The higher-order spectrum of simplicial complexes: a renormalization group approach

Network topology is a flourishing interdisciplinary subject that is relevant for different disciplines including quantum gravity and brain research. The discrete topological objects that are investigated in network topology are simplicial complexes. Simplicial complexes generalize networks by not only taking pairwise interactions into account, but also taking into account many-body interactions between more than two nodes. Higher-order Laplacians are topological operators that describe higher-order diffusion on simplicial complexes and constitute the natural mathematical objects that capture the interplay between network topology and dynamics. Higher-order up and down Laplacians can have a finite spectral dimension, characterizing the long time behaviour of the diffusion process on simplicial complexes. Here we provide a renormalization group theory for the calculation of the higher-order spectral dimension of two deterministic models of simplicial complexes: the Apollonian and the pseudo-fractal simplicial complexes. We show that the RG flow is affected by the fixed point at zero mass, which determines the higher-order spectral dimension $d_S$ of the up-Laplacians of order $m$ with $m\geq 0$. Finally we discuss how the spectral properties of the higher-order up-Laplacian can change if one considers the simplicial complexes generated by the model"Network Geometry with Flavor". These simplicial complexes are random and display a structural topological phase transition as a function of the parameter $\beta$, which is also reflected in the spectrum of higher-order Laplacians.


Introduction
Simplicial complexes [1][2][3][4][5][6] are generalized network structures that capture many-body interactions. They are not just formed by nodes and links like networks but they also include simplices of higher dimensions such as triangles, tetrahedra and so on. Being arXiv:2003.09143v1 [cond-mat.dis-nn] 20 Mar 2020 build by these topological building blocks, simplicial complexes are the ideal discrete structures to investigate emergent geometry [7][8][9][10][11] and can be described by discrete algebraic and combinatorial topology. Topology is a traditional tool of high-energy physics and quantum gravity and recently it has also become increasingly popular to investigate complex systems [12]. In fact topological methods have been shown to be very powerful to analyse datasets, including brain networks and collaboration networks [2,[13][14][15]. Finally there is an increasing interest in revealing the role that the higher-order interactions of simplicial complexes have on their dynamics [16,17,[19][20][21][22].
The network Laplacian [23][24][25][26] is fundamental to understand the interplay between topology and dynamics and its spectral properties are known to affect diffusion and synchronization on network structures. In particular the spectral dimension [27][28][29][30][31][32][33][34][35][36] characterizes the spectral properties of networks with distinct geometrical features and determines the late time behavior of diffusion and more general dynamical processes on networks [37][38][39][40][41]. The spectral dimension can also be defined on simplicial complexes [28] by focusing on their skeleton (the network obtained from a simplicial complex by retaining only its nodes and links). Thus, the spectral dimension is also considered a key mathematical object for investigating the effective dimension of a simplicial quantum geometry as felt by diffusion processes. More in general in quantum gravity the spectral dimension is used for probing the geometry of the simplicial spacetimes [42][43][44] described by different theoretical approaches including Causal-Dynamical-Triangulations (CDT) [45].
Here we focus on three models of pure d-dimensional simplicial complexes called Apollonian simplicial complexes, [46][47][48] pseudo-fractal simplicial complexes [49] and "Network Geometry with Flavor" [8,9,11] respectively. The Apollonian simplicial complexes [46][47][48] are deterministic hyperbolic d-dimensional manifolds that are obtained by an iterative process, whose limit converges to an infinite hyperbolic lattice. The Apollonian simplicial complex in d-dimensions is dominated by the boundary and is closely related to the melonic graphs of tensor networks [50,51], because melonic graphs can be understood as the merging of two identical Apollonian simplicial complexs upon identification of the all their faces at the boundary. The pseudo-fractal simplicial complexes [49], generalise the Apollonian simplicial complexes to simplicial complexes that are not manifolds. These deterministic simplicial complexes have a skeleton which is non-amenable, i.e. they have an infinite isoperimetric dimension and simultaneously have a very small Cheeger constant [52,53]. Additionally, they are small world and scalefree. The "Network Geometry with Flavor" s ∈ {−1, 0, 1} (NGF) [8,9,11] are random simplicial complexes whose structure evolves according to a stochastic process, where the set of possible simplices is restricted to be a subset of the faces of the Apollonian simplicial complex for s = −1, or a subset of the pseudo-fractal simplicial complexes for s = 0, s = 1. Therefore, NGFs can be interpreted as the result of an Invasion Percolation process [54] defined over the aforementioned simplicial complex topologies. Given the fact that Apollonian and pseudo-fractal simplicial complexes are highly geometrical, deterministic and hierarchical, these structures and their generalizations [55,56] are very suitable for conducting renormalization group (RG) calculations analytically. Examples of dynamical processes already studied with the RG in related simplicial complex models include percolation [57][58][59][60][61][62], spin models [63] and Gaussian models [28][29][30].
In this paper we investigate the properties of higher-order Laplacians [17,52,53,[64][65][66] on the considered simplicial complexes. The higher-order Laplacians describe diffusion processes occurring on higher-order simplices [17,64] and are key mathematical objects to define the higher-order Kuramoto model [18]. Higher order Laplacians are also closely related to approximate Killing vector fields, which are currently being investigated on quantum geometries in CDT [67]. It has been recently shown numerically [17], that the higher-order up-Laplacian and down-Laplacian can display a finite spectral dimension. Here we use renormalization group (RG) theory [28][29][30] to analytically calculate the spectral dimension of higher-order up-Laplacians of Apollonian and pseudo-fractal simplicial complexes. We find that each simplicial complex belonging to the considered class of models, is characterized by a set of analytically predicted spectral dimensions. Each spectral dimension corresponds to the spectrum of a higherorder up-Laplacian of different order m. The values of the predicted spectral dimensions are compared to direct numerical results for d = 3 and d = 4 simplicial complexes. Additionally, the numerical investigation of the higher-order spectrum of the "Network Geometry with Flavor" demonstrates the effect that topological phase transition induced by stochastic growth can have on higher order spectra.
The paper is structured as follows: in Sec. II we introduce simplicial complexes and their higher-order Laplacians, in Sec.III we present the hyperbolic and non-amenable simplicial complex models considered in this work; in Sec. III we give the necessary background for deriving the higher-order spectrum of the Apollonian and pseudo-fractal simplicial complexes using the RG approach; In Sec. IV and in Sec. V we derive the RG equations and the RG flow for the Apollonian simplicial complexes; In Sec. VI and Sec. VII we derive the RG equations and the RG flow for the pseudo-fractal simplicial complexes. In Sec. VIII we summarize the main analytical results and we will compare with numerical results on all the considered simplicial complex models. Finally, in Sec. IX we will provide the conclusions.

