Abstract
Higher order networks are able to characterize data as different as functional brain networks, protein interaction networks and social networks beyond the framework of pairwise interactions. Most notably higher order networks include simplicial complexes formed not only by nodes and links but also by triangles, tetrahedra, etc. More in general, higherorder networks can be cellcomplexes formed by gluing convex polytopes along their faces. Interestingly, higher order networks have a natural geometric interpretation and therefore constitute a natural way to explore the discrete network geometry of complex networks. Here we investigate the rich interplay between emergent network geometry of higher order networks and their complexity in the framework of a nonequilibrium model called Network Geometry with Flavor. This model, originally proposed for capturing the evolution of simplicial complexes, is here extended to cellcomplexes formed by subsequently gluing different copies of an arbitrary regular polytope. We reveal the interplay between complexity and geometry of the higher order networks generated by the model by studying the emergent community structure and the degree distribution as a function of the regular polytope forming its building blocks. Additionally, we discuss the underlying hyperbolic nature of the emergent geometry and we relate the spectral dimension of the higherorder network to the dimension and nature of its building blocks.
Introduction
Network Science [1,2,3,4,5] has allowed an incredible progress in the understanding of the underlying architecture of complex systems and is having profound implications for different fields ranging from brain research [6] and network medicine [7] to global infrastructures [8].
It is widely believed [9] that in order to advance further in our understanding of complex systems it is important to consider generalized networks structures. These include both multilayer networks formed by several interacting networks [10, 11] and higher order networks which allow going beyond the framework of pairwise interactions [12,13,14,15,16,17,18,19,20,21,22,23,24,25,26].
Higher order networks can be essential when analyzing brain networks [12, 27,28,29,30], protein interaction networks [31] or social networks [32, 33]. For instance in brain functional networks, it is important to distinguish between brain regions that interact as a pair, or as a part of a larger complex, yielding their simultaneous coactivation [12]. Similarly, protein interaction networks map the relations between protein complexes of the cell, which are formed by several connected proteins that are able to perform a specific biological function [31]. In social networks simplicial complexes arise in different contexts [32,33,34,35], as for instance in facetoface interacting networks constituted by small groups that form and dissolve in time, usually including more than two people [32, 33].
In many cases the building blocks of a higher order network structures are ddimensional simplices such as triangles, tetrahedra etc., i.e. a set of \((d+1)\) nodes in which each node is interacting with all the others. In this case higher order networks are called simplicial complexes. However, there are some occasions in which it is important to consider higher order networks formed by building blocks that are less densely connected than simplices, i.e cellcomplexes formed by gluing convex polytopes. Cellcomplexes are of fundamental importance for characterizing selfassembled nanostructures [36] or granular materials [37]. However examples where cellcomplexes are relevant also for interdisciplinary applications are not lacking. For instance, a protein complex is formed by a set of connected proteins, but not all proteins necessarily bind to every other protein in the complex. Also in facetoface interactions, a social gathering of people might be organized into small groups, where each group can include people that do not know each other directly. These considerations explain the need to extend the present modelling framework from simplicial complexes to general cellcomplexes formed by regular polytopes such as cubes, octahedra etc.
Modelling frameworks for simplicial complexes [12,13,14,15,16,17,18,19,20,21,22,23,24,25,26] include both equilibrium static models that can be used as null models [20,21,22,23,24,25,26, 38] and nonequilibrium growing models describing their temporal evolution [13,14,15,16,17,18,19]. However, modelling of cellcomplexes has been mostly neglected by the network science community.
Characterizing nonequilibrium growth models of cellcomplexes allows us to investigate the relation between the local geometrical structure of the higher order networks and their global properties, revealing the nature of their emergent geometry and their complexity.
Interestingly, simplices and more in general convex polytopes have a natural geometrical interpretation and are therefore essential for investigating network geometry [13, 15]. As such simplicial complexes are widely adopted in quantum gravity for investigating the geometry of spacetime [39,40,41]. Network geometry is also a topic of increasing interest for network scientists which aim at gaining further understanding of discrete network structures using geometry. This field is expected to provide deep insights and solid mathematical foundation to the characterization of the community structure of networks [42,43,44], contribute in inference problems [45] and shed new light onto the relation between structure (and in specific network geometry) and dynamics [27, 46].
The recent interest in network geometry is reflected in the vibrant research activity which aims at defining the curvature of networks and at extracting geometrical information from network data using these definitions [47,48,49,50,51,52,53]. In a variety of cases [54,55,56,57] it has been claimed that actually the underlying network geometry of complex networks is hyperbolic [58]. This hidden hyperbolic geometry is believed to be very beneficial for routing algorithms and navigability [55, 56, 59]. While several equilibrium and nonequilibrium models imposing an underlying hyperbolic network geometry have been widely studied and applied to real networks [60, 61], recently a significant progress has been made in characterizing emergent hyperbolic network structures [15]. In particular it has been found that Network Geometry with Flavor [14] is a comprehensive theoretical framework that provides a main avenue to explore emergent hyperbolic geometry [15]. This model uses a nonequilibrium evolution of simplicial complexes that is purely combinatorial, i.e. it makes no assumptions on the underlying geometry. The hyperbolic network geometry of the resulting structure is not a priori assumed but instead it is an emergent property of the network evolution.
The theoretical framework of Network Geometry with Flavor shows that nonequilibrium growth dictated by purely combinatorial and probabilistic rules is able to generate an hyperbolic network geometry, and at the same time determines a comprehensive theoretical framework able to generate very different network structures including chains, manifolds, and networks growing with preferential attachment. Most notably this model includes as limiting cases models that until now have been considered to be completely independent such as the Barabási–Albert model [1] and the random Apollonian network [62,63,64,65].
In this paper we extend the Network Geometry with Flavor originally formulated for simplicial complexes to cellcomplexes formed by any type of regular polytopes. In particular we will focus on Network Geometry with Flavor \(s\in \{1,0,1\}\) built by subsequently gluing different copies of a regular polytope along its faces. Note that in this paper we conside cellcomplexes formed by an arbitrary regular polytope but any cellcomplex is pure, i.e. it has only one type of regular polytope forming its building blocks.
Although cellcomplexes can be in several occasions a realistic representation of network data, is not our intention to propose a very realistic model of cellcomplexes. Rather our goal is on one side to propose a very simple theoretical model for emergent geometry and on the other side to investigate the interplay between its geometry and its complexity.
The network geometry is investigated by characterizing the Hausdorff, the spectral [66,67,68,69] and the cellcomplex’ topological dimension, together with the ”holographic” nature of the model. The complexity of the resulting network structures is studied by deriving under which conditions the resulting networks are scalefree and display a nontrivial emergent community structure.
Finally, Network Geometry with Flavor can be considered as the natural extension of the very widely studied framework of nonequilibrium growing complex networks models (with and without preferetial attachment) to characterize network geometry in any dimension. In this respect many nontrivial results are obtained. For instance, we show that when working with simplices scalefree networks can emerge from a dynamical rule that does not contain an explicit preferential attachment mechanism. Additionally, we reveal that even when preferential attachment of the regular polytopes is present, the Network Geometry with Flavor might result in a homogeneous network structure in which the second moment of the degree distribution is finite in the large network limit.
Simplicial Complexes and Higher Order Networks
Simplicial complexes provide the main example of higher order networks where interactions are not only pairwise, but can include more than two nodes. Simplicial complexes are formed by simplices glued along their faces. A simplex of dimension d is a set of \(d+1\) nodes and describes the manybody interaction between these nodes. A simplex admits a natural geometrical interpretation. For instance a simplex of dimension \(d=0,1,2,3\), can be identified with a node, a link, a triangle and a tetrahedron respectively. A \(\delta \)dimensional face of a simplex \(\alpha \) of dimension \(d>\delta \) is a simplex \(\hat{\alpha }\) formed by a subset of \(\delta +1\) of the nodes of \(\alpha \). A simplicial complex \(\mathcal {K}\) of dimension d is formed by a set of simplices of dimension \(\delta \le d\) glued along their faces. Additionally, this set must be closed under the operation of taking faces of any simplex. Therefore, in mathematical terms it must satisfy two conditions:

