Topology and dynamics of higher-order multiplex networks

Higher-order networks are gaining significant scientific attention due to their ability to encode the many-body interactions present in complex systems. However, higher-order networks have the limitation that they only capture many-body interactions of the same type. To address this limitation, we present a mathematical framework that determines the topology of higher-order multiplex networks and illustrates the interplay between their topology and dynamics. Specifically, we examine the diffusion of topological signals associated not only to the nodes but also to the links and to the higher-dimensional simplices of multiplex simplicial complexes. We leverage on the ubiquitous presence of the overlap of the simplices to couple the dynamics among multiplex layers, introducing a definition of multiplex Hodge Laplacians and Dirac operators. We show that the spectral properties of these operators determine higher-order diffusion on higher-order multiplex networks and encode their multiplex Betti numbers. Our numerical investigation of the spectral properties of synthetic and real (connectome, microbiome) multiplex simplicial complexes indicates that the coupling between the layers can either speed up or slow down the higher-order diffusion of topological signals. This mathematical framework is very general and can be applied to study generic higher-order systems with interactions of multiple types. In particular, these results might find applications in brain networks which are understood to be both multilayer and higher-order.


Introduction
Higher-order networks [1][2][3][4][5], and in particular simplicial complexes, capture the many body interactions present in complex systems and display a rich interplay between their topology and their dynamics [6,7].
Despite these important scientific advances, higher-order networks have the limitation that they only consider monolayer simplicial complexes formed by a single higher-order network (layer), while a large variety of complex biological, social and technological systems are described by multilayer structures [48][49][50].
Here we formulate a topological theory of higher-order multiplex simplicial complexes where each layer of the multiplex network includes higher-order interactions of different types between two or more nodes.We define the multiplex Hodge Laplacians and Dirac operators to capture higher-order diffusion on multiplex simplicial complexes whose definition exploit the ubiquitous presence of overlap of simplices [59,60,[67][68][69][70][71] across the different layers of the higher-order multiplex network, i.e. the simultaneous presence of many-body interactions of different types between the same set of nodes.We characterize the spectral properties of multiplex Hodge Laplacian and show that they obey Hodge decomposition.Additionally, we characterize the properties of the resulting higher-order diffusion on random and real multiplex simplicial complex datasets.Our results show that the considered topological coupling between the layers can have opposing effects depending on the topology of the multiplex simplicial complex.Indeed, in some cases it can speed up the dynamics of higher-order topological signals, while in others, it can slow it down.This has potential applications in control [72], where one can modulate the speed of dynamic processes by changing the overlap of simplices, even if only interactions on one layer of the multiplex simplicial complex can be externally modified.
The results provided on this work may be relevant to a variety of complex systems displaying higher- order interactions of different type, including brain networks which are known to be both multilayer and higher-order [17, 33-37, 70, 71, 73].
The paper is organized at follows.In Sec. 2 we introduce the key aspects of the topology of multiplex simplicial complexes; in Sec.3 we define the multiplex Hodge Laplacian and multiplex Dirac operators; in Sec.4 we characterize the higher-order diffusion dynamics of multiplex topological signals; in Sec. 5 we discuss our numerical results on synthetic and real multiplex simplicial complexes.Finally in Sec.7 we will provide the concluding remarks.The paper is enriched by an extensive Appendix providing a mathematically rigorous account of the topology of multiplex simplicial complexes introduced and used in the main body of the paper.

