Topology and spectral interconnectivities of higher-order multilayer networks

Multilayer networks have permeated all the sciences as a powerful mathematical abstraction for interdependent heterogenous complex systems such as multimodal brain connectomes, transportation, ecological systems, and scientific collaboration. But describing such systems through a purely graph-theoretic formalism presupposes that the interactions that define the underlying infrastructures and support their functions are only pairwise-based; a strong assumption likely leading to oversimplifications. Indeed, most interdependent systems intrinsically involve higher-order intra- and inter-layer interactions. For instance, ecological systems involve interactions among groups within and in-between species, collaborations and citations link teams of coauthors to articles and vice versa, interactions might exist among groups of friends from different social networks, etc. While higher-order interactions have been studied for monolayer systems through the language of simplicial complexes and hypergraphs, a broad and systematic formalism incorporating them into the realm of multilayer systems is still lacking. Here, we introduce the concept of crossimplicial multicomplexes as a general formalism for modelling interdependent systems involving higher-order intra- and inter-layer connections. Subsequently, we introduce cross-homology and its spectral counterpart, the cross-Laplacian operators, to establish a rigorous mathematical framework for quantifying global and local intra- and inter-layer topological structures in such systems. When applied to multilayer networks, these cross-Laplacians provide powerful methods for detecting clusters in one layer that are controlled by hubs in another layer. We call such hubs spectral cross-hubs and define spectral persistence as a way to rank them according to their emergence along the cross-Laplacian spectra.


I. INTRODUCTION
Multilayer networks [4,6,17] have emerged over the last decade as a natural instrument in modelling myriads of heterogenous systems.They permeate all areas of science, as they provide a powerful abstraction of real-world phenomena made of interdependent sets of units interacting with each other through various channels.The concepts and computational methods they purvey have been the driving force to recent progress in the understanding of many highly sophisticated structures such as heterogeneous ecological systems [28,34], spatiotemporal and multimodal human brain connectomes [13,24,27], gene-molecule-metabolite interactions [20], and interdisciplinary scientific collaborations [35].This success has led to a growing interdisciplinary research investigating fundamental properties and topological invariants in multilayer networks.Some of the major challenges in the analysis of a multilayer network are to quantify the importance and interdependence among its different components and subsystems, and describe the topological structures of the underlying architecture to better grasp the dynamics and information flows between its different network layers.Various approaches extending concepts, properties, and centrality indices from network science [9,26] have been developed, leading to tremendous results in many areas of sci- * Electronic address: elkaioum.moutuou@concordia.caence [4,8,20,30,33,34,36,40].However, these approaches assume that inter-and intra-communications and relationships between the networks involved in such systems rely solely on node-based interactions.The resulting methods are therefore less insightful when the infrastructure is made up of higher-order intra-and inter-connectivites among node aggregations from different layers -as it is the case for many phenomena.For example, heterogenous ecosystems are made up of interactions among groups of the same or different species, social networks often connect groups of people belonging to different circles, collaborations and citations form a higher-order multilayer network made of teams of co-authors interconnected to articles, etc.Many recent studies have explored higher-order interactions and structures in monolayer networks [1-3, 10, 11, 16, 21-23, 31, 32, 37-39] using different languages such as simplicial complexes and hypergraphs.But a general mathematical formalism for modelling and studying higher-order multilayer networks is still lacking.