Simplicial complexes
A m-dimensional simplex r (also indicated as m-simplex) includes m + 1 nodes and it can be indicated as Therefore, a 0-simplex is a node, a 1-simplex is a link, a 2-simplex a triangle, a 3-simplex a tetrahedron and so on. A m -dimensional face q of a m-dimensional simplicial complex r is a m < m simplex formed by a subset of m + 1 nodes belonging to the simplex r.
In topology, simplices also have an orientation. Two m-simplices differing only by the order in which their nodes are listed are therefore related by where σ(π) indicates the parity of the permutation π of the m + 1 indices of the nodes. A simplicial complex is formed by a set of simplices with the property that the simplicial complex is closed under inclusion of the faces of any of its simplices.
A d-dimensional simplicial complex is a simplicial complex for which the maximum dimension of its simplices is d. Here we are exclusively interested in pure d-dimensional simplicial complexes, which are formed by a set of d-dimensional simplices and all their faces. The skeleton of a simplicial complex is the network formed by the set of all the nodes and links of the simplicial complex. Given a d-dimensional simplex, we indicate the number of its m-simplices with N [m] with 0 ≤ m ≤ d.

Boundary map and incidence matrices
Given a simplicial complex, a m-chain consists of the elements of a free abelian group C m whose base is formed by the set of all the m-simplices of the simplicial complex. The boundary map ∂ m is a linear operator ∂ m : C m → C m−1 whose action is determined by the action on each m-simplex of the simplicial complex. In particular the boundary map ∂ m applied to the m simplex r In words, the boundary map applied to a m-simplex gives a linear combinations of its (m − 1)-dimensional faces.
We say that two m-faces r and q of a simplicial complex are upper adjacent if there is a (m + 1)-simplex τ of which both r and q are faces. The m-faces r and q are upper adjacent with similar orientation if the simplicial complex contains a (m+1)-dimensional simplex τ such that where a, b indicates the inner product. Conversely, they are upper adjacent with opposite orientation if the simplicial complex contains a (m + 1)-dimensional simplex τ such that From the definition of the boundary map ∂ m given by Eq. (3), it follows immediately that for every m-dimensional simplex r which is an important topological property that can be expressed in words with the sentence "the boundary of a boundary is null".
Given a simplicial complex with N [m] m-dimensional simplices we can choose a base for C m by taking an ordered list of its m simplices. If we fix both the base of C m and C m−1 we can represent the the boundary operator ∂ m by a N [m−1] ×N [m] incidence matrix B [m] . In Figure 1 we show an example of a simplicial complex. We choose as bases for C 0 , C 1 and C 2 the ordered list of nodes { [1], [2], [3], [4]}, links { [1,2], [1,3], [2,3], [3,4], [2,4]} and triangles {[123], [234]}. With this choice of bases, the boundary maps ∂ 1 and ∂ 2 can be represented by the incidence matrices B [1] and B [2] with, Figure 1.
An example of a small simplicial complex with the orientation of the simplices induced by the labelling of the nodes.

Higher order Laplacian matrices of simplicial complexes
The graph Laplacian or 0-Laplacian describes the diffusion process over a network and it is an extensively studied topological operator in graph theory [23]. The 0-Laplacian can be also defined for a simplicial complex and describes the diffusion process that goes from a node to another node across shared links. In fact the 0-Laplacian is a N [0] × N [0] matrix and can be expressed in terms of the incidence matrix B [1] , While on networks only the graph Laplacian and its normalized versions can be defined, in simplicial complexes it is possible to define higher-order Laplacians describing diffusion taking place between higher-order simplices. The higher-order Laplacian L [m] with m > 0 (also called combinatorial Laplacians) can be represented as a N [m] × N [m] matrix given by where L down The down-Laplacian L down [m] of order m, describes diffusion process taking place among m simplices across (m − 1) shared simplices. For instance, the down-Laplacian of order 1 describe diffusion from link to link across shared nodes. The up-Laplacian L up [m] of order m describes diffusion processes taking place among m simplices across shared (m + 1) simplices. The up-Laplacian of order 1 for example, describes the diffusion from link to link across shared triangles.
Interestingly the spectral properties of the higher-order Laplacians can be proven to be independent on the orientation of the simplices as long as the orientation is induced by a labelling of the nodes.
One of the main results of Hodge theory [16,17,65] is that the degeneracy of the zero eigenvalues of the m-Laplacian L [m] is equal to the Betti number β m . The corresponding eigenvectors localize around the corresponding m-dimensional cavity of the simplicial complex. It follows that if the simplicial complex has trivial topology, i.e. it is formed by a single connected component, β 0 = 1 and the simplicial complex has no higher-order cavities, (i.e. β m = 0 for all m > 0) then the 0-Laplacian L [0] has a zero eigenvalue that is not degenerate while all the higher-order Laplacians L [m] with m > 0 do not admit any zero eigenvalue.
Let us observe here that Eq. (6) can be expressed in terms of the incidence matrices as . Hodge theory therefore demonstrates (see for instance [16] for a gentle introduction) that the spectrum of the m-Laplacian includes all the non-null eigenvalues of the m-up-Laplacian and all the non-null eigenvalues of the m-down Laplacian. The other eigenvalues of the m-Laplacian can only be zero and their degeneracy is given by the Betti number β m . Therefore the spectrum of the m-Laplacian is completely determined once the spectra of both the m-up-Laplacian and the m-down-Laplacian are known.
Finally we observe that the up-Laplacians and the down-Laplacians are related by transposition Therefore the spectrum of the m-up Laplacian is equal to the spectrum of the (m + 1)down Laplacian. Taking all these consideration together it follows that in order to know the spectrum of all higher-order Laplacians of a simplicial complex it is sufficient to know the spectrum of all its higher-order up-Laplacians.
Therefore in this work, without loss of generality we will focus on the spectral properties of m-up-Laplacians of pure d-dimensional simplicial complexes with order 0 ≤ m < d − 1.

