The spectral dimension of simplicial complexes

# The spectral dimension of simplicial complexes:   a renormalization group theory

## Abstract

Simplicial complexes are increasingly used to study complex system structure and dynamics including diffusion, synchronization and epidemic spreading. The spectral dimension of the graph Laplacian is known to determine the diffusion properties at long time scales. Using the renormalization group here we calculate the spectral dimension of the graph Laplacian of two classes of non-amenable dimensional simplicial complexes: the Apollonian networks and the pseudo-fractal networks. We analyse the scaling of the spectral dimension with the topological dimension for and we point out that randomness such as the one present in Network Geometry with Flavor can diminish the value of the spectral dimension of these structures.

## 1 Introduction

Simplicial complexes [1, 3, 2, 4] are generalized network structures that are ideal to investigate network geometry and topology. They are formed by simplices like nodes, links, triangles, tetrahedra etc. that describe the higher order interactions between the elements of a complex system. The underlying decomposition of simplicial complexes in their geometric building blocks (the simplices) allows to answer novel questions in network topology and geometry. Network geometry and topology are emergent topics in statistical mechanics and applied mathematics that explore the properties of complex interacting systems using geometrical concepts and methods tailored to the discrete setting. Novel equilibrium and non-equilibrium modelling frameworks for simplicial complexes have been proposed recently in the literature [5, 6, 7, 8, 9, 10, 11, 12], and some classic example of deterministic networks such as Apollonian networks [13, 14] and pseudo-fractal networks[15] can be reinterpreted as skeleton of simplicial complexes. Additionally we note that simplicial complexes have been already extensively used in the quantum gravity literature to describe quantum space-time. They constitute, for instance, the underlying structures of Causal Dynamical Triangulations and of Tensor networks [16, 17]. From the applied point of view tools of network topology and geometry span brain research [2, 18], financial networks [19], social science [12] and condensed matter[20].

Network geometry and topology have been recently shown to have a very significant effect on dynamics including synchronization dynamics[21, 22, 23], epidemic spreading [24, 25, 26] and percolation [28, 27, 29]. The spectral properties of network geometries constitute a direct link to the diffusion processes defined on the same structures. The spectral properties of networks have been extensively studied in the literature [30, 31, 32, 33, 34, 35], however the study of the spectral properties of discrete network geometries is only at its infancy. Here we focus on the spectral dimension [36, 37, 38, 39, 40, 41] of the network geometry, which is known to determine the return distribution of a random walk and define universality classes of the Gaussian model. Recently the spectral dimension has been shown to be key to characterize the stability of the synchronized phase in the Kuramoto model defined on simplicial complexes [21, 22]. The spectral dimension [36] is a concept that extends the notion of dimension for a lattice and in fact it is equal to the lattice dimension for Euclidean lattices. The spectral dimension is however distinct from the Hausdorff dimension for a general network [42, 43]. This concept has been introduced to study the diffusion on fractal structures [36] and it has been then applied to a variety of contexts including the characterization of the stability of the 3D folding of proteins [41]. Interestingly, the notion of spectral dimension is used widely in quantum gravity to compare different models of quantum space-time in search for their universal properties[44, 45, 46].

In this work we investigate the spectral properties of the non-amenable skeleton of -dimensional simplicial complexes generated by deterministic and random models: the Apollonian [13], the pseudo-fractal [15] simplicial complexes and the Network Geometry with Flavor (NGF) [7, 6, 8]. The Apollonian simplicial complexes and the NGF with flavor are hyperbolic manifolds, while the other studied simplicial complexes have a non-amenable hierarchical structure. These discrete network structures are ideal to perform real-space renormalization group calculations revealing the critical properties of percolation [28, 27, 29, 49, 48, 50, 51, 52] the Ising model [47] and Gaussian model [40, 37, 38]. Here we use the renormalization group technique proposed in Ref. [37, 38] to predict the spectral dimension of simplicial complexes of different topological dimension for the Apollonian and the pseudo-fractal network. Moreover we will compare numerically the spectral properties of these deterministic network models with the spectral properties of the simplicial complexes generated by the model Network Geometry with Flavor [7] which include some relevant randomness. Our results reveal that the spectral dimension of the deterministic networks can be higher than the topological dimension. Specifically we see that planar Apollonian networks (in ) have a spectral dimension .. Additionally we found that the spectral dimension grows asymptotically for large as for both the Apollonian and the pseudo-fractal network. Finally we show numerically that topological randomness can diminish significantly the spectral dimension of the networks.