(a)
the intersection \(\alpha \cap \tilde{\alpha }\) of two simplices \(\alpha \in {{\mathcal {K}}}\) and \(\tilde{\alpha }\in {{\mathcal {K}}}\) belonging to the simplicial complex is a simplex of the simplicial complex, i.e. \(\alpha \cap \tilde{\alpha }\in {{\mathcal {K}}}\);

(b)
if the simplex \(\alpha \) belongs to the simplicial complex, i.e. \(\alpha \in {{\mathcal {K}}}\), then every simplex \(\hat{\alpha }\) which is a face of \(\alpha \) (i.e. \(\hat{\alpha }\subset \alpha \)) must also belong to the simplicial complex, i.e. \(\hat{\alpha }\in {{\mathcal {K}}}\).
Among simplicial complexes we distinguish pure ddimensional simplicial complexes which are formed exclusively by ddimensional simplices and their faces.
Here we consider not only simplicial complexes, but we treat higher order networks including simplicial complexes and also cell complexes, which differ from simplicial complexes because they are formed by subsequently gluing convex polytopes along their faces. In particular we will focus on cellcomplexes \({{\mathcal {Q}}}\) formed by identical ddimensional regular polytopes glued along their \((d1)\)faces, called here pure cellcomplexes. A pure cellcomplex reduce to pure ddimensional simplicial complex if the regular polytope that constitute its building blocks is a ddimensional simplex [70].
A regular polytope of dimension d is a maximally symmetric ddimensional polytope having F identical \((d1)\)dimensional faces and M nodes. Each node of a regular polytope has degree v and it is incident to the same number f of \((d1)\)dimensional faces. Each \((d1)\)face includes m nodes. A ddimensional simplex is a regular polytope. However, the number of regular polytopes in dimension \(d\ge 2\) is larger than one. In Table 1 we report the complete list of regular polytopes and their properties.

(1)
Dimension \(d=1\) This is the trivial case in which the regular polytope is just a single link.

(2)
Dimension \(d=2\) In dimension \(d=2\) the regular polytopes are the regular polygons, i.e. triangles, squares, pentagons, hexagons etc.

(3)
Dimension \(d=3\) In dimension \(d=3\) the regular polytopes are the Platonic solids, namely the tetrahedron, the cube, the octahedron, the dodecahedron and the icosahedron. The 5 Platonic solids are shown in Fig. 1. These solids have an underlying network structure which is planar as shown in Fig. 2.

(4)
Dimension \(d= 4\) In dimension \(d=4\) the number of regular polytopes is 6, namely the pentachoron, the tesseract, the hexadecacoron, the 24cell, the 120cell and the 600cell.

(5)
Dimension \(d\ge 5\) In dimension \(d\ge 5\) the number of regular polytopes is 3, i.e. the dsimplex, the dhypercube and the dorthoplex.
Here in the following we introduce some structural properties of the higher order networks (simplicial complexes and cellcomplexes) that will play a key role in the following paragraphs.
Let us assign to each \(\delta \)dimensional face \(\alpha \) of the pure cellcomplex a generalized degree \(\kappa _{d,\delta }(\alpha )\) indicating how many ddimensional regular polytopes are incident to the face. Additionally we associate to each \((d1)\) face of the cellcomplex an incidence number \(n_{\alpha }\) equal to the generalized degree \(\kappa _{d,d1}(\alpha )\) of the same \((d1)\)face \(\alpha \) minus one, i.e.
Being a 0dimensional face, every node \(i\in \{1,2,\ldots , N\}\) of the cellcomplex is also assigned a generalized degree \(\kappa _{d,0}(i)\) indicating how may ddimensional regular polytopes are incident to it. The degree \(k_i\) of node i is related to the generalized degree \(\kappa _{d,0}(i)\) by
where v is the degree of each node in the regular polytope. Finally we note that here we will focus mainly on network of pairwise interactions induced by the higher order network, i.e. we will mostly focus on its skeleton.
Network Geometry with Flavor
The Network Geometry with Flavor [14, 15] is a nonequilibrium model describing the evolution of higher order networks. Originally this model has been formulated to study the evolution and the emergent geometry of simplicial complexes, here we extend the model to pure cellcomplexes formed by identical regular ddimensional polytopes.
The Network Geometry with Flavor depends on the specific regular polytope that form its building blocks and in particular on its dimension d. Moreover it also depends on a parameter s called flavor taking values \(s\in \{1,0,1\}\).
The algorithm generating the Network Geometry with Flavor, is simply stated.
At time \(t=1\) the higherorder network \({{\mathcal {Q}}}\) is formed by a single regular polytope.
At each time \(t>1\) we choose a \((d1)\)dimensional face \(\alpha \) of the higher order network with probability
with
and we glue a new regular polytope to it.
In this model, the necessary and sufficient (combinatorial) condition to get a discrete manifold is that every \((d1)\)face of the higher order network has incidence network \(n_{\alpha }\in \{0,1\}\). The higher network topology generated by this model depends on the flavor s and on specific type of regular polytope that forms the building block of the structure. Here we discuss the major effect of considering different flavors.