Up-Laplacians and their spectral dimension
For a simplicial complex of dimension d > m it is possible to define both a normalized and an un-normalized higher-order up m-Laplacian. The un-normalized higher order up-Laplacian L up where δ x,y indicates the Kronecker delta. In Eq. (14) we have used the oriented upper incidence matrices a where we note that in this expression we use the convention 0/0 = 0. The normalized m-up-LaplacianL up [m] has elements In this work we will focus on the spectral properties of the normalized up-Laplacians. The spectrum of the normalized and un-normalized m-up-Laplacians is in general distinct for simplicial complexes in which k r is dependent on r. However, we anticipate that when they both display a spectral dimension, their spectral dimension is the same [31].
The density of eigenvaluesρ(µ) of the normalized m-up-Laplacian has a density of eigenvalues that includes a singular part formed by a delta function at µ = 0 and a regular part ρ(µ), i.e.
where we useδ(x) to denote the delta function. The emergence of the delta peak at µ = 0 can be easily explained. First let us observe that Eq. (15) implies that the number of zero eigenvalues of the normalized and un-normalized m-up-Laplacians is the same. Secondly let us note that the spectrum of the m-up-Laplacian L up ). In particular for simplicial complexes with trivial topology, the Hodge decomposition [16] implies that the number of non-zero eigenvalues of the m-up-Laplacian with m > 0 are given byN [m] = min(N [m] , N [m+1] ). It follows that all the other eigenvalues are zero. Therefore for m > 0 the degeneracy of the zero eigenvalue can be extensive, while for m = 0 the degeneracy of the zero eigenvalue is given by the Betti number β 0 , where β 0 = 1 for a trivial topology. For a trivial topology the density of eigenvalues at µ = 0 of the graph Laplacian (m-up-Laplacian with m = 0) is zero in the large network limit, while it can be greater than zero for m > 0.
The normalized m-up Laplacian displays a finite spectral dimension d S when the regular part of its density of eigenvaluesρ(µ) obeys the asymptotic behaviour where µ 1 and C is independent of µ. From this scaling it directly follows that the cumulative distribution ρ c (µ) of the regular part of the density of eigenvalues ρ(µ), which is the integral of the the density of eigenvalues 0 < µ ≤ µ, follows the scaling for µ 1. This relation will prove useful in the following, when we will numerically compare the predicted spectral dimension with the numerical results.

Apollonian simplicial complexes of any dimension
A d-dimensional Apollonian simplicial complex [46,47] (with d ≥ 2) is generated iteratively by starting from a single d-simplex at generation n = 0 and adding a dsimplex at each generation n > 0 to every (d − 1)-dimensional face introduced at the previous generation. At generation n = 0 there are N The number N [m] n of m-dimensional faces at generation n is given by In these Apollonian simplicial complexes, there are N [m] n m-dimensional simplicial complexes at generation n with Finally we note here that in the following we will used the notation Q [m] to indicate the set of m-simplices of the Apollonian simplicial complex.
The Apollonian simplicial complex are small-world, i.e. their skeleton has an infinite Hausdorff dimension, therefore at each generation their diameter grows logarithmically with the total number of nodes of the network. Moreover, the Apollonian simplicial complex of dimension d are manifolds that define discrete hyperbolic lattices including for d = 2 the Farey graph. Let us add here a pair of additional combinatorial properties of Apollonian simplicial complex that will be useful later. At each generation n we call simplices of type the simplices added at generation n = n − . At generation n, the number of d-simplices of generation n attached to simplices of dimension m (with m < d) of type > 0 is given by Moreover, we observe that the number of (m + 1)-dimensional simplices of generation n incident to m-simplices added at generation n = n − is given by w [m] for > 0 and

Pseudo-fractal simplicial complexes of any dimension
A pseudo-fractal simplicial complex [49] of dimension d with d ≥ 2 is constructed iteratively. At generation n = 0 the simplicial complex is formed by a single d-simplex (with d ≥ 2). At each generation n > 0 we glue a d-simplex to every (d − 1)-dimensional face introduced at generation n ≥ 0. At generation n = 0 the number of m-dimensional simplices N [m] 0 is given by The number N [m] n of m-dimensional faces added at generation n > 0 is given by The number m-dimensional faces N [m] n at generation n is The pseudo-fractal simplicial complexes differs from Apollonian simplicial complexes significantly as they are not discrete manifolds. However both simplicial complexes have an underlying non-amenable network structure and are characterized by having a small Cheeger constant. Moreover the pseudo-fractal simplicial complexes, as the Apollonian simplicial complexes, have a small-world skeleton, i.e. their underlying networks have an infinite Hausdorff dimension For pseudo-fractal simplicial complexes we use the same notation as for Apollonian simplicial complex and we indicate Q [m] the set of m-simplices of the pseudo-fractal simplicial complex. Additionally we indicate as simplices of type the simplices added at generation n = n − , in the pseudo-fractal simplicial complex evolved up to generation n. We make the following useful remark: at generation n the number of d-simplices of generation n attached to m-simplices (with m < d) of type > 0 is given bŷ Finally the number of (m + 1)-simplices of generation n added to m-simplices of generation n = n − is given byŵ [m] for > 0 andŵ