The paper is organized as follows: In Sec. 2 we define simplicial complexes and introduce the simplicial complex models investigated in this work. In Sec. 3 we define the spectral dimension and the relation between the spectral dimension and the Gaussian model. In Sec. 4 we introduce the real space renormalization group approach used in this work. In Sec. 5 we derive the RG equations for the Gaussian model on the Apollonian network, we theoretically predict the spectral dimension of Apollonian networks of any dimension and we compare the spectral properties of Apollonian networks with the spectral properties of NGFs with flavor . In Sec. 6 we use the RG approach to predict the spectral dimension of pseudo-fractal networks of any topological dimension and we compare the results with numerical result of both pseudo-fractal networks and NGFs with flavor and . Finally in Sec. 7 we provide the conclusions.

## 2 Simplicial complexes under study

### 2.2 Simplicial complexes

A simplicial complex of nodes can be used to describe complex interacting systems including higher order interactions. A simplicial complex is formed by simplices glued along their faces. A -dimensional simplex is a set of nodes characterizing a single many body interaction. A -simplex is a node, a -simplex is a link, a -simplex is a triangle, and so on. For instance a -simplex in a collaboration network can indicate that three authors have co-authored a paper, or a -simplex in a face-to-face interaction indicates a group of four people in a conversation. A -dimensional face of a -simplex , is a simplex formed by a subset of nodes of , i.e. .

A -dimensional simplicial complex is formed by a set of simplices of dimensions (including at least a -dimensional simplex) that satisfy the two conditions:

• if a simplex belongs to the simplicial complex, i.e. then also all its faces belong to the simplicial complex, i.e. ;

• if two simplices and belong to the simplicial complex, i.e. and , then either the two simplices do not intercept or their intersection is a face of the simplicial complex, i.e. .

A pure -dimensional simplicial complex is only formed by -dimensional simplices and their faces.

The -skeleton of a simplicial complex is the network formed exclusively by the nodes and the links or the simplicial complex.

Here we will focus exclusively on the skeleton of pure -dimensional simplicial complexes. The simplicial complexes that we will consider are the Apollonian, the pseudo-fractal simplicial complexes and the Network Geometry with Flavor. In the following paragraphs we will introduce each one of these models.

### 2.3 Apollonian network

A -dimensional Apollonian network [13, 14] (with ) is the skeleton of a simplicial complex that is generated iteratively by starting from a single -simplex at iteration and at each iterations adding a -simplex to every -dimensional face introduced at the previous generation. Therefore in these Apollonian networks, at generation there are nodes, and nodes added at iteration with

 Nn = (d+1)dn+d−2d−1, Nn = (d+1)dn−1. (1)

The Apollonian networks are small-world, i.e. their Hausdorff dimension is infinity,

 dH=∞, (2)

therefore at each iteration their diameter grows logarithmically with the total number of nodes of the network. Moreover, the Apollonian networks of dimension are manifolds that define discrete hyperbolic lattices.

Let us add here a pair of additional combinatorial properties of Apollonian networks that will be useful in the future. At each iteration we call links of type the links added at generation . At generation , the number of -simplices of generation attached to links of type is given by

 wℓ=(d−1)(d−2)ℓ−1. (3)

The number of -dimensional simplices of generation incident to nodes added at generation is given by

 vℓ=d(d−1)ℓ−1 (4)

for . Moreover it can be easily shown that the number of links of generation incident to nodes added at generation is given by for and for .