(1)
Flavor \(s=1\) In this case we can attach a ddimensional regular polytope only to a face with \(n_{\alpha }=0\). In fact for \(n_{\alpha }=1\) we have \({\varPi }_{\alpha }^{[1]}=0\). Therefore each face of the higher order network will have a incidence number \(n_{\alpha }\in \{0,1\}\) resulting in a discrete manifold structure. We call these networks Complex Network Manifolds [16].

(2)
Flavor \(s=0\) In this case \({\varPi }_{\alpha }^{[0]}\) is constant for each face of the higher order network. Therefore the attachment probability enforces a uniform attachment in which every face has the same probability to attract new regular polytopes. Consequently the incidence number can take any value \(n_{\alpha }\in \mathbb {N}^0\).

(3)
Flavor \(s=1\) In this case the probability \({\varPi }_{\alpha }^{[1]}\) to attach a new regular polytope to the face \(\alpha \) is proportional to the generalized degree of the face \(\kappa _{d,d1}(\alpha )=1+n_{\alpha }\), resulting in a explicit preferential attachment mechanism. Consequently the incidence number can take any value \(n_{\alpha }\in \mathbb {N}^{0}\).
The Network Geometry with Flavor been proposed in Ref. [14] and Complex Network Manifolds have been first introduced in [16, 17] for describing the evolution and growth of simplicial complexes. However the Network Geometry with Flavor reduces to other known models in some specific limits.

(1)
Dimension \(d=1\) In dimension \(d=1\) the Network Geometry with Flavor is a growing tree and reduces for \(s=1\) to a growing chain, for \(s=0\) to a tree growing by uniform attachment, and for \(s=1\) it reduces to the Barabási–Albert model with preferential attachment [1].

(2)
Dimension \(d=2\) The Network Geometry with Flavor \(s=0\) having triangles as building blocks has been first proposed in Ref. [71].

(3)
Dimension \(d=3\) In dimension \(d=3\) the Network Geometry with Flavor \(s=1\) reduces to a random Apollonian network [62,63,64,65].
Therefore the Network Geometry with Flavor can be considered as a theoretical framework which unifies and extends several well known network models. Moreover as we will see in the next section it reveals an important mechanism for emergent hyperbolic network geometry.
We observe that variations of this model can be envisaged in the following directions:

(i)
The present choice of the values \(s\in \{1,0,1\}\) for the flavor s is driven by the need to explore regions of the possible parameter space with very distinct dynamics. Note however that the model can be as well studied by taking any real positive value of s (which will enforce a preferential attachment with initial attractivness of the faces [3]) or any rational negative value of s with \(1\le s<0\) (which will enforce a upper limit to the number of polytopes that are incident to any given face).

(ii)
The model can be easily extended to cellcomplexes that are not pure by allowing the gluing of regular polytopes having the same \((d1)\)faces. For instance it is possible to consider a variation of the \(d=3\) Network Geometry with Flavor in which tetrahedra, octahedra and icosahedra can be glued along their triangular faces.

