Network Geometry and Complexity
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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.
Keywords
Higher order networks Network geometry Hyperbolic geometry Complexity1 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.
2 Simplicial Complexes and Higher Order Networks
 (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. \(\hat{\alpha }\in {{\mathcal {K}}}\).
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].
 (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.
Properties of all regular polytopes in dimension d: M number of nodes, F number of faces, f number of \((d1)\) faces incident to a node, m number of nodes incident to a single \((d1)\) face, v degree of a node
F  M  f  m  v  

\(d=1\)  
Link  1  2  1  1  1 
\(d=2\)  
pPolygon  p  p  2  2  2 
\(d=3\)  
Tetrahedron  4  4  3  3  3 
Cube  6  8  3  4  3 
Octahedron  8  6  4  3  4 
Dodecahedron  12  20  3  5  3 
Icosahedron  20  12  5  3  5 
\(d=4\)  
Pentachoron  5  5  4  4  4 
Tesseract  8  16  4  8  4 
Hexadecachoron  16  8  8  4  6 
24Cell  24  24  6  6  8 
120Cell  120  600  4  20  4 
600Cell  600  120  20  4  12 
\(d>4\)  
Simplex  \(d+1\)  \(d+1\)  d  d  d 
Cube  2d  \(2^d\)  d  \(2^{d1}\)  d 
Orthoplex  \(2^d\)  2d  \(2^{d1}\)  d  \(2(d1)\) 
3 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.
 (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 \(n_{\alpha }\in \mathbb {N}^{0}\).
 (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].
 (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.
4 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 \(d_H=1\).
5 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).
6 Complexity and Degree Distribution
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].
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].
Degree distribution of Network Geometry with Flavor as a function of the flavor s and the dimension d
Flavor  \(s=1\)  \(s=0\)  \(s=1\) 

\(d=1\)  Bimodal  Exponential  Powerlaw 
\(d=2\)  Exponential  Powerlaw  Powerlaw 
\(d\ge 3\)  Powerlaw  Powerlaw  Powerlaw 
Powerlaw exponent \(\gamma \) of the degree distribution of Network Geometry with Flavor s built by gluing regular, convex polytopes in dimension d
\(\gamma \)  \(s= 1\)  \(s=0\)  \(s=1\) 

\(d=1\)  
Link  N/A  N/A  3 
\(d=2\)  
pPolygon  N/A  p  \(1+\frac{p}{2}\) 
\(d=3\)  
Tetrahedron  3  \(2\frac{1}{2}\)  \(2\frac{1}{3}\) 
Cube  5  \(3\frac{1}{2}\)  3 
Octahedron  4  \(3\frac{1}{3}\)  3 
Dodecahedron  11  \(6\frac{1}{2}\)  5 
Icosahedron  7  \(5\frac{3}{4}\)  5 
\(d=4\)  
Pentachoron  \(2\frac{1}{2}\)  \(2\frac{1}{3}\)  \(2\frac{1}{4}\) 
Tesseract  4  \(3\frac{1}{3}\)  3 
Hexadecachoron  \(3\frac{1}{3}\)  \(3\frac{1}{7}\)  3 
24Cell  \(6\frac{1}{2}\)  \(5\frac{3}{5}\)  5 
120Cell  60  \(40\frac{2}{3}\)  31 
600Cell  \(34\frac{2}{9}\)  \(32\frac{10}{19}\)  31 
\(d>4\)  
Simplex  \(2+\frac{1}{d2}\)  \(2+\frac{1}{d1}\)  \(2+\frac{1}{d}\) 
Cube  \(3+\frac{2}{d2}\)  \(3+\frac{1}{d1}\)  3 
Orthoplex  \(3+\frac{1}{2^{(d2)}1}\)  \(3+\frac{1}{2^{d1}1}\)  3 
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.
 (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.

7 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 \(d=5\).
8 Spectral Dimension of Network Geometry with Flavor
\(\bar{a}\)  \(\bar{b}\)  \(\bar{c}\)  

Simplices  \(s=1\)  0.09(1)  0.4(1)  0.8(1) 
\(s=0\)  0.11(2)  0.3(1)  1.5(2)  
\(s=1\)  0.07(1)  0.8(1)  1.0(1)  
\(\tilde{a}\)  \(\tilde{b}\)  \(\tilde{c}\)  
Hypercubes  \(s=1\)  2.38(1)  1.4(2)  1.4(2) 
\(s=0\)  2.49(3)  1.0(2)  1.7(4)  
\(s=1\)  2.55(3)  1.0(4)  2(1)  
Orthoplexes  \(s=1\)  2.79(4)  1.9(2)  0.4(1) 
\(s=0\)  2.78(1)  1.7(1)  0.51(2)  
\(s=1\)  2.76(1)  1.4(1)  0.54(3) 
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.
9 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.
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