Multiplex simplicial complexes and multisimplices
A multiplex network [48][49][50] can be defined as a network of networks (layers) where the nodes of each network are in one-to-one correspondence.
Here we define a multiplex simplicial complex the encode higher-order interactions of different types.
A multiplex simplicial complex K (see Figure 1) is the union of M simplicial complexes, K [α] with α ∈ {1, 2, . . ., M }, which we will also called the layers of the multiplex simplicial complex.In the brain, this construction can allow one to distinguish between different types of higher-order interactions between the same regions of the brain.
Each simplicial complex K [α] is formed by a set of simplices indicating higher-order interactions.An n-dimensional simplex σ [α] ∈ K [α] is set of n + 1 nodes describing a n-th order interaction of type α.For instance a 0-simplex is a node, a 1-simplex is a link and a 2-simplex is a triangle.A face of n-dimensional simplex σ [α] is a simplex of dimension 0 ≤ n ′ < n formed by a proper subset of the nodes of σ [α] .Any simplicial complex K [α] is closed under the inclusion of its faces, i.e. if a simplex σ [α] belongs to K [α] , then all its faces also belong to the simplicial complex as well.We will call the dimension d of the multiplex simplicial complex, the largest dimension of any of its simplices.
An important property of multiplex networks that we extend to multiplex simplicial complexes is that nodes of different layers are in one to one correspondence.In particular corresponding nodes belonging to different layers are called replica nodes.In the higher-order setting, this notion extends to higherorder simplices as well and one can therefore speak of replica links, replica triangles and so on.These replica simplices are representations of the simplices σ that belong to the monolayer simplicial complex K formed N nodes and all the possible simplices that can be constructed among these N nodes.Hence, each node σ = [v 0 ] ∈ K is represented by the replica nodes σ = (σ [1] , σ [2] , . . ., σ [M ] ) where σ [α] indicates the replica node σ [α] = [v 0 ; α] which is the representation of node v 0 in layer α.Similarly, each higher dimensional simplex σ = [v 0 , v 1 . . ., v n ] among n nodes can be represented as a vector of replica simplices σ = (σ [1] , σ [2] , . . ., σ [M ] ), where σ [α] = [v 0 ; α, v 1 ; α . . .v n ; α] represents the simplex σ in layer α.Note that replica simplices are 'potential simplices' defined on all layers, but not all replica simplices may exist on all layers.In order to define the actual structure of the multiplex simplicial complexes and determine which replica simplices exist in a given multiplex simplicial complex we will need the notion of multisimplices that we will introduce in the following.
Link overlap is a very important and well known property of real multiplex networks which refers to the case where two nodes are connected in more than one layer.Link overlap is described by multilinks [59,60,[67][68][69][70][71], which capture all the possible ways in which two nodes can be connected in different layers.In order to capture this important property in our framework we extend the notion of multilinks on multiplex networks to multiplex simplicial complexes.While on a monoplex simplicial complex each simplex can be either present or not, in a multiplex simplicial complex any given simplex can be present or not in each of the different layers.In order to account for all the possible types of connections between a given set of n nodes, we associate to each set of replica simplices σ = (σ [1] , σ [2] , . . ., σ [M ] ) a vector ⃗ m σ = (m σ , . . ., m [M ] σ ) that captures whether the simplex σ [α] is present in the layer α (m σ = 0).In this way we define multisimplices of type ⃗ m that can represent every possible given pattern of connection among the same set of nodes across the different layers.In the case of two layers we can hence observe non trivial multi-0-simplices (multinodes) of type ⃗ m given by (1, 0), (0, 1), and (1, 1) indicating whether the node is present (i.e. it is connected) only in the first, only in the second or in both layers.Moreover we can observe multi-1-simplices (multilinks) and multi-2-simplicies (multitriangles) of type ⃗ m given by (1, 0), (0, 1) and (1, 1) indicating whether links or triangles are present only in the first, only in the second or in both layers respectively (see Figure 1).
In multilayer network literature [51][52][53], diffusion among the layers of the multiplex network is often treated with interlinks, i.e. links connecting the replica nodes in different layers.However interlinks do not have a clear and natural higher-order generalization.Therefore here we do not use interlinks; rather, we harness multisimplices to define a way to topologically couple the layers of the multiplex simplicial complexes.
To this end, we extend weighted algebraic topology operators [24,27,30] to describe the topology of multiplex simplicial complexes.
For the full mathematical framework we refer the reader to our Appendix, providing a more informal and up to the point account of our results below.

Multiplex cochains, and coboundary operators
Topological signals are dynamical variables sustained by the simplices of a simplicial complex.Typically topological signals sustained by n-dimensional simplices are represented as n-cochains.In a multiplex network, we consider multiplex n-cochains f ∈ C n , where C n is the set of all multiplex cochains, uniquely defined by the vector f taking a real value on each n-dimensional replica simplex.In particular, choosing a basis in which the n-dimensional simplices are listed consecutively for different layers we will have f [2]  . . .
where f [α] is a vector defined on the n-dimensional replica simplices of layer α.
In order to start defining the topology of multiplex simplicial complexes, generalizing the topology of monolayer simplicial complexes [74], our first step is to introduce the multiplex boundary operator and coboundary operators.Here we take the simple approach of defining the multiplex coboundary operator as the direct sum of the coboundary operators of each individual layer (see Appendix).Thus, we have that the multiplex coboundary operator δ n−1 can be expressed in matrix form by the n-th coboundary matrix B n of block diagonal form given by where n is the n-coboundary matrix matrix for layer α.For instance for a 2-layer multiplex simplicial complex the n-th coboundary matrix is given by The multiplex boundary operator ∂ n is instead captured by the boundary matrix B ⊤ n .An important property of the boundary operators, which is also shared by our definition of multiplex operators, is that "the boundary of the boundary is null", i.e.
for any n ≥ 1 (see Appendix for details).