### 2.4 Pseudo-fractal network of any dimension

The pseudo-fractal network [15] is the skeleton of a simplicial complex constructed iteratively starting at iteration from a single -simplex (here and in the following we take ) At each time we attach a -simplex to every -dimensional face introduced a time . At iteration the number of nodes and the number of links added at iteration is

 Nn=d+1d[(d+1)n−1], Nn=(d+1)n (5)

The pseudo-fractal networks are small-world, i.e. their Hausdorff dimension is infinity,

 dH=∞. (6)

As for the Apollonian network, also for the pseudo-fractal networks, at each iteration we call link of type the links added at generation . We observe that at generation the number of -simplices of generation attached to links of type is given by

 ^wℓ=(d−1)ℓ∑ℓ′=0(d−2)ℓ′−1. (7)

It is also possible to show with straightforward combinatorial arguments that the number of -dimensional simplices added to nodes of generation is

 ^vℓ=dℓ∑ℓ′=0(d−1)ℓ′−1. (8)

for . Finally the number of links of iteration added to nodes of generation is given by for and for .

### 2.5 Network Geometry with Flavor

Network Geometry with Flavor (NGF) [7, 6] generates growing -dimensional simplicial complexes as Apollonian and pseudo-fractal simplicial complexes. However the dynamics of the NGF is not deterministic but random.

We assign to every -dimensional face of a simplex an incidence number equal to the number of -dimensional simplices incident to it minus one. Therefore we note that the incidence number can change with time.

The evolution of the NGF is dictated by a parameter called flavor The algorithm that determines the NGF evolution assumes that at time the simplicial complex is formed by a single -simplex. At each time a -face is chosen with probability

 Πd,d−1(α)=(1+snα)Z[s], (9)

where is called the partition function of the NGF and is given by

 Z[s](t)=∑α′∈Sd,d−1(1+snα′). (10)

We note here that the Hausdorff dimension of NGFs defined above is always

 dH=∞, (11)

as these networks are small-world for any value of the flavor . In the case the NGF evolves as a subgraph of the Apollonian network connected to the initial -dimensional simplex. In this case we obtain a random Apollonian network [14]. Therefore it is interesting to compare the spectral properties of the NGF with to the spectral properties of the Apollonian network. The NGF with flavor describe emergent hyperbolic geometries [6]. In fact they are hyperbolic networks emerging from a fully stochastic dynamics that makes no reference to their underlying geometry. Indeed NGF with flavor form a subset of the Apollonian networks of the same dimension . In the case and every -dimensional face can be incident to an arbitrary number of -dimensional simplices. Therefore it is interesting to compare the spectral properties of the NGF with and to the spectral properties of the pseudo-fractal network. Note that the Network Geometry with Flavor [7] was originally defined with an additional dependence to another parameter called inverse temperature . Here we focus only on the case , therefore we do not need to introduce this additional parameter in this work.

## 3 Laplacian spectrum and the Gaussian model

### 3.1 Spectral dimension

For a network it is possible to defined both a normalized and a un-normalized Laplacian. The un-normalized Laplacian has elements

 ^Lrq=krδr,q−arq, (12)

where is the generic element of their adjacency matrix and is the degree of node . The normalized Laplacian of a network has instead elements

 Lrq=δr,q−arq√krkq. (13)

Their spectrum is in general distinct for non-regular networks having nodes of different degree. However as we will observe later, their spectral dimension is the same in the large network limit.

Here we start from the normalized Laplacian and we predict the spectral dimension analytically. This analytical calculation will be done using the renormalization group which acts on a more general class of graphs in which the links can be weighted, so in general we are interested to study the fixed-point properties of spectrum of weighted normalized Laplacian matrices of elements

 Lrq=δrq−wrq√srsq, (14)

where indicates the weight of link and indicates the strength of node , i.e. .

The spectral dimension determines the scaling of the density of eigenvalues of the normalized Laplacian of networks with distinct geometrical properties. In particular, in presence of the spectral dimension for we observe the asymptotic behavior

 ρ(μ)≅CμdS/2−1, (15)

where is independent of .