Our goal in this study is twofold.First, we propose a mathematical formalism that is rich enough to model and analyze multilayer complex systems involving higher-order connectivities within and in-between their subsystems.Second, we establish a unified framework for studying topological structures in such systems.This is done by introducing the concepts of crossimplicial multicomplex, cross-homology, cross-Betti vectors, and cross-Laplacians.Before we dive deeper into these notions, we shall give the intuition behind them by considering the simple case of an undirected two-layered network Γ; here Γ consists of two graphs ( FIG. 1: Shcematic of a 2-dimensional crossimplicial multicomplex X with 3 layers and 30 nodes in total; X consists of the vertex sets V 1 ,V 2 ,V 3 and the three CSBs X 1,2 , X 1,3 , X 2,3 defined respectively on the products where V 1 ,V 2 are the node sets of Γ, E s ⊆ V s × V s , s = 1, 2 are the sets of intra-layer edges, and a set E 1,2 ⊆ V 1 × V 2 of interlayer edges.Intuitively, Γ might be seen as a system of interactions between two networks.And what that means is that the node set V 1 interacts not only with V 2 but also with the edge set E 2 and vice versa.Similarly, intra-layer edges in one layer interact with edges and triads in the other layer, and so on.This view suggests a more combinatorial representation by some kind of two-dimensional generalization of the fundamental notion of simplicial complex from Algebraic Topology [14,18].The idea of crossimplicial multicomplex defined in the present work allows such a representation.In particular, when applied to a pairwise based multilayer network, this concept allows to incorporate, on the one hand, the clique complexes [19,31] corresponding to the network layers, and on the other, the clique complex representing the inter-layer relationships between the different layers into one single mathematical object.Morever, Γ can be regarded through different lenses, and each view displays different kind of topological structures.The most naive perspective flattens the whole structure into a monolayer network without segregating the nodes and links from one layer or the other.Another viewpoint is of two networks with independent or interdependent topologies communicating with each other through the interlayer links.The rationale for defining cross-homology and the cross-Laplacians is to view Γ as different systems each with its own intrinsic topology but in which nodes, links, etc., from one system have some restructuring power that allows them to impose and control additional topologies on the other.This means that in a multilayer system, one layer network might display different topological structures depending on whether we look at it from its own point of view, from the lens of the other layers, or as a part of a whole aggregated structure.We describe this phenomenon by focusing on the spectra and eigenvectors of the lower degree cross-Laplacians.We shall however remark that our aim here is not to address a particular real-world problem but to provide broader mathematical settings that reveal and quantify the emergence of these structures in any type of multilayer network.

II. CROSSIMPLICIAL MULTICOMPLEXES
General definitions.Given two finite sets V 1 and V 2 and a pair of integers , and its crossfaces are its subsets of the form {v 1 0 , . . ., Note that here we have used the conventions that which is closed under the inclusion of crossfaces; i.e., the crossface of a crossimplex is also a crossimplex.A crossimplex is maximal if it is not the crossface of any other crossimplex.V 1 and V 2 are called the vertex sets of X .
Given a CSB X , for fixed integers k, l ≥ 0 we denote by X k,l the subset of all its (k, l )-crossimplices.We also use the notations and X −1,−1 = .And recursively, X k,−1 will denote the subset of crossimplices of the form {v 1 0 , . . ., v 1 k } ⊂ V k+1 1 and X −1,l as the subset of crossimplices of the form {v 2 0 , . . ., v 2 l } ⊂ V l +1 2 .Such crossimplices will be referred to as intralayer simplices or horizonal simplices.
We then obtain two simplicial complexes [14] X •,−1 and X −1,• that we will refer to as the intralayer complexes, and whose vertex sets are respectively V 1 and V 2 .In particular, X 1,−1 and X −1,1 are graphs with vertex sets V 1 and V 2 , respectively.
The dimension of a (k, l )-crossimplex is k + l + 1, and the dimension of the CSB X is the dimension of its crossimplices of highest dimension.The n-skeleton of X is the restriction of X to the (k, l )-crossimplices such that k + l + 1 ≤ n.In particular, the 1-skeleton of a CSB is a 2-layered network, with X 0,0 being the set of interlayer links.Conversely, given a 2-layered network Γ formed by two graphs where σ 1 is a k-clique in Γ 1 and σ 2 is an l -clique in Γ 2 with the property that (i , j ) ∈ E 1,2 for every i ∈ σ 1 and j ∈ σ 2 .We define the cross-clique bicomplex X associated to Γ by letting X k,l to be the set of all (k + 1, l + 1)-cliques in Γ.Now a crossimplicial multicomplex (CSM) X consists of a family of finite sets V s , s ∈ S ⊆ N, and a CSB X s,t for each pair of distinct indices s, t ∈ S. It is undirected if the the sets of crossimplices in X s,t and X t ,s are in one-to-one correspondence.In such a case, X is completely defined by the family of CSB X s,t with s < t (see Fig. 1 for a visualization of a 3-layer CSM).(1, 1)-crossimplex (a cross-tetrahedron); and (e) a (0, 2)-crossimplex (also a cross-tetrahedron).Notice that cross-edges are always oriented from the vertex of the top layer to the one in the bottom layer.Therefore, cross-edges belonging to a cross-triangle are always of opposite orientations with respect to any orientation of the cross-triangle.There are two types of cross-triangles: the (1, 0)-crossimplices (top cross-triangles) and the (0, 1)-crossimplices (the bottom cross-triangle).Moreover, there are three types of cross-tetrahedrons: the (0, 2)-crossimplices, the (2, 0)-crossimplices, and the (1, 1)-crossimplices.