(iii)
The model can be extended by associating a fitness to the faces of the cellcomplexes and modifying the attachment probability along the lines proposed in Refs. [14, 15]. This modification can lead to very interesting topological phase transitions.
Despite the fact that these modifications of the model have significant potential for understanding network geometry, in this paper, due to space limitation, we limit our study to pure cellcomplexes described by the Network Geometry with Flavor \(s\in \{1,0,1\}\) in which we do not consider the effect of the fitness of the faces.
Emergent Hyperbolic Geometry
While the definition of the Network Geometry with Flavor is purely topological, the emergent geometry is observed when one attributes equal length to all the links. Attributing the same length to each link consists of making the least biased assumption on their length. Therefore this procedure defines the main path to explore the emergent hidden network geometry of the Network Geometry with Flavor which is a combinatorial network model that makes no explicit use of the hidden geometry.
The Network Geometry with Flavor are small world [2] for every flavor s and any dimension d except from the special case \(s=1,d=1\) in which the resulting network is a chain [46]. Specifically in Network Geometry with Flavor both the diameter D and the average shortest distance increase logarithmically with the total number of nodes N. This implies that the number of nodes in the network N increases exponentially with its diameter D, i.e. \(N\simeq e^{\alpha D}\) where \(\alpha >0\). Consequently, as long as we do not allow ”crossing” of the simplices, their emergent geometry cannot be an Euclidean geometry with finite Hausdorff dimension \(d_H\) because in this case, we would observe the powerlaw scaling \(N\simeq D^{d_H}\). This observation implies that actually the Hausdorff dimension \(d_H\) of the Network Geometry with Flavor is infinite \(d_H=\infty \), with the only exception of the case \(s=1,d=1\) in which \(d_H=1\).
The emergent network geometry of the Network Geometry with Flavor is hyperbolic [15] as long as \((s,d)\ne (1,1)\). This can be shown by constructing the natural hyperbolic embedding of the Network Geometry with Flavor in the hyperbolic spaces \({\mathbb {H}}^d\), and specifically the Poincaré ball model [15]. Let us consider a Poincaré ball model of \(\mathbb {H}^d\). The Poincaré ball model includes all the points of the unit ball \(B^n=\{\mathbf{x}\in \mathbb {R}^d: \mathbf{x}<1\}\), with \(\ldots \) indicating the Euclidean norm. The Poicaré ball model is associated to the hyperbolic metric \(d_B\) assigning to each pair of points \(\mathbf{x,y}\in \mathbb {R}^d\) the distance
Here we identify every ddimensional polytope of our cell complex with an ideal regular polytope of the Poincaré ball model. An ideal regular polytope has all its nodes at the boundary of the hyperbolic ball, so all the nodes i have a position \(\mathbf{r}_i\in \mathbb {R}^d\) satisfying \(\mathbf{r}_i=1\). This construction allows for having all the connected nodes at equal hyperbolic distance. Note however this distance is infinite which is the condition we need to satisfy for having an embedding that for infinite network size fills the entire hyperbolic space. In Figs. 3 and 4 we show some examples of the hyperbolic embedding of Network Geometry with Flavor of dimension \(d=2\) and dimension \(d=3\) respectively. For dimension \(d=2\) we have considered the Network Geometry with Flavor \(s=1\) formed by triangles or squares, for dimensions \(d=3\) we have considered the Network Geometry with Flavor \(s=1\) formed by tetrahedra and cubes.
Complex Network Manifolds Topological Dimensions
The Network Geometry with Flavor has a topological dimension d given by the dimension of the d dimensional regular polytope that forms its building blocks.
In particular Complex Network Manifolds made by ddimensional simplices are ddimensional manifolds with boundary having all their nodes residing at the boundary of the manifold. Additionally the Complex Network Manifolds are \((d1)\)connected meaning that each d dimensional regular polytope can be connected to any other d dimensional regular polytope by paths that go from one d dimensional polytope to another one if they share a \((d1)\)face. Given these properties, the Complex Network Manifolds can be interpreted as \((d1)\)dimensional manifolds without a boundary by considering the cellcomplex formed by all the \((d1)\)faces with \(n_{\alpha }=0\) and all their lowerdimensional faces. In this way the ddimensional manifold can be projected on its \((d1)\)dimensional boundary without losing any information about the network skeleton, i.e. while keeping all the links.
For example Complex Network Manifolds builded by 3dimensional regular polytopes can be reduced to 2dimensional closed manifolds. Specifically a Complex Network Manifold build from tetrahedra can be reduced to a closed \(d=2\) manifold whose faces are initially four identical triangles which evolve though a sequence of successive triangulations forming a generalized Apollonian network (see Fig. 5).
These properties of Complex Network Manifolds reveal the ”holographic character” of this model and indicate that these structures are interesting for the study of network geometry of complex networks and are closely related to tensor networks which have been attracting large interest in the quantum information community (see for instance their use in [72]).
Complexity and Degree Distribution
In order to characterize the emerging complexity of the Network Geometry with Flavor, in this paragraph we derive the degree distribution \(P_{d}^{[s]}(k)\) of the Network Geometry with Flavor s and dimension d. In particular here our aim to to explore under which conditions on the flavor s, the dimensionality d and the nature of the regular polytope we observe that the Network Geometry with Flavor has a scalefree topology. A scalefree network topology is observed when the degree distribution can be approximated for \(k\gg 1 \) as a powerlaw
with powerlaw exponent
This range of powerlaw exponents indicates that the network is dominated by hubs nodes and the second moment of the degree distribution \(\langle {k^2}\rangle \) is diverging as the network size N grows, i.e. \(\langle {k^2}\rangle \rightarrow \infty \) as \(N\rightarrow \infty \) also if the average degree \(\langle {k}\rangle \) is independent of N [1, 3,4,5]. Most notably these networks are widely represented in real complex systems and have dynamical properties strongly affected by their underlying complex scalefree topology [1, 3,4,5].