"Network Geometry with Flavor"
The simplicial complex model "Network Geometry with Flavor" (NGF) [8,9,11] generates random d-dimensional simplicial complexes which can be interpreted as the result of an Invasion Percolation process [54] on an Apollonian simplicial complex (for flavor s = −1) or on a pseudo-fractal simplicial complex (for s = 0 and s = 1). Let us indicate with Q [m] the set of m-simplices belonging to the NGF. To every m-dimensional face q ∈ Q [m] of the simplicial complex with m < d we associate an incidence number n q equal to the number of d-dimensional simplices incident to it minus one. Every m-simplex r is assigned an energy r . In particular each node r ∈ Q [0] is assigned an energy r from a distribution g( ). Each m-simplex q ∈ Q [m] with m > 0 is assigned the energy As the simplicial complex evolves by the addition of new simplices, the incidence numbers of its faces also change with time.
The evolution of the NGF is determined by two parameters: the flavor s ∈ {−1, 0, 1} and the inverse temperature β ≥ 0.
The dynamics of the NGF is dictated by the following algorithm: at time t = 1 the simplicial complex is formed by a single d-simplex. At each time t > 1 a (d − 1)-face α is chosen with probability where Z [s] is called the partition function of the NGF and is given by The NGFs have two phases: a stationary phase for β < β c (s) and a non-stationary phase for β > β c (s) where β c (s) is a flavor dependent critical value for β. In the stationary phase the NGF are small-world, i.e. have Hausdorff dimension The maximum distance D of the nodes from the initial simplicial complex therefore scales with the total number of nodes Moreover, the topology of the network can be characterized by the distribution obeyed by the incidence numbers. In particular it has been found in Ref. [9,11] that the average value of the incidence number of m-simplices with energy , denoted by n| , follows the Fermi-Dirac distribution, the Boltzmann distribution or the Bose-Einstein distribution depending on the values of d, m and s (see Table 1). The phase transition at β = β c (s) is characterized by the breakdown of the selfconsistent hypothesis that is responsible for the emergence of the quantum statistics in these simplicial complexes. Together with the breakdown of the stationarity, for β > β c (s) the skeleton of the NGF display topological phase transitions. For s = −1 it show deviations from the small-world scaling defined in Eq. (34). In particular for s = −1 and β > β c the network develops a finite Hausdorff dimension d H (see Figure  2). Therefore the maximum distance D of the nodes from the initial simplex scales with the number of nodes For s = 0 and s = 1 with β > β c (s) the NGFs develop hub nodes that are connected to a large number of nodes (see Figure 3).

The ensemble of weighted normalized Laplacians
In this section our goal is to define the theoretical framework of a real space RG approach to calculate the spectrum of the normalized m-dimensional up-Laplacian Table 1. The average n| of the incidence numbers n of m-faces with energy in a d-dimensional NGF of flavor s follows either the Fermi-Dirac, the Boltzmann or the Bose-Einstein statistics depending on the values of the dimensions d and m [9]. of the Apollonian and the pseudo-fractal simplicial complexes. The renormalization group acts on a weighted simplicial complex in which we attribute a weight p τ to each (m+1)-dimensional simplex τ while the topology of the simplicial complex remains fixed. Therefore in the RG approach we investigate the RG flow defined over the ensemble of weighted normalized up-Laplacian matricesL up [m] of elements where p τ (qr) indicates the weight of the (m + 1)-dimensional simplex τ incident to both r and q and s r indicates the strength of the simplex r, i.e. s r = N [m+1] n τ ⊃r p τ . From here on, we will focus on finding the density of eigenvalues of the up-Laplacian of order m. In the following sections we will therefore adopt a simplified notation, dropping the ↑↓ as a ↑↓ , a [m] as a and so on.

Gaussian models and Laplacian spectrum
The density of eigenvalues of a symmetric matrix can be derived analytically using the properties of the Gaussian model following a standard procedure of statistical mechanics [29] quite common in Random Matrix Theory [?]. Therefore if we want to derive the density of eigenvalues of the m-dimensional up-LaplacianL which for generation n will be a N n × N n symmetric matrix we should consider the Gaussian model whose partition function reads where µ r are the eigenvalues of the normalized up-Laplacian matrixL and the differential Dψ stands for By changing variables and putting φ = ψ/ √ s r the partition function can be rewritten as with where r, q are both m-simplices, i.e. r, q ∈ Q [m] . The spectral densityρ(µ) of the normalized Laplacian matrix can be found using the relation [68] ρ where f is the free-energy density defined as In fact, inserting Eq. (37) in the Eq. (42) we obtain Therefore we can show that Eq. (41) is correct by plugging the final expression for f in Eq.(41),ρ

The general RG approach
As was the case in Ref. [28], where the spectrum of the 0-Laplacian was derived using the RG flow, the parameters p and µ are renormalized differently for faces of different type when we study the spectrum of the m-dimensional up-Laplacian. The partition function Z n (ω) corresponding to the Gaussian model of the simplicial complex evolved up to generation n is a function of the parameters ω = ({µ }, {p }), and can be expressed as where with Q [m+1] n ( ) indicating the set of (m+1)-dimensional simplices of type in a simplicial complex evolved up to generation n and with r, q ∈ Q [m] . The Gibbs measure of this Gaussian model is given in terms of the Hamiltonian H({φ}) defined in Eq. (46) as In order to calculate the partition function Z n (ω) we adopt a real space renormalization group approach. We will first integrate the Gaussian fields corresponding to the N n m-dimensional simplices added to the simplicial complex at generation n and then iteratively integrate over the simplices added at generation n − 1 and so forth, until all the integrals in the definition of the partition function Z n (ω) are performed. More specifically we consider the following real space renormalization group procedure. We start with initial conditions µ = µ and p = 1 for all values of > 0. At each RG iteration, we integrate over the Gaussian variables φr associated to simplicesr ∈ N n and we rescale the remaining Gaussian variables in order to obtain the renormalized Gibbs measure P ({φ }) of the same type as Eq. (47) but with rescaled parameters where The fields are rescaled in a way that keeps p 1 = 1 at each iteration of the RG flow, i.e. the weight of the (m + 1)-dimensional faces of type = 1 is always fixed to one. It follows that at each step of the RG transformation we have where, This procedure allows us to determine the renormalization group transformation R acting on the model parameters ω = ({µ }, {p }), Under the renormalization group flow, the partition function transforms according to By using Eq. (21) and Eq. (26), the free energy density at generation n can be approximated as for the Apollonian simplicial complexes, and as for the pseudo-fractal simplicial complexes. We will show in the next section that the RG flow for this Gaussian model is determined by the fixed point at µ = 0. This implies that the spectral dimension of higher-order up-Laplacians is universal [31], i.e. it is the same for normalized and un-normalized up-Laplacians.