In -dimensional Euclidean lattices the spectral dimension coincides with the Hausdorff dimension . More generally, it can be shown that is related to the Hausdorff dimension of the network by the inequalities [42, 43]

 dH≥dS≥2dHdH+1. (16)

It follows that for small-world networks, having infinite Hausdorff dimension , it is only possible to have finite spectral dimension greater or equal than two, i.e.

 dS≥2. (17)

Additionally we mention here that in presence of a finite spectral dimension, the cumulative distribution evaluating the density of eigenvalues follows the scaling

 ρc(μ)≅~CμdS/2, (18)

for . This relation it is useful to evaluate the spectral dimension numerically, as we will do in order to compare our analytical results with numerical results.

### 3.2 Gaussian model

In order to predict the spectral dimension of a network it is useful to consider [37] the corresponding Gaussian model whose partition function reads

 Z(μ)=∫Dψexp[iμ∑rψ2r−i∑rqLrqψrψq]=(iπ)Nn/2√∏r(μ−μr), (19)

where are the eigenvalue of the normalized Laplacian matrix and

 Dψ=Nn∏r=1(dψr√2π). (20)

By changing variables and putting the partition function can be rewritten as

 Z(μ)=∏r√sr∫Dϕexp⎡⎣∑(r,q)∈E−i(1−μ)wrq(ϕ2r+ϕ2q)−iwrqϕrϕq⎤⎦, (21)

where indicates the set of links of the network. The spectral density of the normalized Laplacian matrix can be found using the relation

 ρ(μ)=−2πIm∂f∂μ, (22)

where is the free-energy density defined as

 f=−limn→∞1NnlnZ(μ). (23)

In fact we can use Eq. (19) to show that

 f=−limn→∞1NnNn∑r=112ln(μ−μr)+12ln(iπ). (24)

Using Eq.(23) we get

 ρ(μ)=−2πIm∂f∂μ=1πlimn→∞1NnImNn∑r=11μ−μr=limn→∞1NnNn∑r=1δ(μ−μr). (25)

## 4 Renormalization group approach

Under the renormalization flow, and are renormalized. A closer look to the problem reveals that the parameters and are renormalized differently for links of different type . Therefore we parametrize the partition function describing the partition function of the Gaussian model over a network grown up to iteration with the parameters , i.e.

 Zn(ω)=∫DϕM∏ℓ=1∏(r,q)∈E(ℓ)nz(ℓ)n(ϕr,ϕq), (26)

where indicates the total number of iterations and where

 z(ℓ)n(ϕr,ϕq) = exp[−i(1−μℓ)pℓ(ϕ2r+ϕ2q)+2ipℓϕrϕq],

with indicating the set of links of type in a network evolved up to iteration . The Gibbs measure of this Gaussian model reads

 Pn({ϕ})=1Z(ω)n∏ℓ=1∏(r,q)∈E(ℓ)nz(ℓ)n(ϕr,ϕq)=1Z(ω)e−iH({ϕ}), (28)

where the Hamiltonian is given by

 (29)

Let us indicate with the nodes added at iteration . We consider the following real space renormalization group procedure to calculate the free energy of the Gaussian model. We start with initial conditions and for all values of . At each RG iteration, we integrate over the Gaussian variables associated to nodes and we rescale the remaining Gaussian variables in order to obtain the renormalized Gibbs measure of the same type as Eq. (28) but with rescaled parameters , i.e.

 Pn−1({ϕ′})=∫Dϕ(n)Pn({ϕ})∣∣∣ϕ′=F({ϕ}), (30)

where

 Dϕ(n)=∏r∈Nn(dϕr√2π). (31)

The rescaling of the field is done in such a way that for a -dimensional deterministic simplicial complex, at each iteration of the RG flow. Then at each step of the RG transformation we have

 H({ϕ})→H′({ϕ′}), (32)

where

 H′({ϕ})=n−1∑ℓ=1∑(r,q)∈E(ℓ)n−1{−(1−μ′ℓ)p′ℓ[(ϕ′r)2+(ϕ′q)2]+2p′ℓϕ′rϕ′q}. (33)

In this way we define a group transformation acting on the model parameters so that

 ω′=Rω. (34)

Under this renormalization flow, the partition function transforms in the following way:

 Zn(ω)=e−Nng(ω)Zn−1(ω′). (35)

Using Eq. (1) and Eq. (5) for the number of nodes at iteration respectively for the Apollonian and the pseudo-fractal network, the free energy density

 f = −limn→∞1NnlnZn(ω) (36)

can be written as

 f ≃ ∞∑τ=0g(R(τ)ω)dτ (37)

for the Apollonian network and as

 f ≃ ∞∑τ=0g(R(τ)ω)(d+1)τ (38)

for the pseudo-fractal network.