Orientation on crossimplices.An orientation of a (k, l )-crossimplex is an ordering choice over its vertices.When equipped with an orientation, the crossimplex is said to be oriented and will be represented as ) if k ≥ 0 and l = −1 (resp.k = −1 and l ≤ 0).We shall note that an orientation on crossimplices is just a choice purely made for computational purposes.Extending geometric representations from simplicial complexes, crossimplices can be represented as geometric objects.Specifically, a (0, −1)-crossimplex is a vertex in the top layer; (0, 0)-crossimplex is a cross-edge between layers V 1 and V 2 ; a (1, −1)-crossimplex (resp.(−1, 1)-crossimplex) is a horizontal edge on V 1 (resp.V 2 ); a (0, 1)-crossimplex or a (1, 0)-crossimplex is a cross-triangle; a (2, −1)-crossimplex or (−1, 2)-crossimplex is a horizontal triangle on layer V 1 or V 2 ; a (3, −1)-crossimplex or (−1, 3)-crossimplex is a horizontal tetrahedron on V 1 or V 2 ; a (1, 1)-crossimplex, a (2, 0)crossimplex, or a (0, 2)-crossimplex, is a cross-tetrahedron; and so on (see Fig. 2 for illustrations).On the other hand, horizontal edges, triangles, tetrahedron, and so on, are just usual simplices on the horizontal complexes.One can think of a cross-edge as a connection between a vertex from one layer to a vertex on the other layer.In the same vein, a crosstriangle can be thought of as a connection between one vertex from one layer and two vertices on the other, and a cross-tetrahedron as a connection between either two vertices from one layer and two vertices on the other, or one vertex from one layer to three vertices on the other.
Weighted CSBs.A weight on a CSB X is a positive function w : k,l X k,l −→ R + that does not depend on the orientations of crossimplices.A weighted CSB is one that is endowed with a weight function.The weight of a crossimplex a ∈ X is the number w(a).

Cross-boundaries.
A CSB X defines a bisimplicial set [12,25] by considering respectively the top and bottom crossface maps d (1)  i |k,l : X k,l −→ X k−1,l and d (2)  i |k,l : where the hat over a vertex means dropping the vertex.Moreover, for a fixed l ≥ −1, X •,l = (X k,l ) k≥−1 is a simplicial complex.Similarly, X k,• = (X k,l ) l ≥−1 is a simplicial complex.Observe that if a = {v 1 0 , . . ., v 1 k , v 2 0 , . . ., v 2 l } ∈ X k,l , then a (1) = {v 1 0 , . . ., v 1 k } ∈ X k,−1 and a (2) = {v 2 0 , . . ., v 2 l } ∈ X −1,l .We will refer to a (1) and a (2) as the top horizontal face and the bottom horizontal face of a, respectively.Conversely, two horizontal simplices v 1 ∈ X k,−1 and v 2 ∈ X −1,l are said to be interconnected in X if they are respectively the top and bottom horizontal faces of a (k, l )-crossimplex a.We then write v 1 ∼ v 2 .This is basically equivalent to requiring that if , we define its top cross-boundary ∂ (1) a as the subset of X k−1,l consisting of all the top crossfaces of a; i.e., all the (k − 1, l )-crossimplices of the form d (1)  i |k,l [a] for i = 0, . . ., k. Analogously, its bottom cross-boundary ∂ (2) a ⊆ X k,l −1 is the subset of all its bottom crossfaces d (2)  i |k,l [a], i = 0, . . ., l .Now two (k, l )-crossimplices a, b ∈ X k,l are said to be: • top-outer (TO) adjacent, and we write a (1) b or a (1)   c b, if both are top crossfaces of a (k + 1, l )-crossimplex c; in other words a, b ∈ ∂ (1) c; • top-inner (TI) adjacent, and we write a (1) b or a d which is a top crossface of both a and b; i.e., d ∈ ∂ (1) a ∩ ∂ (1) b; • bottom-outer (BO) adjacent, and we write a (2) b or a (2)  c b, if both are bottom crossfaces of a (k, l + 1)crossimplex c ∈ X k,l +1 ; that is to say a, b ∈ ∂ (2) c; and • bottom-inner (BI) adjacent, and we write a (2) b or a d (2) b, if there exists a (k, l −1)-crossimplex f ∈ X k,l −1 which is a bottom face of both a and b; that is d ∈ ∂ (2) a ∩ ∂ (2) b.