In order to find the degree distribution \(P_d^{[s]}(k)\) let us first derive the expression for the probability \(\tilde{P}_{d}^{[s]}(\kappa )\) that a random node has generalized degree \(\kappa \) using the master equation approach [3].
For a realization of the Network Geometry with Flavor s and dimension d, let us indicate with \(N^{[s]}_{d}(\kappa ,t)\), the average number of nodes that at time t have generalized degree \(\kappa \). This quantity obeys the master equation
where \(\tilde{{\varPi }}(\kappa )\) is the probability that we attach a regular polytope to a \((d1)\)face including a node with generalized degree \(\kappa \), \(Mm\) is the number of nodes added to the Network Geometry with Flavor at every time step and \(\delta _{x,y}\) is the Kronecker delta.
It can be shown (see Appendix for details of the derivation) that \(\tilde{{\varPi }}(\kappa )\) is approximated for \(t\gg 1\) by
This expression indicates that in Network Geometry with Flavor, we can observe an emergent preferential attachment. In fact for \(f1+s>0\) the probability \(\tilde{{\varPi }}(\kappa )\) increases linearly with the generalized degree \(\kappa \) also when the flavor \(s\in \{1,0\}\), i.e. also when the model definition does not contain an explicit preferential attachment. Therefore in this case the emergent preferential attachment is an outcome of the network geometry.
We observe that emergent preferential attachment occurs if and only if the dimension d satisfies \(d1+s>0\). In fact the condition \(d1+s>0\) is equivalent to the condition \(f1+s>0\) (see Table 1 for the values of f as a function of d). Moreover we have \(f1+s=0\) only for \(d1+s=0\), i.e. only for \((d,s)=(2,1)\) and \((d,s)=(1,0)\). Finally only for dimension \(d=1\) and flavor \(s=1\) we can have \(f1+s=1\). This case should be consider somewhat separately because the network evolution produces a one dimensional chain having only two nodes with generalized degree \(\kappa =1\) and all the other nodes with generalized degree \(\kappa =2\). In fact \(\tilde{{\varPi }}(\kappa )>0\) only for \(\kappa =1\).
For parameter values \((s,d)\ne (1,1)\) the number of nodes that can be incident to the new polytope increases with the network size, generating a smallworld topology. In this case we can solve the master equation using techniques extensively used for growing network models [3].
Using Eq. (8) and imposing that for parameter values \((s,d)\ne (1,1)\) in the large network limit the number of nodes with generalized degree \(\kappa \) grows as
we can solve the master equation (Eq. (7)) finding the exact asymptotic result for the generalized degree distribution \(\tilde{P}_{d}^{[s]}(\kappa )\) valid as \(t\rightarrow \infty \).
In this way it can be shown that for \(d1+s=0\) the generalized degree distribution \(\tilde{P}_{d}^{[s]}(\kappa )\) is exponential and given by
For \((d1+s)>0\), however the generalized degree distribution \(\tilde{P}_{d}^{[s]}(\kappa )\) is given by
where \(\tilde{C}\) is a constant given by
Using these results of the generalized degree distribution \(\tilde{P}_d^{[s]}(\kappa )\) let us now derive the degree distribution \(\tilde{P}_d^{[s]}(k)\).
In the case \(s=1,d=1\) the Network Geometry with Flavor is a onedimensional chain and it is easy to see that the degree distribution is bimodal and given by
with \(P_d^{[s]}(k)=0\) for every \(k>2\). In fact in a chain only two nodes have degree \(k=1\) and all the other nodes have degree \(k=2\).
For \((s,d)\ne (1,1)\), using Eq. (1) we can derive the expression for the degree distribution \(P_d^{[s]}(k)\) in terms of the generalized degree distribution \(\tilde{P}_d^{[s]}(\kappa )\)
Therefore for \(d1+s=0\) (for which \(f=d\)) the degree distribution is exponential and given by
while for \(d1+s>0\) it is given by
Therefore for \(d1+s>0\) the degree distribution decays as a powerlaw, i.e. follows Eq. (5) with powerlaw exponent \(\gamma \) given by
In Figs. 6, 7 and 8 we show the agreement between these theoretical expectations and the degree distribution of simulated Network Geometry with Flavor \(s\in \{1,0,1\}\) and dimension \(d\le 5\) built by simplices, hypercubes and orthoplexes respectively.
From this analytical derivation of the degree distribution it follows that the Network Geometry with Flavor has power powerlaw degree distribution if and only if
In Table 2 we summarize the functional form (Bimodal, Exponential, Powerlaw) of the Network Geometry with Flavor as a function of the dimension d and the flavor s. Note that this classification is valid for any Network Geometry with Flavor having dimension d and flavor s independently of the specific regular polytope that forms its building blocks.
If we make the distinction between scalefree degree distributions with powerlaw exponents \(\gamma \in (2,3]\) and more homogeneous powerlaw exponents \(\gamma >3\) we notice that not only the dimensionality of the regular polytope but also its geometry has important consequences (see Table 3).
In the case of simplicial complexes, the powerlaw degree distributions of the Network Geometry with Flavor are always scalefree. This implies that explicit preferential attachment imposed by the flavor \(s=1\) always gives rise to scalefree simplicial complexes topologies with powerlaw exponent \(\gamma \in (2,3]\). Moreover this result indicates that the observed emergent preferential attachment occurring for \(s\in \{0,1\}\) implies that both simplicial Complex Network Manifolds (flavor \(s=1\)) and simplicial complexes evolving by uniform attachment (flavor \(s=0\)) are scalefree, provided that the dimension is sufficiently high. In fact the emergent preferential attachment is observed only for \(d1+s>0\).
However when we include the treatment of Network Geometry with Flavor formed by any type of regular polytope the rich interplay between network geometry and complexity is revealed and a much more nuanced scenario emerges.
The expression for the powerlaw exponent \(\gamma \) (Eq. (17)) together with the condition to get a scalefree distribution (Eq. (6)) indicates that the Network Geometry with Flavor s and dimension \(d>1s\) are scalefree only if
This relation implies the following dependence of the scalefree property with the dimension d and the flavor s.