Interestingly, we anticipate here that the RG flow will be determined by a fixed point having . This result reveals that indeed the spectral dimension here calculated for normalized Laplacian is universal, i.e. in the large network limit, the spectral dimension of the normalized Laplacian is the same as the spectral dimension of the un-normalized Laplacian as already observed in Ref. [39].

## 5 Apollonian network

### 5.1 General RG equations

The renormalization group equations for the Apollonian networks of arbitrary topological dimension can be obtained using the general renormalization group approach described in the previous paragraph. Therefore at each renormalization group step we need first to integrate over the fields with and subsequently perform a rescaling of the fields to guarantee that at the next iteration. Since any node added at generation is only connected to nodes at the previous generations the integration over the corresponding field can be done independently for any node .

The integration over a single Gaussian variable with can be easily done and is given by

 I=∫dϕrd∏q=1z(1)(ϕq,ϕr) = (−πi/d)1/2G(μ1)−1/2exp[−i(1−μ1)d∑q=1ϕ2q] (39) ×exp⎡⎢ ⎢ ⎢⎣i(∑dq=1ϕq)2d(1−μ1)⎤⎥ ⎥ ⎥⎦,

where

 G(μ) = 1−μ. (40)

Of course, at each step of the RG procedure we will need to integrate over each node . Each of these integrations will contribute to the Hamiltonian by a term

 [(21d(1−μ1))ϕqϕq′] (41)

for any link incident to the -simplex added at iteration and including node . Since in the Apollonian network, there are simplices of iteration incident to a link added at iteration , the overall contribution to the link is

 [(21d(1−μ1))wℓϕqϕq′]. (42)

If we focus on the overall contribution to the Hamiltonian proportional to the product of the two field and before rescaling we get

 {2[pℓ+1+(1d(1−μ1))wℓ]ϕqϕq′}. (43)

After rescaling of the fields and defining the Gibbs measure over the renormalized network formed by generation, i.e. putting we should have

 {2[pℓ+1+(1d(1−μ1))wℓ]ϕqϕq′}={2p′ℓϕ′qϕ′q′}. (44)

Therefore in order to ensure that the value of the parameter remains fixed at after each RG iteration we need to rescale the fields by considering the rescaled variables

 ϕ′ = ϕ[p2+d−1d(1−μ1)]1/2, (45)

where we have used . Finally by using Eq. (3) for , the RG equation for reads

 p′ℓ = [pℓ+1+(d−1)(d−2)ℓ−1d(1−μ1)][p2+d−1d(1−μ1)]−1. (46)

Now we will proceed similarly to find the RG equations for . Each integration over the Gaussian variable of a node contributes a factor

 [(−(1−μ1)+1d(1−μ1))ϕ2q] (47)

to the Hamiltonian for any node belonging to the -simplex added at iteration and including node . Since there are simplicies of iteration incident to a node added at iteration the integration over the Gaussian variable at iteration provides, for each node , a contribution to the Hamiltonian given by

 [(−(1−μ1)+1d(1−μ1))vℓϕ2q]. (48)

By identifying the overall term of the Hamiltonian that is proportional to before and after the rescaling of the fields we obtain the equation

 {−ℓ∑ℓ′=1(1−μℓ′+1)pℓ′+1vℓ−ℓ′+(−(1−μ1)+1d(1−μ1))vℓ(ϕ′q)2}= {−ℓ∑ℓ′=1(1−μ′ℓ′)p′ℓ′vℓ−ℓ′(ϕ′q)2}. (49)

We now make a useful combinatorial observation and we note that the coefficient can be written as

 vℓ=ℓ∑ℓ′=1vℓ−ℓ′cℓ′ (50)

where is given by

 cℓ=(d−2)ℓ−1. (51)