The integral
To derive the renormalization group equations for the Apollonian simplicial complex we need to perform the integration over the Gaussian fields associated to the m-simplices added at generation n. In the Apollonian simplicial complex, any d-simplex of generation n is only incident to d-simplices added at previous generations. Specifically, every new d-simplex contains a single new node and shares exactly one of its (d − 1)faces with the Apollonian simplicial complex at the previous iteration. Therefore the integrations over all m-simplices added at iteration n can be performed independently by separately considering the Gaussian fields corresponding to m-simplices belonging to different d-simplices added at iteration n. Consequently in this paragraph we only focus on the integration over the Gaussian fields associated to m-simplices belonging to a single d-simplex of generation n.
In order to perform this integral let us define some notation. Given the generic d-simplexr added at iteration n, i.e.r ∈ N n , we indicate with j its most recent node, i.e the single node j ⊂r of type = 0. Each d-simplexr added at generation n contains d m new m-simplices added at generation n. All these simplices include the node j and m other nodes out of the d nodes of type > 0 belonging tor. We will denote the set of these m-simplices by M n and the Gaussian fields associated to the m-simplices q ∈ M n byψ q . Additionally, the simplexr contains d m+1 m-faces formed exclusively by nodes of type > 0. We will denote the set of these simplices by R n and the Gaussian fields associated to the m-simplices q ∈ R n by φ q . Finally, let us define Q [m+1] to be the set of all (m + 1)-dimensional faces of the simplexr added at iteration n. With this notation, the integral over the fields {ψr} reads, where H 0 ({ψ}, {φ}) is given by and H 1 ({ψ}, {φ}) is given by Here A rq is given by and Dψ is defined by The integral I m is given by where We note that for m = d − 1, the cardinality of the set M n equals one. Therefore the integral I d−1 simplifies to and G(µ 1 ) given by Eq.(63) simplifies to Given the different structure of the integral I m for m ≤ d − 2 and for m = d − 1, we will treat the case m ≤ d − 2 and the case m = d − 1 separately in the subsequent paragraphs.

The RG equations for
In this section we will show that the RG equations for the Apollonian simplicial complex for m ≤ d − 2 have the explicit expression, for all ≥ 1. The initial conditions for all ≥ 1 are (µ , p ) = (µ, 1) with µ 1. This result generalizes the RG equations that were found in Ref. [28] and can be derived using a similar procedure. The results derived in Ref. [28] correspond to the case of m = 0 in Eqs. (67).
According to the renormalization group procedure explained in the previous section, we have to integrate over each m simplexr ∈ N n at each iteration of the RG procedure. Each integration over the generic simplexr performed in Eq.(62) contributes to the If we just focus on the term coupling different Gaussian fields for any (m + 1)dimensional simplex which include both q and r the contribution is, In the Apollonian simplicial complex, there are w [m+1] d-simplices of iteration n incident to a (m+1)-simplex of type , including both the m simplex q and simplex r. The overall contribution to the term proportional to φ r φ q in H ({φ }) is It follows that, before rescaling, the overall contribution of the integrals overr ∈ N n to the term of the Hamiltonian H ({φ }) proportional to φ r φ q is given by The real space RG procedure prescribes that after rescaling of the fields φ q → φ q , we should have The correct rescaling of the fields that ensures p 1 = p 1 = 1 is given by Here we have used w In order to find the RG equations for µ , we need to consider the contribution to the rescaled Hamiltonian coming from the integral I m in Eq. (68) that is proportional to φ 2 q . This contribution is, Since there are w [m] d-simplicies of generation n incident to the m-simplex q added at generation n = n − , the integration over the Gaussian fields corresponding to the simplices added at generation n contributes, to the Hamiltonian for each m-dimensional simplex q. Let us now equate the term proportional to φ 2 q in the Hamiltonian before and after the rescaling of the fields, i.e.
We observe that the coefficients w [m] can be written as where c is given by After rescaling the fields according to Eq. (73), using Eq. (78) and Eq. (77) we get the RG equation for µ , This completes our derivation of the RG equations Eq.(67).