Metric matrices and coupling among different layers
So far we have described the multiplex simplicial complex as a simplicial complex built by concatenating all the simplices present in every layer.However in this way the simplices of different layers are uncoupled.
In order to couple the layers we exploit the overlap of the simplices including nodes, links, triangles and so on and we introduce metric matrices that couple the overlapping simplices among the layers of the multiplex simplicial complexes.
The metric matrices G −1 n define non-degenerate scalar products between n-dimensional multiplex cochains.In particular, given f 1 , f 2 ∈ C n their scalar product is defined as where G n are positive definite and non-singular matrices.Typically, for a single simplicial complex, the metric matrices are taken to be diagonal, with diagonal elements given by the affinity weight of the corresponding n-dimensional simplex.Here, in order to couple the different layers of the multiplex simplicial complex, we consider non-diagonal metric matrices G n that exploit the overlap of the simplices whose coupling among the layers is modulated by the parameters γ n and W n .Specifically, the elements of the matrix G n can be non-zero only among replica simplices.For two n-dimensional replica simplices σ [α] , σ [α ′ ] in layer α and in layer α ′ respectively these matrix elements are defined as: with the parameters γ n and W n taking values in the range γ n ∈ [0, 1) and W n > 0. In particular the parameters γ n determine the coupling between the layers for γ n ̸ = 0 while when γ n = 0 for all values of n, the layers of the multiplex simplicial complex are completely uncoupled.Moreover W n > 0 are real parameters that can be used to tune the scale of the metric matrices for different dimensions n.In the definition of the metric matrices G n , C indicates the normalization constant that ensures that the matrix indicating the multiplicity of overlap of the n-dimensional simplex σ, i.e. the number of layers in which the simplex σ is present.It follows that the constant C is given by Note that according to the definition adopted for the metric matrices we have It is instructive to express the matrix elements of G n for the simple case of overlapping replica simplices σ = (σ [1] , σ [2] ) in a multiplex simplicial complex with M = 2 layers, For three layers, with σ containing the set of replica simplices σ = (σ [1] , σ [2] , σ [3] ) we have instead, where σ is the multiplicity of overlap of the simplex.