In fact, by substituting the explicit expression for Eq. (50) follows directly from the expression

 d(d−1)r=r−1∑k=0d(d−1)r−1−k(d−2)k+d(d−2)r. (52)

Using Eq. (50) and the expression for the rescaled field, Eq. (45), in Eq. (49) we get the RG equation for given by

 (1−μ′ℓ)p′ℓ = ((1−μ1)(d−2)ℓ−1+(1−μℓ+1)pℓ+1−(d−2)ℓ−1d(1−μ1)) (53) ×[p2+d−1d(1−μ1)]−1.

In summary, in this paragraph we have derived the RG equation for any -dimensional Apollonian networks which we rewrite here for completeness,

 (1−μ′ℓ)p′ℓ=[(1−μ1)(d−2)ℓ−1+(1−μℓ+1)pℓ+1−(d−2)ℓ−1d(1−μ1)] ×[p2+d−1d(1−μ1)]−1, p′ℓ=[pℓ+1+(d−1)(d−2)ℓ−1d(1−μ1)][p2+d−1d(1−μ1)]−1. (54)

Under the renormalization group the partition function follows Eq. (35) with

 g(ω)=Nn2NnlnG(μ1)+Nn−12Nnln[p2+1d(1−μ1)]+c, (55)

where is a constant. The first term comes directly from each integration over the variables with given by Eq. (39) and the second term comes from the rescaling of the fields. In the following paragraphs we will first study the RG flows in the cases and and afterwards we investigate the RG flow in any arbitrary dimension .

### 5.2 d=2 Farey graph

For the Apollonian network reduces to a Farey graphs and the RG Eqs. (110) simplify greatly. In fact we have

 μℓ=μ2, pℓ=p

for all . The renormalization group transformations read then

 (1−μ′1)=((1−μ1)+(1−μ2)p−12(1−μ1))[p+12(1−μ1)]−1, μ′2=μ2, p′=p[p+12(1−μ1)]−1. (57)

Under the renormalization group the partition function follows Eq. (35) with given by Eq. (55).

By putting in the zero order approximation the renormalization group equations (57) have three fixed points:

 (μ⋆,p⋆) = (0,0), (μ⋆,p⋆) = (0,1/2), (μ⋆,p⋆) = (3/2,0),

Since we are interested in the RG flow starting from an initial condition we focus on the fixed points with . The fixed point is unstable as it has two eigenvalues given by and . The fixed point is associated to the eigenvalues and . If we have initial condition with and the renormalization flow will first move toward the fixed point and then will move away from it along its repulsive direction. Close to the fixed point, putting , the linearized RG equations read

 (μ′1p′−1/2)=(20−1/41/2)(μ1p−1/2)+μ2(10). (59)

At iteration of the RG flow we have

 μ(τ) = (λτ−1)μ2, (60) p(τ) = 12−14τ∑τ′=12−(τ−τ′)μ(τ′)+2−(τ+1). (61)

Therefore for large the RG flow runs away from the RG fixed point and we can approximate

 μ(τ) ≃ λτμ2, (62) p(τ) ≃ 12−16λτμ. (63)

The RG flow is shown in Fig. 1 where we have set initially . Using Eq. (36) free-energy density can be therefore written as

 f = ∞∑τ=0g(R(τ)ω)dτ (64) ≃∞∑τ=01dτ⎧⎨⎩(d−1)2dln(1−μ(τ)1)+12dln⎡⎣p(τ)+1d(1−μ(τ)1)⎤⎦⎫⎬⎭.

Therefore we have the spectral density given by

 ρ(μ)≃2πIm∞∑τ=01dτ∂g(μτ1,1)∂μ ≃2πIm∞∑τ=0λτdτ⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩(d−1)2d11−μ(τ)1+12d1p(τ)+1/[d(1−μ(τ)1)]⎛⎜ ⎜⎝−13+1d(1−μ(τ)1)2⎞⎟ ⎟⎠⎫⎪ ⎪ ⎪⎬⎪ ⎪ ⎪⎭.