(1)
Flavor \(s=1\) In dimension \(d\ge 3\) the Network Geometry with Flavor are powerlaw distributed. However only the simplicial complexes are scalefree.

(1)
Flavor \(s=0\) In dimension \(d\ge 2\) the Network Geometry with Flavor are powerlaw distributed. However only the simplicial complexes are scalefree.

(3)
Flavor \(s=1\) The Network Geometry with Flavor are always powerlaw distributed. For dimension \(d=1\) and \(d\ge 4\) Network Geometry with Flavor \(s=1\) implying an explicit preferential attachment are always scalefree. However for dimension \(d\in \{2,3,4\}\) they are not always scalefree.

For \(d=2\) the Network Geometry with Flavor \(s=1\) are not scalefree if they are formed by polygons different from triangles and squares.

For \(d=3\) the Network Geometry with Flavor \(s=1\) are not scalefree if they are formed by dodecahedra and icosahedra.

For \(d=4\) the Network Geometry with Flavor \(s=1\) are not scalefree if they are formed by the 24cell, the 120cell and the 600cell.

Complexity and Emergent Community Structure
An important signature of the complexity of the Network Geometry with Flavor is its emergent community structure. In fact this model, constraining the microscopic structure of the network formed by identical, highly clusterised building blocks (the regular polytopes), spontaneously generates a mesoscale structure organized in communities of nodes more densely connected with each other than with the other nodes of the network [15, 43]. In order to characterize the emergent mesoscale structure of the Network Geometry with Flavor we have estimated the maximal modularity M [4] of the network by averaging the results obtained using the GenLouvain algorithm [73, 74] over different realizations of the Network Geometry with Flavor having up to dimension \(d=5\) (see Fig. 9). From Fig. 9 it is possible to appreciate that while the modularity M decreases as the topological dimensions d increases, its values remain significant for every flavor s up to dimension \(d=5\).
A nontrivial community structure is observed very widely in network data. Therefore the emergent community structure of the Network Geometry with Flavor is a desired property for the modelling of real complex networks observed also in other growing network models [15, 43, 44, 75]. However the community structures of real datasets can display significant differences for different networks. Therefore here, it is not our intention to fit the Network Geometry with Flavor to any specific real data, rather our aim is to indicate that the Network Geometry with Flavor can provide a simple stylized mechanism to generate a discrete network structure with communities.
Spectral Dimension of Network Geometry with Flavor
The spectral dimension \(d_S\) [66,67,68,69, 76] of a networks characterizes how the structure of the network and its underlying network geometry affects the property of diffusion and has profound implications for quantum networks as well [77]. The Laplacian matrix \(\mathbf{L}\) of the network of elements
where \(\mathbf{a}\) indicates the adjacency matrix of the network characterizes fully the properties of diffusion on a given network. In fact the probability diffusion of a given continuous variable \(x_i\) defined on each node i of the network follows
with given initial condition \(x_i(0)=\delta _{i,i_0}\) describing the initial concentration of the continuous variable \(\mathbf{x}\) on the node \(i_0\). The spectral properties of the Laplacian fully determine the diffusion properties. The Laplacian has real spectrum with eigenvalues \(0=\lambda _1\le \lambda _2\ldots \le \lambda _N\). The degeneracy of the zero eigenvalue is equal to the number of connected components of the network. Therefore for Network Geometry with Flavor the zero eigenvalue is not degenerate and \(\lambda _2>0\). Let us indicate with \(\rho (\lambda )\) the density of eigenvalues. The spectral dimension, if it exist, characterizes the powerlaw scaling of \(\rho (\lambda )\) as a function of \(\lambda \)
valid for \(0<\lambda \ll 1\). In particular network models with finite spectral dimension must have \(\lambda _2\rightarrow 0\) as \(N\rightarrow \infty \).
In discrete lattices the spectral dimension \(d_S\) is known to be equal to the Hausdorff dimension \(d_H\) of the lattice and the dimension d of the unitary cell of the lattice, however the spectral dimension of a network in general is not equal to its Hausdorff dimensional and satisfies \(d_S\le d_H\) [66]. Additionally if we consider the skeleton of a ddimensional simplicial complex in general we will not find that the spectral dimension \(d_S\) is equal to d.
Note that not every network has a spectral dimension. Most notably in networks in which the smallest nonzero eigenvalue \(\lambda _2\) is well separated from the smallest eigenvalue \(\lambda _1=0\), the spectral dimension is not defined and we say in that case that the network has a spectral gap (technically a model having a spectral gap means that \(\lambda _2\) is not vanishing in the large network limit). However the presence of a spectral dimension is the rule in networks with a nontrivial underlying geometry like lattices and fractal structures [66,67,68,69, 76]. While in presence of the spectral gap, convergence to the steady state of the diffusion dynamics is exponentially fast with a typical time scale \(\tau =1/\lambda _2\), in absence of a spectral gap it can be much slower. In fact when the spectral gap closes and the network has a finite spectral dimension \(d_S\) the density distribution at the starting node asymptotically in time decays as
In presence of a spectral dimension, given that Eq. (22) holds, the cumulative density of eigenvalues of the Laplacian \(\rho _c(\lambda )\) obeys the scaling relation
for \(0<\lambda \ll 1\). Figure 10 shows \(\rho _c(\lambda )\) for Network Geometry with Flavor \(s\in \{1,0,1\}\) formed by ddimensional simplices, ddimensional hypercubes and ddimensional orthoplexes up to \(d=7\). From this figure it is apparent that these cellcomplexes have a finite spectral dimension.
Starting from \(\rho _c(\lambda )\) we have numerically evaluated the spectral dimension \(d_S\) of the Network Geometry with Flavor (see Fig. 11) finding that the spectral dimension \(d_S\) of the Network Geometry with Flavor can be larger or smaller than d depending on the value of the flavor s and the nature of the polytopes that form its building blocks. Moreover our result indicate that while for simplicial complexes the spectral dimension \(d_S\) increases faster than linearly with d, for cellcomplexes formed by hypercubes or orthoplexes the spectral dimension tends to saturate. Specifically for simplicial complexes the spectral dimension \(d_S\) can be well fitted by
with coefficients \(\bar{a},\bar{b},\bar{c}\) depending on the flavor s as shown in Table 4. Note that we have compared the quadratic fit of \(d_S\) versus d to a simpler linear fit, performing a Ftest, which yields very small pvalues (\(p < 0.01\)) for all values of s, confirming the validity of the quadratic fit. For cellcomplexes formed by ddimensional hypercubes or ddimensional orthoplexes, the spectral dimension \(d_S\) can be fitted by
with coefficients \(\tilde{a},\tilde{b},\tilde{c}\) shown in Table 4.
These results point out the important role of the regular polytope forming the building blocks of the Network Geometry with Flavor in determining its geometrical properties.
Conclusions