Multiplex Hodge Laplacians and multiplex Dirac operator
Having defined the metric matrices among multiplex cochains, one can now define the Hodge dual of the coboundary operator that will be instrumental in defining the multiplex Hodge Laplacians and Dirac operators.In matrix form, the multiplex Hodge dual coboundary operator δ ⋆ n−1 is expressed (see Appendix for details) by the matrix B ⋆ n which depends on the multiplex coboundary matrix B n and by the metric matrices G n and G n−1 with We are now in the position to define the multiplex n-Hodge Laplacian that generalizes the n-Hodge Laplacian of monolayer simplicial complexes [23][24][25][26][27][28] and characterizes the diffusion of higher-order topological signals from n-dimensional replica simplices to n-dimensional replica simplices through (n + 1)-dimensional and (n − 1)-dimensional replica simplices.In matrix form the multiplex Hodge Laplacian is expressed as with where L down 0 = 0. Therefore, in a d = 2 dimensional multiplex simplicial complex, in which each layer contains nodes, links, triangles, we have the following Hodge Laplacian operators: Interestingly, it can be easily proven that these multiplex Hodge-Laplacians, although asymmetric, have a real, non-negative spectrum (see Appendix).
Moreover, from the definition of the multiplex Hodge Laplacians we notice that since the boundary and the coboundary matrices obey Eq.( 4), the multiplex Hodge Laplacians satisfy which implies Hodge decomposition.A consequence of Hodge decomposition is that not only do L up n and L down n commute, but also for any left eigenvector of the multiplex Hodge Laplacian of L n corresponding to the eigenvalue λ there are only three options: An important spectral property of the multiplex n-Hodge Laplacian is that the dimension of its kernel is given by the multiplex n-Betti number, which is given by the sum of the sum of the n-Betti numbers of each individual layer (see Appendix for details).
We also define the multiplex Dirac operators that generalize the Dirac operators of simplicial complexes [29][30][31] that are algebraic topological operators that are emerging as an important tool to couple topological signals of different dimensions [13,14,32,38,47].Extending the definition of the weighted Dirac operator [30], here we define the multiplex Dirac operators.The multiplex Dirac operator can be expressed as sums of partial Dirac operators acting exclusively on (n − 1) dimensional and n-dimensional signals, D n : These partial Dirac operators can be used to define the multiplex Dirac operator of the entire multiplex simplicial complex.The Dirac operator acts on the direct sum of topological signals of any given dimension, allowing their cross-talk.On a multiplex simplicial complex of dimension d = 2 the Dirac operator D : A key property of the Dirac operator is that they can be interpreted as the "square root" of the Hodge Laplacians.Indeed, on a 2-dimensional multiplex simplicial complex we have Consequently, we obtain that the eigenvalues η of the Dirac operator are given by the square root of the eigenvalues of the Hodge Laplacians taken with both positive and negative sign, i.e.
where λ is the generic eigenvalue of any of the Hodge Laplacians of the simplicial complex.Thus it follows that while the Hodge Laplacians are positive semi-definite, the Dirac operator is not.Layer 3) of the microbiome multiplex network and the C.elegans connectome (data from [75]).The summary statistics of their multisimplices are listed in Table 1 and Table 3.
define the multiplex higher-order diffusion as Since the multiplex Hodge Laplacian obeys Hodge decomposition, every n topological signal X can be decomposed in a unique way into where X 1 describes the irrotational component of the signal, X 2 its solenoidal component and X harm its harmonic component, i.e., This is analogous to the case of the monoplex 1−Hodge Laplacian, where Hodge decomposition results in three orthogonal signals that are indeed curl-free (irrotational), divergence-free (solenoidal), and both curl-free and divergence-free (harmonic).Note however that due to the presence of the non-zero metric matrices the irrotational and the soleinodal components of the multiplex network signals may not in general be irrotational and solenoidal on each individual layer.
From the Hodge decomposition of the multiplex topological signals it follows that the multiplex higherorder diffusion can be captured by the three uncoupled set of equations: It follows that the harmonic component is not affected by the diffusion dynamics and if present will remain unchanged in time.However, the irrotational component X 2 and the solenoidal component X 2 will independently relax to zero asymptotically in time with different time-scales.In particular we can distinguish between two different Fiedler eigenvalues of the topological dynamics, one that describes the relaxation of the irrotational component λ − F (and is the smallest non zero eigenvalue of L down n ) and one that describes the relaxation to equilibrium of the solenoidal component λ + F (and is the smallest non zero eigenvalue of L up n ).We have that λ − F is a function of γ n−1 and γ n , and λ + F is a function of γ n and γ n+1 .For each value of n, monitoring the dependence of the values of these two Fiedler eigenvalues on the coupling constants γ n we can assess whether the introduced coupling between the layers speeds up or slows down the relaxation dynamics with respect to the case in which the layers are uncoupled (γ n = 0).Moreover λ − F is linear in W n while λ + F if linear in W n+1 .Therefore their relative value can be tuned freely by changing the relative values of W n and W n+1 .This implies that by changing the values of the parameters W n and W n+1 we can allow either the irrotational or the solenoidal component of the signal to relax faster.
We note that the Dirac operator can also be used to describe dynamics of topological signals.However given that the Dirac operator is not positive definite, the appropriate dynamics should consider complexvalues topological signals obeying the equation where on a d-dimensional multiplex simplicial complex, Y ∈ d n=1 C n should be a vector given by the direct sum of topological signals defined on replica simplices of every dimension.This is a very interesting dynamics, which may be relevant in the framework of continuous-time quantum walks and the recent interest in complex weights [76][77][78].The above dynamics, clearly, will not relax to the harmonic eigenstates due to the presence of the imaginary unit on the right hand side.Therefore, the dynamics in this case will depend on the whole spectrum of the multiplex simplicial complex rather than just on its Fiedler eigenvalues.Despite being interesting, the analysis of this dynamics requires a detailed investigation, which may be a topic of future interest.1 and Table 3 respectively.

Results on synthetic models
We investigate the spectral and diffusion properties of some simple synthetic models of multiplex simplicial complexes.As a first example we consider a multiplex simplicial complex of M = 2 two layers and N = 3 nodes, formed by a filled triangle in each layer (see Figure 2).
By investigating the spectrum of the 1-multiplex Hodge Laplacian we observe that the Fieder eigenvalue of the 1-up Hodge Laplacian display a maximum for γ 1 = γ 2 where there is a level crossing of the two non-zero eigenvalues.This phenomenon is also observed in other more complex synthetic multiplex simplicial complexes (see Figure 3), where we plot the Fiedler eigenvalue λ + F of the 1-up Hodge Laplacian as a function of γ 2 − γ 1 showing that in presence of simplices overlap the maximum for γ 1 = γ 2 .In this case, it is observed that the removal of some simplices in one layer can have different effects increasing or even decreasing the Fiedler eigenvalue while always decreasing the sharpness of the maximum.