By approximating the sum over with an integral and changing the variable of integration to upon using the theorem of residues to solve the integral, we can derive the asymptotic scaling of the density of eigenvalues . This asymptotic scaling for is given by

 ρ(μ) ≃ CμdS/2−1, (65)

where the spectral dimension is given by

 dS=2lndlnλ=2. (66)

### 5.3 d=3 Apollonian graph

For the RG Eqs. (110) simplify significantly. In fact we have

 μℓ=μ1 (67)

for all and

 pℓ=p (68)

for all . The RG equations differ from the ones derived in the case , and they read

 (1−μ′1)=((1−μ1)+(1−μ2)p−1d(1−μ1))[p+d−12(1−μ1)]−1, p′=1. (69)

Under the renormalization group the partition function follows Eq. (35) with given by Eq. (55). The renormalization group equations (69) give and reduce to a single non trivial RG equation for ,

 (1−^μ′) = ((1−^μ)+(1−^μ)−1d(1−^μ))[1+d−1d(1−^μ)]−1,

which has two fixed points:

 μ⋆ = 0 μ⋆ = 4/3. (71)

For the relevant fixed point is which has a non-trivial associated eigenvalue given by . Therefore under the RG flow we have that at iteration of the RG flow

 (μ(τ)1,p(τ))=(λτμ,1) (72)

with indicating the initial condition . Using Eq. (36) free-energy density can be therefore written as

 f = ∞∑τ=0g(R(τ)ω)dτ (73) ≃∞∑τ=01dτ⎧⎨⎩(d−1)2dln(1−μ(τ)1)+12dln⎡⎣1+d−1d(1−μ(τ)1)⎤⎦⎫⎬⎭.

Therefore we have the spectral density given by

 ρ(μ)≃2πIm∞∑τ=01dτ∂g(μτ1,1)∂μ ≃2πIm∞∑τ=0λτdτ⎧⎪ ⎪⎨⎪ ⎪⎩(d−1)2d11−μ(τ)1+12d1[d(1−μ(τ)1)+d−1]d−1(1−μ(τ1)⎫⎪ ⎪⎬⎪ ⎪⎭.

By proceeding similarly to the case and approximating the sum over with an integral, we obtain the asymptotics

 ρ(μ) ≅ Cμds/2−1 (74)

valid for with the spectral dimension given by

 ds=2lndlnλ=2lndln9/5=3.73813…. (75)

Interestingly, Apollonian networks in are planar. As we will see when comparing the spectral dimension of Apollonian network with the spectral dimension of NGFs, the randomness introduced by the NGF constructions always lower the spectral dimension of the network.

### 5.4 d>3 dimensional Apollonian graph

Let us now determine the RG flow in the general case of a -dimensional Apollonian network. By putting

 xℓ=(1−μℓ)pℓ, (76)

the RG Eqs. (110) relating the parameters at iteration of the RG flow with the parameters at iteration of the RG flow read

 x(τ+1)ℓ=⎡⎣x(τ)ℓ+⎛⎝x(τ)1−1dx(τ)1⎞⎠(d−2)ℓ−1⎤⎦⎡⎣p(τ)2+d−1dx(τ)1⎤⎦−1, p(τ+1)ℓ=⎡⎣p(τ)ℓ+1+(d−1)(d−2)ℓ−1dx(τ)1⎤⎦⎡⎣p(τ)2+d−1dx(τ)1⎤⎦−1. (77)

In order to solve these equations we use the auxiliary variable defined as

 y(τ+1)1=p(τ+1)2+d−1dx(τ+1)1. (78)

The explicit solution of the RG equations (77) equations read

 p(τ+1)2 = τ∏m=11y(m)1+d−1dτ∑m=1(d−2)τ−m+1x(m)τ∏m′=m1y(m′)1, y(τ+1)1 = p(τ+1)2+(d−1)dx(τ+1)1 = τ∏m=11y(m)1+d−1dτ∑m=1(d−2)τ−m+1x(m)τ∏m′=m1y(m′)1+(d−1)dx(τ+1)1, x(τ+1)1 = x(1)1