In this paper we have characterized the Network Geometry with Flavor \(s\in \{1,0,1\}\) which are cell complexes built by gluing identical regular polytopes along their faces. The flavor \(s=1\) imposes that the cell complexes generated by the Network Geometry with Flavor \(s=1\) are manifolds also called Complex Network Manifolds. The flavor \(s=0\) indicates that the cell complexes grow by uniform attachment of the new polytope to a random \((d1)\)face. The flavor \(s=1\) indicates that the model includes an explicit preferential attachment of the new polytopes to \((d1)\)faces that have large number of polytopes already attached to them.
This purely topological model generates cell complexes with emergent hyperbolic network geometry revealed by imposing that every link has equal length. Here we characterize the interplay between the emergent geometry of Network Geometry with Flavor and complexity. Specifically we characterize under which conditions the Network Geometry with Flavor are scalefree. We observe that Network Geometry with Flavor can display or not display a scalefree degree distribution depending on the dimension d flavor s and specific type of regular polytope that forms its building blocks. Interestingly the Network Geometry with Flavor which is made by simplices (and are therefore simplicial complexes) has notable properties that makes it different from other realizations of the Network Geometry formed by other types of regular polytopes. In fact in dimension \(d>2\) the simplicial complexes are scalefree for every flavor \(s\in \{1,0,1\}\) while for Network Geometry formed by other types of regular polytopes not even in presence of an explicit preferential attachment (flavor \(s=1\)) we are always guaranteed to obtain a scalefree degree distribution. Additionally Network Geometry with Flavor displays another important signature of complexity, i.e. they have a nontrivial emergent community structure.
Interestingly the special role of simplicial complexes is also revealed by the spectral properties of the Network Geometry with Flavor which depend on the nature of the specific regular polytope that forms its building block. For instance if the building block is a dsimplex we have found that the spectral dimension \(d_S\) increases quadratically with the dimension d, while if the building block is an ddimensional hypercube and for the ddimensional orthoplex the spectral dimensions tend to saturate as the dimension d increases.
This work can be extended in different directions. First of all there are very clear paths leading to possible generalizations of the model including other values of the flavor, the introduction of a fitness of the faces of the polytope or the extention of the model beyond pure cellcomplexes. Secondly this theoretical framework provides an ideal setting to study the interplay between network geometry and dynamics such as frustrated synchronization [46]. Finally this framework is very promising for establishing close connections between growing network models and tensor networks.
References
 1.
Barabási, A.L., Albert, R.: Emergence of scaling in random networks. Science 286, 509 (2009)
 2.
Watts, D.J., Strogatz, S.H.: Collective dynamics of smallworldnetworks. Nature 393, 440 (1998)
 3.
Dorogovtsev, S.N., Mendes, J.F.F.: Evolution of networks. Adv. Phys. 51, 1079 (2002)
 4.
Newman, M.E.J.: Networks: An Introduction. Oxford University Press, Oxford (2010)
 5.
Barabási, A.L.: Network Science. Cambridge University Press, Cambridge (2016)
 6.
Bassett, D.S., Sporns, O.: Network neuroscience. Nat. Neurosci. 20, 353–364 (2017)
 7.
Barabási, A.L., Gulbahce, N., Loscalzo, J.: Network medicine: a networkbased approach to human disease. Nat. Rev. Gen. 12, nrg2918 (2010)
 8.
Buldyrev, S.V., Parshani, R., Paul, G., Stanley, H.E., Havlin, S.: Catastrophic cascade of failures in interdependent networks. Nature 464, 1025 (2010)
 9.
Bianconi, G.: Interdisciplinary and physics challenges of network theory. EPL 111, 56001 (2015)
 10.
Boccaletti, S., et al.: The structure and dynamics of multilayer networks. Phys. Rep. 544, 1 (2014)
 11.
Kivelä, M., Arenas, A., Barthelemy, M., Gleeson, J.P., Moreno, Y., Porter, M.A.: Multilayer networks. J. Comp. Netw. 2, 203 (2014)
 12.
Giusti, C., Ghrist, R., Bassett, D.S.: Two’s company, three (or more) is a simplex. J. Comput. Neurosci. 41, 1 (2016)
 13.
Wu, Z., Menichetti, G., Rahmede, C., Bianconi, G.: Emergent complex network geometry. Sci. Rep. 5, 10073 (2014)
 14.
Bianconi, G., Rahmede, C.: Network geometry with flavor: from complexity to quantum geometry. Phys. Rev. E 93, 032315 (2016)
 15.
Bianconi, G., Rahmede, C.: Emergent hyperbolic network geometry. Sci. Rep. 7, 41974 (2017)
 16.
Bianconi, G., Rahmede, C.: Complex quantum network manifolds in dimension \(d> 2\) are scalefree. Sci. Rep. 5, 13979 (2015)
 17.
Bianconi, G., Rahmede, C., Wu, Z.: Complex quantum network geometries: evolution and phase transitions. Phys. Rev. E 92, 022815 (2015)
 18.
Courtney, O.T., Bianconi, G.: Weighted growing simplicial complexes. Phys. Rev. E 95, 062301 (2017)
 19.
da Silva, D.C., Bianconi, G., da Costa, R.A., Dorogovtsev, S.N., Mendes, J.F.F.: Complex network view of evolving manifolds. Phys. Rev. E 97, 032316 (2018)
 20.
Ghoshal, G., Zlatić, V., Caldarelli, G., Newman, M.E.J.: Random hypergraphs and their applications. Phys. Rev. E 79, 066118 (2009)
 21.
Courtney, O.T., Bianconi, G.: Generalized network structures: the configuration model and the canonical ensemble of simplicial complexes. Phys. Rev. E 93, 062311 (2016)
 22.
Zuev, K., Eisenberg, O., Krioukov, D.: Exponential random simplicial complexes. J. Phys. A 48, 465002 (2015)
 23.
Avetisov, V., Hovhannisyan, M., Gorsky, A., Nechaev, S., Tamm, M., Valba, O.: Eigenvalue tunneling and decay of quenched random network. Phys. Rev. E 94, 062313 (2016)
 24.
Kahle, M.: Topology of random clique complexes. Discret. Math. 309, 1658 (2009)
 25.
Costa, A., Farber, M.: Random simplicial complexes. In: Configuration Spaces, pp. 129–153. Springer, Berlin (2016)
 26.
Cohen, D., Costa, A., Farber, M., Kappeler, T.: Topology of random 2complexes. Discret. Comput. Geom. 47, 117 (2012)
 27.
Severino, F.P.U., Ban, J., Song, Q., Tang, M., Bianconi, G., Cheng, G., Torre, V.: The role of dimensionality in neuronal network dynamics. Sci. Rep. 6, 29640 (2016)
 28.
Petri, G., Scolamiero, M., Donato, I., Vaccarino, F.: Topological strata of weighted complex networks. PLoS ONE 8, e66506 (2013)
 29.
Petri, G., et al.: Homological scaffolds of brain functional networks. J. R. Soc. Interface 11, 20140873 (2014)
 30.
Giusti, C., Pastalkova, E., Curto, C., Itskov, V.: Clique topology reveals intrinsic geometric structure in neural correlations. PNAS 112, 13455 (2015)
 31.
Wan, C., et al.: Panorama of ancient metazoan macromolecular complexes. Nature 525, 339 (2015)
 32.
Stehlé, J., Barrat, A., Bianconi, G.: Dynamical and bursty interactions in social networks. Phys. Rev. E 81, 035101 (2010)
 33.
Zhao, K., Stehlé, J., Bianconi, G., Barrat, A.: Social network dynamics of facetoface interactions. Phys. Rev. E 83, 056109 (2011)
 34.
Carstens, C.J., Horadam, K. J.: Persistent homology of collaboration networks. Mathematical Problems in Engineering 2013 (2013)
 35.
Patania, A., Vaccarino, F., Petri, G.: Topological analysis of data. EPJ Data Sci. 6, 7 (2017)
 36.
\(\breve{{\rm S}}\)uvakov, M., Andjelković, M., Tadić, B.: Hidden geometries in networks arising from cooperative selfassembly. Sci. Rep. 8, 1987 (2018)
 37.
Papadopoulos, L., Porter, M.A., Daniels, K.E., Bassett, D.S.: Network Analysis of Particles and Grains. arXiv preprint arXiv:1708.08080 (2017)
 38.
Young, J.G., Petri, G., Vaccarino, F., Patania, A.: Construction of and efficient sampling from the simplicial configuration model. Phys. Rev. E 96, 032312 (2017)
 39.
Ambjorn, J., Jurkiewicz, J., Loll, R.: Reconstructing the universe. Phys. Rev. D 72, 064014 (2005)
 40.
Oriti, D.: Spacetime geometry from algebra: spin foam models for nonperturbative quantum gravity. Rep. Prog. Phys. 64, 1703 (2001)