Degrees of crossimplices.
Given a weight function w on X , we define the following degrees of a (k, l )-crossimplex a relative to w.
• The TO degree of a is the number a∈∂ (1) a w(a ). (2) • Similarly, the TI degree of a is defined as • Analogously, the BO degree of a is given by • And the BO degree of a is Observe that in the particular case where the weight function is everywhere equal to one, the TO degree of a is precisely the number of (k + 1, l )-crossimplices in X of which a is top cross-face, while deg T I (a) is the number of top cross-faces of a, which equals to k + 1. Analogous observation can be made about the BO and BI degrees.
Cross-homology groups.Define the space C k,l of (k, l )cross-chains as the real vector space generated by all oriented (k, l )-crossimplexes in X .The top and bottom crossboundary operators ∂ (1)  k,l : then defined as follows by the formula for s = 1, 2 and a generator a ∈ X k,l , where sgn(b, , then sgn(b, ∂ (1) a) := (−1) i , and we define sgn(b, ∂ (2) a) in a similar fashion.
It is straightforward to see that in particular are the usual boundary maps of simplicial complexes.For this reason, we will put more focus on the mixed case where both l and k are non-negative.We will often drop the indices and just write ∂ (1) and ∂ (2) to avoid cumbersome notations.
Cross-Betti vectors.The cross-homology groups are completely determined by their dimensions, the top and bottom In particular, β (1)  k,−1 and β (2)  −1,l are the usual Betti numbers for the horizontal simplicial complexes [14].The couple β k,l = (β (1)  k,l , β (2)  k,l ) is the (k, l )-cross-Betti vector of X and can be computed using basic Linear Algebra.These vectors are descriptors of the topologies of both the horizontal complexes and their inter-connections.For instance, β 0,−1 and β −1,0 encode the connectivities within and in-between the 1-skeletons of the horizontal complexes associated to X .Precisely, β (1)  0,−1 is the number of connected components of the graph X 1,−1 and β (2)  0,−1 is the number of nodes in V 1 with no interconnections with any nodes in V 2 .Similarly, β (1)  −1,0 is the number of nodes in V 2 with no interconnections with any nodes in V 1 , while β (2)  −1,0 is the number of connected components of the bottom horizontal graph X −1,1 .Furthermore, β 1,−1 counts simultaneously the number of loops in X 1,−1 and the number of its intralayer links that do not belong to cross-triangles formed with the graph X −1,1 .Analogous topological information is provided by β −1,1 .Also, β 0,0 measures the extent to which individual nodes of one complex layer serve as communication channels between different hubs from the other layer.More precisely, an element in H (1)  0,0 (X ) represents either an interlayer 1-dimensional loop formed by a path in X 1,−1 whose end-nodes interconnect with the same node in V 2 , or two connected components in the top complex communicating with each other through a node in the bottom complex.In fact, β 0,0 counts the shortest paths of length 2 between nodes within one layer passing through a node from the other layer and not belonging to the crossboundaries of cross-triangles; we call such paths cones.Put differently, β 0,0 quantifies node clusters in one layer that are "controlled" by nodes in the other layer.Detailed proof of this description is provided in Appendix A.  1.The table quantifies the connectedness of the three horizontal complexes, the number of cycles in each of them, the number of nodes in each layer that are not connected to the other layers, the number of intra-layer edges not belonging to any cross-triangles, as well as the number of paths of length 2 connecting nodes in one layer and passing through a node from another layer.Now, given a CSM X , its cross-Betti table β ⊗ k,l is obtained by computing all the cross-Betti vectors of all its underlying CSB's.Computation of the cross-Betti table of the CSM of Fig. 1 is presented in Table I.