Results on real multiplex datasets
In this section we consider the spectral properties of real multiplex simplicial complexes and their effect on the higher-order diffusion of the topological signals.We consider the multiplex microbiome dataset [79,80] (available at [75]) showing interactions present among of different types of microbial communities in the human body.We build five multiplex simplical complexes (4 with M = 2 layers-duplex simplicial complexes-and 1 with M = 3) combining 4 distinct human body microbial communities (see Table 1 for the layer composition).We also consider multiplex simplicial complexes built from the multiplex C. elegans connectome [81,82], (available at [75]) where the multiplex consists of layers corresponding to: electric (gap-junction) connection ("ElectrJ"), chemical monadic ("MonoSyn"), and polyadic ("PolySyn") synaptic connections.Specifically, out of the multilayer c.elegans connectome we built 3 multiplex simplicial complexes of M = 2 layers-duplex simplicial complexes-and one multiplex simplicial complex of M = 3 layers (see Table 1 for the layer composition).All these multiplex simplicial complexes are built from the multiplex networks by considering the d = 2 dimensional clique complex of each layer, i.e. filling all the cliques of the multiplex network structure.Interestingly the distribution of multisimplices in these multiplex simplicial complexes is non trivial, often showing a significant overlap of nodes, links and triangles of different type ⃗ m.This can be observed from the Tables 2 and 3 displaying the total number of multinodes N ⃗ m the total number of multilinks L ⃗ m and the total number of multitriangles T ⃗ m for the considered multiplex simplicial complexes of M = 2 and M = 3 layers respectively.
We perform a large scale study of the dependence of the irrotational and solenoidal Fiedler eigenvalues of the n = 1 multiplex Hodge Laplacian for the considered real multiplex simplicial complexes on the coupling among the layers.In our analysis we have chosen W n = 1 for all value of n but actually changing their values, for instance changing the ratio W 2 keeping W 1 and W 0 constant will modulate the scale of the solenoidal spectrum and hence can modulate the value of the Fiedler eigenvalue λ + F allowing this to acquire larger or smaller values.
Our analysis (see Figures (see Figures 4 − 7) reveal very diverse functional behavior of the Fiedler eigenvalues with the coupling constant γ n .Indeed in some cases the multiplex Fiedler eigenvalue can be larger than the uncoupled one obtained for γ n = 0 for any value of n, indicating that the proposed topological coupling between the layers can speed up the higher-order diffusion dynamics.However we also find instances of multiplex simplicial complexes in which the opposite behavior is observed and the coupling between the layers leads to a slower dynamics (smaller Fiedler eigenvalue that in the uncoupled scenario).This is particularly interesting for the study of real world complex networks where controlling diffusion of solenoidal or irrotational signals is desirable.

Conclusions
Characterizing the topology of multiplex higher-order networks is a key challenge in order to capture the higher-order structure and dynamics of multilayer data.Here we propose a framework that exploits the overlap of simplices in multiple layers in order to couple the dynamics occurring among the layers.We do this through introducing multiplex Hodge Laplacians and Dirac operators, an extension of the monoplex Hodge Laplacians and Dirac operator, that preserve important topological and spectral properties of simplicial complexes in the multiplex case.Specifically, we assume that the interlayer coupling is enforced by generalized multiplex metric matrices that couple overlapping simplices.Our analysis can be applied to synthetic multiplex simplicial complexes and to clique complexes of real multiplex networks as well.Additionally, our spectral properties of the multiplex simplicial complexes reveals that the multiplex diffusion dynamics can be either sped up or slowed down by the coupling among different layers.We provide evidence of this result on synthetic multiplex simplicial complexes in which we tune the overlap among the simplices as well as on real multiplex simplicial complexes.We believe that this work introduces a fundamental way of formulating a mathematical framework for multiplex simplicial complexes and demonstrates applicability to a wide range of data-driven complex systems.This work opens new perspectives for uncovering the topology of multiplex higher-order simplicial complexes and it is our hope that the multiplex Hodge Laplacians and multiplex Dirac operators defined hereby could find further applications in the modelling and control of higher-order nonlinear dynamics on multiplex structures.with c i n ∈ Z.Note that a chain c n ∈ C n will in general include linear combinations of simplices belonging to different layers and having non-zero coefficients.In particular the group of multiplex n-chains C n is distinct from the group of n-chains in layer α indicated as C n has the same definition as the multiplex boundary operator but its action is restricted to the n-chains of layer α only, i.e. ∂ n .Since each simplex in layer α admits as faces only simplices belonging to layer α, it follows that the multiplex boundary operator is given by the direct sum of the boundary operators acting on simplicies of each distinct layer, i.e.