 41.
Lionni, L.: Colored discrete spaces: higher dimensional combinatorial maps and quantum gravity. arXiv preprint arXiv:1710.03663 (2017)
 42.
Fortunato, S.: Community detection in graphs. Phys. Rep. 486, 75 (2010)
 43.
Bianconi, G., Darst, R.K., Iacovacci, J., Fortunato, S.: Triadic closure as a basic generating mechanism of communities in complex networks. Phys. Rev. E 90, 042806 (2014)
 44.
Krapivsky, P.L., Redner, S.: Emergent network modularity. J Stat Mech Theory Exp. 7, 073405 (2017)
 45.
Daminelli, S., Thomas, J.M., Durán, C., Cannistraci, C.V.: Common neighbours and the localcommunityparadigm for topological link prediction in bipartite networks. New J. Phys. 17, 113037 (2015)
 46.
Millán, A.P., Torres, J.J., Bianconi, G.: Complex Network Geometry and Frustrated Synchronization. arXiv preprint arXiv:1802.00297 (2018)
 47.
Lin, Y., Lu, L., Yau, S.T.: Ricci curvature of graphs. Tohoku Math. J. 63, 605 (2011)
 48.
Lin, Y., Yau, S.T.: Ricci curvature and eigenvalue estimate on locally finite graphs. Math. Res. Lett 17, 343 (2010)
 49.
Gromov, M.: Hyperbolic Groups. Springer, Berlin (1987)
 50.
Bauer, F., Jost, J., Liu, S.: OllivierRicci curvature and the spectrum of the normalized graph Laplace operator. Math. Res. Lett. 19, 1185 (2012)
 51.
Sreejith, R.P., Mohanraj, K., Jost, J., Saucan, E., Samal, A.: Forman curvature for complex networks. J. Stat. Mech. 2016, 063206 (2016)
 52.
Weber, M., Jost, J., Saucan, E.: FormanRicci flow for change detection in large dynamic data sets. Axioms 5, 26 (2016)
 53.
Klitgaard, N., Loll, R.: Introducing quantum Ricci curvature. Phys. Rev. D 97, 046008 (2018)
 54.
Aste, T., Di Matteo, T., Hyde, S.T.: Complex networks on hyperbolic surfaces. Physica A 346, 20 (2005)
 55.
Kleinberg, R.: Geographic routing using hyperbolic space. In: INFOCOM 2007. 26th IEEE International Conference on Computer Communications. IEEE, 1902 (2007)
 56.
Boguñá, M., Papadopoulos, F., Krioukov, D.: Sustaining the internet with hyperbolic mapping. Nat. Commun. 1, 62 (2010)
 57.
Serrano, M.A., Boguñá, M., Sagués, F.: Uncovering the hidden geometry behind metabolic networks. Mol. BioSyst. 8, 843 (2012)
 58.
Nechaev, S.: NonEuclidean geometry in nature. arXiv preprint arXiv:1705.08013 (2017)
 59.
Boguñá, M., Krioukov, D., Claffy, K.C.: Navigability of complex networks. Nat. Phys. 5, 74 (2008)
 60.
Krioukov, D., et al.: Hyperbolic geometry of complex networks. Phys. Rev. E 82, 036106 (2010)
 61.
Papadopoulos, F., Kitsak, M., Serrano, M.A., Boguñá, M., Krioukov, D.: Popularity versus similarity in growing networks. Nature 489, 537 (2012)
 62.
Andrade Jr., J.S., Herrmann, H.J., Andrade, R.F.S., Da Silva, L.R.: Apollonian networks: simultaneously scalefree, small world, euclidean, space filling, and with matching graphs. Phys. Rev. Lett. 94, 018702 (2005)
 63.
Andrade, R.F.S., Herrmann, H.J.: Magnetic models on Apollonian networks. Phys. Rev. E 71, 056131 (2005)
 64.
Söderberg, B.: Apollonian tiling, the Lorentz group, and regular trees. Phys. Rev. A 46, 1859 (1992)
 65.
Graham, R., et al.: Apollonian circle packings: geometry and group theory I. The Apollonian group. Discret. Comput. Geom. 34, 547 (2005)
 66.
Burioni, R., Cassi, D.: Random walks on graphs: ideas, techniques and results. J. Phys. A 38, R45 (2005)
 67.
Rammal, R., Toulouse, G.: Random walks on fractal structures and percolation clusters. J. Phys. Lett. 44, 13 (1983)
 68.
Benedetti, D.: Fractal properties of quantum spacetime. Phys. Rev. Lett. 102, 111303 (2009)
 69.
Sotiriou, T.P., Visser, M., Weinfurtner, S.: Spectral dimension as a probe of the ultraviolet continuum regime of causal dynamical triangulations. Phys. Rev. Lett. 107, 131303 (2011)
 70.
Note that cellcomplexes in general can be formed by using any convex polytope, and that a given cellcomplex might be not pure, i.e. it can be formed by different types of convex polytopes. However in this paper we restrict out attention to pure cellcomplexes formed by a single type of regular polytope
 71.
Dorogovtsev, S.N., Mendes, J.F., Samukhin, A.N.: Sizedependent degree distribution of a scalefree growing network. Phys. Rev. E 63, 062101 (2001)
 72.
Orús, R.: A practical introduction to tensor networks: matrix product states and projected entangled Shi, YY., Duan, L.pair states. Ann. Phys. 349, 117 (2014)
 73.
Mucha, P.J., Richardson, T., Macon, K., Porter, M.A., Onnela, J.P.: Community structure in timedependent, multiscale, and multiplex networks. Science 328, 876 (2010)
 74.
Blondel, V.D., Guillaume, J.L., Lambiotte, R., Lefebvre, E.: Fast unfolding of communities in large networks. J. Stat. Mech. Theory Exp. 10, P10008 (2008)
 75.
Bagrow, J.P., Brockmann, D.: Natural emergence of clusters and bursts in network evolution. Phys. Rev. X 3, 021016 (2013)
 76.
Burioni, R., Cassi, D.: Universal properties of spectral dimension. Phys. Rev. Lett. 76, 1091 (1996)
 77.
Nokkala, J., Galve, F., Zambrini, R., Maniscalco, S., Piilo, J.: Complex quantum networks as structured environments: engineering and probing. Sci. Rep. 6, srep26861 (2016)
Author information
Affiliations
Corresponding author
Appendix: Derivation of Eq. (8)
Appendix: Derivation of Eq. (8)
In this appendix our goal is to derive Eq. (8) providing the expression for the probability \(\tilde{{\varPi }}(\kappa )\) to glue a new regular polytope which increases the generalized degree of a node having generalized degree \(\kappa \). Since in the Network Geometry with Flavor one polytope is added at each time step, the probability \(\tilde{{\varPi }}_i\) that we glue a new regular polytope to a \((d1)\)face \(\alpha \) incident to a node i (i.e. \(i\subset \alpha \)) is given by
For \(t\gg 1\) we note that we can approximate \(Z^{[s]}\) is given by
where the last expression is the approximate expression for \(t\gg 1\). In fact for \(s=1\) each new regular polytope introduces \(F1\) new \((d1)\)faces each one contributing one to \(Z^{[s]}\). However the \((d1)\)face \(\alpha \) to which we attach the new polytope acquires incidence number \(n_{\alpha }=1\) and therefore its contribution should be removed from \(Z^{[s]}\). Therefore \(Z^{[1]}\simeq (F2)t\). For \(s=0\), \(Z^{[0]}\) simply counts the total number of different faces, therefore each new polytope contributes by a term given by \((F1)\) to the sum corresponding the the number of novel \((d1)\) faces that each regular polytope introduces. Therefore \(Z^{[1]}\simeq (F1)t\). Finally for \(s=1\), each face is counted proportional to the number of polytopes that are incident to it. Therefore any new regular polytope contributes by a term given by F and \(Z^{[1]}\simeq Ft\).
If a node has generalized degree \(\kappa _{d,0}(i)=\kappa =1\) then it must be incident to f faces \(\alpha \) each one belonging to the same regular polytope. It follows that in this case the numerator of the left hand side of Eq. (27) reads
For nodes with generalized degree \(\kappa _{d,0}(i)=\kappa >1\), by following the same line of arguments presented above for deriving Eq. (28) it can be easily shown that
In fact each new regular polytope attached to the node after the initial one introduces \(f1\) new \((d1)\)faces incident to node i and therefore contributes by a term \((f1+s)\) to the sum.
Therefore the probability \(\tilde{{\varPi }}(\kappa )\) that we add a new polytope to a node with generalized degree \(\kappa _i=\kappa \) is given for \(t\gg 1\) by
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Mulder, D., Bianconi, G. Network Geometry and Complexity. J Stat Phys 173, 783–805 (2018). https://doi.org/10.1007/s1095501821159
Received:
Accepted:
Published:
Issue Date:
Keywords
 Higher order networks
 Network geometry
 Hyperbolic geometry
 Complexity