5.
3. The free-energy density and spectral dimension for m ≤ d − 2 Using the renormalization group and in particular equation Eq. (53) for the partition function, we can calculate the function g(ω) where c indicates a constant. The first term on the right hand side of this equation comes from the result of the integral I m in Eq. (62). The second term is the contribution due to the rescaling of the fields given by Eq. (73). Given this expression for g(ω) the free energy density f can be obtained from Eq. (55), Anticipating that the relevant fixed point at (µ , p 2 ) = (0, p ) is repulsive, we assume that close to this fixed point the RG flow can be described by the equations where µ indicate the value of µ 1 and p 2 at the iteration τ of the RG transformation, and where λ > 1 is the largest eigenvalue of the RG equations linearlised close to the relevant fixed point. Therefore using Eq. (41) the spectral densityρ(µ) can be found by, We notice that for m > 0 the spectrum acquires a delta peak at µ = 0, corresponding to the finite density of zero eigenvalues of the up-Laplacian, i.e.
In fact by using the relation 1 π and the RG flow given by Eq. (134), we have The regular part of the density of eigenvalues ρ(µ) is given by This expression can be approximated by substituting the sum over τ with an integral. Upon changing the variable of this integral to z = λ τ we can use the theorem of residues at µ (τ ) to solve the integral, obtaining the asymptotic scaling where the spectral dimension d S is given by,

RG equations for
In this paragraph we will show that for m = d − 1, the RG equations read, for all ≥ 1 with initial conditions (µ , p ) = (µ, 1) with µ 1. First we observe that for m = d − 1 the contribution of the integral I d−1 to the Hamiltonian H (φ ) is given by This contribution does not contain any term proportional to φ r φ q . This observation automatically indicates that p = 1 for all and that the rescaling of the fields is trivial, i.e. φ q = φ q . The RG equations for µ can be obtained by proceeding as for the case m < d − 1 and investigating the contributions of the integral I d−1 to the Hamiltonian. In particular, if q is a type = 1 simplex, the term proportional to φ 2 q transforms as, If instead the (d − 1)-simplex q is of type > 1, after one RG step we have, In fact, any (d − 1)-dimensional simplex q of type > 1 after the RG step is incident exclusively to a d-dimensional simplex of type and another d-dimensional simplex of type − 1. Eqs. (93) and (94) can be solved and reduce to the single RG equation valid for m = d − 1 5.5. The free-energy density and spectral dimension for m = d − 1 For m = d − 1 the RG flow is dictated by the Eqs. (91) and there is no rescaling of the fields. In this case the free-energy can be calculated using Eq. (55) with g(ω) given by where c is a constant. Note that this expression for g(ω) differs from Eq. (81) as it does not contain the terms related to rescaling of the fields. Using this expression and the Eq. (55) we can approximate the free energy f by, with G(µ 1 ) given by Eq.(65). Using Eq. (41) we can deduce the spectral densityρ(µ) given by

RG flow for the Apollonian simplicial complex
In this section we will investigate the RG flow for the spectrum of the m dimensional up-Laplacians on a d-dimensional Apollonian simplicial complex and we will derive its density of eigenvalues and its spectral dimension. Interestingly, the RG equations can be easily treated in full generality by considering the cases m = d − 1, m = d − 2 and m ≤ d − 3.

Case
The RG equations for the case m = d − 1 are given by Eq.(91), which we will repeat here for convenience The initial condition is µ = µ 1. From these equations we can obtain the recursive equation for µ where µ (0) 1 = µ 1. The fixed points of this RG flow are given by The relevant fixed point is defined in Eq.(101), the derivative of the recursive RG equation close to this fixed point at µ 1 is given by Since λ 1 < 1 it follows that the fixed point µ 1 defined in Eq.(101) is attractive. Consequently, the RG flow starting from µ 1 converges fast towards the fixed point µ 1 defined in Eq. (101). The fixed point µ 1 is of the same order of magnitude as the initial condition for µ.
In this case the fixed point is not at zero but at µ 1 = O(µ). Moreover, the fixed point is attractive. This constitute a rather special scenario that we will not find for smaller values of m. A careful study of the equation (98) for the spectral density ρ(µ) reveals that in this case the corresponding up-Laplacian does not display a finite spectral dimension.

Case m = d − 2
For m = d − 2 the RG Eqs. (67) imply that for all ≥ 1, while µ 1 and p obey the following recursive RG equations, with initial condition (µ , p ) = (µ, 1) with µ 1 for all ≥ 1. In the zero order approximation we can put µ 2 = µ = 0. Therefore the renormalization group equations (107) have three fixed points: Close to the fixed point defined in Eq.(109) the linearised RG equations read, It follows that the eigenvalues of the Jacobian are Using Eq. (90) one would expect the spectral dimension d S is given by However, this is incorrect, because the RG flow is affected by the third fixed point close to the pole at µ  iteration. This set of equations is given by with initial conditions x (0) = 1 − µ and p (0) = 1. In order to find the solution of these equations we use the auxiliary variable y (τ ) 1 given by The explicit solution of the RG equations (116) reads with τ ≤ τ . This solution therefore seems to indicate that in order to calculate the value of x (τ +1) and p (τ +1) the knowledge of the entire RG flow up to iteration τ is necessary. However, one can recover some Markovian recursive equations by introducing the additional auxiliary variables called A (τ ) , B (τ ) and C (τ ) . The auxiliary variables A (τ ) , B (τ ) and C (τ ) are defined as The variables y can be simply expressed in terms of A (τ ) , B (τ ) and C (τ ) by The solution of the RG equations can be written as the following set of recursive equations for A (τ ) , B (τ ) and C (τ ) This set of equations can be written as a closed set of equations for A (τ ) , B (τ ) and C (τ ) using Eq. (118), with initial conditions A (0) = 1, The fixed point of these RG equations at µ = 0 is The Jacobian matrix of these RG equations has eigenvalues λ 1 > λ 2 > λ 3 given by , with λ 1 > 1 and λ 2 < 1. The right eigenvectors corresponding to these eigenvalues are (1, 0, 0) , where c 1 and c 3 are normalization constants. The left eigenvectors corresponding to these eigenvalues are where d 1 , d 2 , d 3 are normalization constants. In order to solve Eqs.(119) we indicate with X (τ ) the column vector By linearizing Eqs.(119) near the fixed point X given by we obtain For the the leading order term, we have where the scalar product is, We therefore have proved that for µ 1 we have, Using Eq. (90) it follows that for m ≤ d − 3 the spectral dimension d S decreases with increasing m and is given by Finally, we observe that in the limit d → ∞ and m d the spectral dimension scales like The spectral dimension therefore grows faster than linearly with the topological dimension d.

