an unknown form of matter interacting with ordinary matter only through gravity and 68% dark energy, attributed typically to the cosmological constant discussed above.

There are two possible avenues to tackle the dark matter problem. One is to posit a new type of particle, with essentially only gravitational interactions, which is the mainstream belief today, the other is to posit a modification of classical GR. In [17, 35], however, it was proposed that dark matter is a macroscopic manifestation of quantum gravity. We now focus on how dark matter naturally fits in the present model.

We have shown that the holographic graph can have two phases, a random phase, identified with ordinary matter, in which there are no short loops and a geometric phase, representing space, in which each vertex i can exist in four possible states, corresponding to the number of squares surrounding it. The number determines the quantized curvature at this vertex as shown above. Until now we have focused on uniform curvature configurations in which all vertices in the geometric phase have the same number of squares and we have shown that this curvature is inherited in the emergent (1+3)-dimensional space-time.

What about inhomogeneous space configurations with different values of ? Imagine, in particular that the coupling g decreases from the critical coupling so that the Universe emerges from an all-random state at an extremely high curvature , representing the big bang. As always when lowering the coupling in continuous phase transitions, there typically survive domains of higher free energy. These are metastable states, which, depending on their free energy difference to the minimum and the barrier in between, may be extremely long-lived. These states are called allotropes and the most famous example are the two states of carbon, graphite, the free energy minimum and diamond the higher free energy allotrope (for a review see [100]).

In the present case, we can have metastable domains characterized by a higher number than the prevailing configuration and therefore having a larger free energy than the space around them at a given g. Because of this higher free energy they have also a higher curvature (in the Lorentzian picture) and interact gravitationally. But they are neither empty space nor ordinary matter. These domains are natural candidates to represent dark matter in the Universe. Dark matter, in this model, appears like crystal allotropy in the network fabric of space-time. An example is shown in figure 9.

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Of course, for couplings g midway between values favouring two different number of squares as the free energy minima, small domains of lower absolute curvature can form before the whole space-time transitions to this configuration. These would correspond to antigravity domains within a uniform metastable Universe.