( 1 )( 2 )
If λ = 0 the eigenvector is in both a left eigenvector in the kernel of L up n and a left eigenvector in the kernel of L down n .If λ = 0 only two mutually excluding options exists: (i) the left eigenvector is also a left eigenvector of L up n corresponding to eigenvalue λ and simultaneously is a left eigenvector of L down n corresponding to an eigenvalue zero, (ii) the eigenvector is also a left eigenvector of L down n corresponding to eigenvalue λ and simultaneously is a left eigenvector of L up n corresponding to the zero eigenvalue, Similar options apply for the right eigenvectors of L n .

Figure 2 : 1 , L up 1 and L 1 Figure 3 :
Figure 2: Example of d = 2-dimensional multiplex simplicial complexes and its spectrum.A simple multiplex simplicial complex is shown (on the left) together with the corresponding spectrum of the L down 1 , L up 1 and L 1 multiplex Hodge Laplacians (on the right).The eigenvalues λ down (panel (a)), λ up (panel (b)) and λ (panel (c)) of L down 1 , L up 1 and L 1 respectively are displayed as a function of γ 1 for γ 2 = 0.5.

Figure 4 :
Figure 4: Irrotational and solenoidal Fiedler eigenvalues as a function of the layer couplings for the Microbiome duplex simplicial complexes.The irrotational and solenoidal Fiedler eigenvalues of n = 1-multiplex Hodge Laplacians are plotted as a functions of the parameters γn showing non trivial dependence with the coupling between the layers.The results are here shown from the top to the bottom for the duplex simplicial complexes: Microbiome A, B, C, D whose layer composition and multisimplices statistics are reported inTable1 and Table 2 respectively.

Figure 5 :
Figure 5: Irrotational and solenoidal Fiedler eigenvalues as a function of the layer couplings for the C.elegans duplex simplicial complexes.The irrotational and solenoidal Fiedler eigenvalues of n = 1-multiplex Hodge Laplacians are plotted as a functions of the parameters γn showing non trivial dependence with the coupling between the layers.The results are here shown from the top to the bottom for the duplex simplicial complexes: C.elegans A, B, C whose layer composition and multisimplices statistics are reported inTable1 and Table 2 respectively.

Figure 6 :
Figure 6: Irrotational and solenoidal Fiedler eigenvalues as a function of the layer couplings for the Microbiome E multiplex simplicial complex.The irrotational and solenoidal Fiedler eigenvalues of n = 1-multiplex Hodge Laplacians are plotted as a functions of the parameters γn showing non trivial dependence with the coupling between the layers.The results are here shown for the multiplex simplicial complexes Microbiome E whose layer composition and multisimplices statistics are reported in Table1and Table3respectively.

Figure 7 :
Figure 7: Irrotational and solenoidal Fiedler eigenvalues as a function of the layer couplings for the C.elegans D multiplex simplicial complex.The irrotational and solenoidal Fiedler eigenvalues of n = 1-multiplex Hodge Laplacians are plotted as a functions of the parameters γn showing non trivial dependence with the coupling between the layers.The results are here shown for the multiplex simplicial complexes C.elegans D whose layer composition and multisimplices statistics are reported inTable1 and Table 3 respectively.

1 .(− 1 )
Therefore we have∂ α n [v 0 , . . ., v n ] = n p=0 p [v 0 , . . ., vp , . . ., v n ] (A.5) with σ n i = [v 0 , . . ., v n ] ∈ Q [α] . the multiplex boundary of the multiplex boundary is null.This is a direct consequence of the definition of the multiplex boundary operator and the property ∂ individual layer α of the multiplex network.A multiplex n-cochain f ∈ C n is a homeomorphism between the multiplex n-chains C n and the reals R, i.e. f :C n → R such that f (c n ) = σ n i ∈Qn c i n f (σ n i ).(A.9)

Table
1 and Table 2 respectively.

Table
1 and Table 2 respectively.

Table
1 and Table 3 respectively.