The RG equations
In a d-dimensional pseudo-fractal simplicial complex at each iteration n each (d − 1)simplex is glued to a new d-dimensional simplex. The difference with the algorithm generating the Apollonian simplicial complexes is that in the case of the Apollonian simplicial complex at each iteration n only the (d − 1)-simplices of the last generation are glued to a new d-dimensional simplex. Given the structure of the pseudo-fractal simplicial complex and its relation to the Apollonian simplicial complex, which was already noted in Ref. [28], the general RG equations for the pseudo-fractal simplicial complex can be easily derived from those for the Apollonian simplicial complex. In fact it is sufficient to observe that in the pseudo-fractal simplicial complex each simplex of type receives the sum of the contributions coming from the integration of the Gaussian variables associated to the d-simplices added at the last generation. The RG equations for m ≤ d − 2 are therefore given by for all ≥ 1, with initial conditions (µ , p ) = (µ, 1) with µ 1 for all ≥ 1. For m = d − 1 every m-simplex of type is connected to a d-simplex of generation n and the RG equations for m = d − 1 and ≥ 1 read and with initial conditions (µ , p ) = (µ, 1) with µ 1 for all ≥ 1.

The free energy density and the spectral dimension
The free energy is given by Eq. (56). By using a procedure similar to the one used to derive the corresponding expression for the Apollonian simplicial complex we easily find for m ≤ d − 2 where c is a constant. Given this expression for g(ω), the free energy density f obtained from Eq. (56) reads For the pseudo-fractal complex, we expect to find a relevant repulsive fixed point at (µ , p 2 ) = (0, p ). Under this hypothesis the RG flow is described by close to the relevant fixed point, where λ > 1 is the largest eigenvalue of the linearized RG equations close to the relevant fixed point. Using Eq. (41), the spectral densitȳ ρ(µ) can be expressed as where y = d − m − 1. In the pseudo-fractal simplicial complex, the spectrum of the up-Laplacian of order m acquires a delta peak at µ = 0 as well. This corresponds to the finite density of zero eigenvalues of the up-Laplacian, i.e.
whereρ(0) given bȳ and the regular part of the spectrum is given by By approximating this expression with an integral over τ and by changing the variable of this integral to z = λ τ we can approximate ρ(µ) by using the residue theorem at the pole µ (τ ) The spectral dimension d S is then given by For m = d − 1 the Gaussian fields are not rescaled and g(ω) is given by where c is a constant. Using this expression and Eq. (56) we can approximate the free energy f by with G(µ 1 ) given by Eq.(65). Using Eq. (41), we can deduce that the spectral densitȳ ρ(µ) is given by

RG flow for the pseudo-fractal simplicial complex
In this section we will treat the RG flow for the pseudo-fractal simplicial complex. We consider the cases m = d − 1, m = d − 2, m = d − 3 and m < d − 3.

Case
The RG equations for m = d − 1 are given by Eqs.(132), which can be used to derive the following recursive equation for µ 1 , with initial condition µ 1 = µ. The fixed points of this equations are At the fixed point at µ = 0 the recursive equation Eqs.(144) has eigenvalue so µ = 0 is a repulsive fixed point. The RG flow starts from µ 1 = µ 1 and runs away from µ = 0 according to In doing so, the RG flow approaches the point µ 1 = d + 1 and changes its trend, in some cases even changing sign. This scenario is apparent from Figure 5 were the absolute values of µ for ≥ 1 and for all ≥ 2. The resulting RG equations are with initial conditions (µ , p ) = (µ, 1) with µ 1 for all ≥ 1. The fixed point is (µ , p ) = (0, 1). The eigenvalue of this system of equations is The fixed point is µ 1 = 0 and p = 1 with eigenvalue λ = 2. Using Eq.(140) we can predict the spectral dimension with initial conditions x (0) = 1 − µ and p (0) = 1. Equations (154) can be solved in terms of the auxiliary variable and we obtain Also in the pseudo-fractal case these non-Markovian equations can be turned to Markovian iterative relations by expressing the variable at iteration τ + 1 exclusively in terms of the variable at iteration τ . This is achieved by introducing the auxiliary variables A (τ ) , B (τ ) , C (τ ) , D (τ ) and E (τ ) defined as These auxiliary variables are related to y (τ ) The recursive Markovian RG equations for the case m = d − 3 read , with initial conditions A (1) , B (1) , C (1) , D (1) and E (1) , which can be found by inserting x The relevant fixed point of these equations is Close to this fixed point, the RG equations (159) have the relevant eigenvalue wherex is the largest positive real root of the equation Using Eq.(140) we obtain that the spectral dimension d S is therefore given by In this paragraph we study the RG flow for the pseudo-fractal simplicial complex for m < d − 3. By expressing Eqs. (130) in terms of the variables x (τ ) defined in Eq. (115) and the variables p (τ ) calculated at iteration τ , we obtain the recursive equations with initial conditions x (0) = 1 − µ and p (0) = 1. These equations can be solved in terms of the variables y (τ ) 1 defined as In particular the solution of Eqs. (164) is given by In order to turn this system of equations into a Markovian system of equations, we again express the variables at iteration τ + 1 only in terms of variables at iteration τ . We then have with A (τ ) , B (τ ) , C (τ ) , D (τ ) , E (τ ) given by The RG flow can therefore be cast in a set of recursive equations for A (τ ) , B (τ ) , C (τ ) , D (τ ) and E (τ ) given by  Table  3).
Here we make an additional useful observation. As is true for the specific case m = 0 and d > 3 (see Ref. [28]) and in the more general case investigated here with m < d − 3, we observe that the RG Eqs.(164) of the pseudo-fractal simplicial complex have the same leading term of the RG Eqs.(116) valid for the Apollonian simplicial complex with m < d − 3. Therefore the leading eigenvalue λ of the Eqs.(164) is given It follows that for d 1 and m finite, the spectral dimension d S obeys the asymptotic scaling i.e. it grows faster than linearly with d. and pseudo-fractal simplicial complexes (m ≤ d − 2 ) up to dimension d = 9 are shown in Table 2 and Table 3 respectively. In Figure 6 and Figure 7) we compare the spectra obtained by numerical diagonalization of the of the higher-order up-Laplacians for Apollonian and pseudo-fractal simplicial complexes of dimension d = 3 and d = 4. We find a very good agreement with our exact analytical results. In addition we can fit the numerical data finding the spectral dimensions for the case m = d − 1 of the Apollonian simplicial complex and the case m = d − 2 of the pseudo-fractal simplicial complex. From our RG calculations of the spectrum of higher-order up-Laplacians of Apollonian simplicial complexes and pseudo-fractal simplicial complexes and its numerical validation we draw the following main conclusions: (1) Higher-order up-Laplacians of order m on Apollonian and pseudo-fractal simplicial complexes display a finite spectral dimension with the only exception of the case of m = d − 1 for the Apollonian simplicial complex. A single simplicial complex generated by the over-mentioned models is therefore not just characterized by a single spectral dimension but by multiple spectral dimensions corresponding to different orders m.
(2) The analytical prediction of the spectrum of the m-order up-Laplacian on ddimensional Apollonian and pseudo-fractal simplicial complexes shows that the spectral dimension d S decreases with increasing m as long as m ≤ d − 3 for the Apollanian simplicial complexes and as long as m ≤ d − 2 for the pseudo-fractal simplicial complex.
(3) The symmetries of the simplicial complex do not only induce degenerate eigenvalues for the graph Laplacian [28] but also for their higher-dimensional counterparts. Indeed, from our numerical results ( Figures 6) and 7)) we observe that the higherorder up-Laplacian have several eigenvalues that are highly degenerate.

Higher-order spectral dimensions of "Network Geometry with Flavor"
In this paragraph we discuss the numerical results concerning the spectrum of the higher-order up-Laplacians of order m on "Network Geometries with Flavor" (NGF) above and below the phase transition at β = β c (s). This shed some light on the role that the stochastic evolution of the simplicial complex can have on the spectrum of the higher-order Laplacians. In Figure 8 we display the spectrum of the up-Laplacians of order m of a NGF of flavor s ∈ {−1, 0, 1}, dimension d = 4 and inverse temperature β = 0.1 < β c (s) and β = 8 > β c (S). By comparing the results obtained for NGFs above and below the phase transition and by comparing the spectrum of NGFs with s = −1 to Apollonian simplicial complexes and the spectrum of NGFs with s = 0 and s = 1 to the spectrum of pseudo-fractal simplicial complex we make the following observations: (1) The higher-order spectral dimension of the NGFs change with respect to the spectral dimensions of the Apollonian and pseudo-fractal simplicial complexes.
(2) The discontinuities observed in the cumulative distribution of eigenvalues ρ c (λ) for the Apollonian and the pseudo-fractal simplicial complexes are strongly suppressed in the NGFs indicating that these randomly growing simplicial complexes are less symmetric than their deterministic counterpart. A similar observation was already made in Ref. [28] by considering the cumulative density of eigenvalues of the graph Laplacian of Apollonian simplicial complexes and NGFs.
(3) The higher-order spectral dimension of NGFs depends on the value of the flavor s, the dimension d and the inverse temperature β.
(4) For β > β c (s) the asymptotic scaling of the cumulative distribution is perturbed and the numerical results do not give conclusive results regarding the existence of a very well defined spectral dimension. Possibly one could interpret these results as showing a scale-dependent spectral dimension. For s = −1 the spectrum of the graph-Laplacian ((m = 0)) appears to develop a spectral dimension less than two, while the up-Laplacian with m > 0 appears to develop a spectral gap. For s = 0 and s = 1 we observe the emergence of a very degenerate eigenvalue. In the case of s = 1, we observe that the spectrum of the up-Laplacian of order (m = d − 1) seems to develop a small spectral dimension, while the up-Laplacian of order m < d − 1 seems to develop a spectral gap.

Conclusions
Higher-order Laplacians are important topological objects that generalize graph Laplacians and extend the notion of diffusion to higher dimension. Here we show that three classes of non-amenable simplicial complex models (the Apollonian simplicial complex, the pseudo-fractal simplicial complex and the "Network Geometry with Flavor") display finite higher-order spectral dimensions d S . We observe that a single simplicial complex can be characterized by a set of spectral dimensions corresponding to the spectrum of the up-Laplacians of different order m. We have used renormalization group methods applied to a Gaussian model to predict the higher-order spectral dimension d S of up-Laplacians of order m of the Apollonian simplicial complex and the pseudo-fractal simplicial complex of arbitrary dimension d. With our RG approach it is possible to analytically calculate the spectral dimension d S for order m ≤ d − 3 for the Apollonian simplicial complexes and for order m ≤ d − 2 for pseudo-fractal simplicial complexes. In these cases the spectral dimensions are determined by the scaling of the RG flow away from the repulsive fixed point at zero mass, i.e. at (µ 1 , p 2 ) = (0, p ). Additionally we have found that in the range of values of m for which we can predict the spectral dimension, the spectral dimension d S up-Laplacians of order m decreases as m increases. Our analytical calculations are validated by numerical results. Finally we study numerically how randomness can affect these results by investigating the higher-order spectral properties of simplicial complexes generated by the model called "Network Geometry with Flavor"(NGF). These simplicial complexes can be seen as random counterparts of the Apollonian simplicial complexes and the pseudo-fractal simplcial complexes. We observe that the role of randomness can be moderate, perturbing essentially the value of the higher-order spectral dimension, or drastic, in correspondence to the topological phase transitions of NGFs. However, the role of the topological phase transitions in changing the spectral properties of simplicial complexes is only mentioned here. A full understanding of these phenomena are beyond the goal of the present work. We hope that this work can stimulate further research on higher-order spectral dimensions and topological phase transitions in different fields related to network topology including quantum gravity and brain research.