To illustrate what the cross-Betti vectors represent, we consider the simple 2-dimensional CSB X of Fig. 3.We get β (1)  0,−1 = 2, β (1) 1,−1 = 1, and −1,1 = 0; which reflects the fact the top layer has 2 connected components and 1 cycle, while the bottom one has one component and no cycles.Moreover, 3 top nodes are not interconnected to the bottom complex, 6 top edges are not top faces of cross-triangles, 2 bottom nodes are not interconnect to the top layer, and 5 bottom edges are not bottom faces of cross-triangles.This information is encoded in β 0,−1 = (2, 3), β 1,−1 = (1, 6), β −1,0 = (2, 1) and β −1,1 = (5, 0).There are 3 generating interlayer cycles, two of which are formed by an intralayer path in the bottom layer and a node in the top layer (v 1  4 and v 1  6 ), and the FIG. 3: Cross-Betti vectors.Schematic of a 2-dimensional CSB with 14 nodes in total, and whose oriented maximal crossimplices are the intralayer triangle , and the intralayer edge other one is formed by an intralayer path in the top layer and a node (v 2 1 ) in the bottom layer.Moreover, the two nodes v 2  1 and v 2 4 of V 2 interconnect the two separated components of the top layer; they serve as cross-hubs: removing both nodes eliminates all communications between the two components of the top layer.Cross-hubs and these types of interlayer cycles are exactly what β 0,0 encodes.Specifically, by computing the cross-homology of X we get β (1)  0,0 = 3 which count the cycle 2 and the nodes v 2  4 and v 2 1 that interconnect v 1 4 to v 1 6 and v 1 2 to v 1 6 , β (2)  0,0 = 2 counting the interlayer cycles In each of these cycles, the top node allows a shortest (interlayer) path between the end-points of the involved intralayer path.
Using algebraic-topological methods to calculate the cross-Betti vectors for larger multicomplexes can quickly become computationally heavy.We provide powerful linearalgebraic tools that not only allow to compute easily the β k,l 's, but also tell exactly where the topological structures being counted are located within the multicomplex.

IV. SPECTRAL DESCRIPTORS
Cross-forms.Denote by C k,l := C k,l (X , R) the dual space Hom R (C k,l , R) of the real vector space C k,l .Namely, C k,l is the vector space of real linear functionals φ : C k,l −→ R.
Notice that a natural basis of C k,l is given by the set of linear forms which naturally identify C k,l with C k,l .Now, define the maps δ (1)  k,l : C k,l −→ C k+1,l and δ (2)  k,l : C k,l −→ C k,l +1 by the following equations δ (1)  k,l φ([a]) = b∈∂ (1) a sgn(b, for φ ∈ C k,l , a ∈ X k+1,l and c ∈ X k,l +1 .Next, given a weight w on X , we get an inner-product on cross-forms by setting 〈φ, ψ〉 k,l := It can been seen that, with respect to this inner-product, elementary cross-forms form an orthogonal basis, and by simple calculations, the dual maps are given by (δ (1) for φ ∈ C k+1,l , a ∈ X k,l .And obviously we get a similar formula for the dual (δ (2)  k,l ) * .The cross-Laplacian operators.Identifying C k,l with C k,l and equipping it with an inner product as (8), we define the following self-adjoint linear operators on C k,l for all k, l ≥ −1: -the top (k, l )-cross-Laplacian -and the bottom (k, l )-cross-Laplacian Being defined on finite dimensional spaces, these operators can be represented as square matrices indexed over crossimplices.Specifically, denoting k,l is represented by positive definite N k,l × N k,l -matrices (see Appendix B 2) for the general expressions).
Moreover, the null-spaces, the elements of which we call harrmonic cross-forms, are easily seen to be in one-to-one correspondence with cross-cycles on X .Namely, we have the following isomorphisms (see Appendix B 1 for the proof) H (1)  k,l (X ) It follows that in order to compute the cross-Betti vectors, it suffices to determine the dimensions of the eigenspaces of the zero-eigenvalues of the cross-Laplacians.
It should be noted that in addition to being much easier to implement, the spectral method to compute cross-homology has the advantage of providing a geometric representation of the cross-Betti numbers through eigenvectors.But before we see how this works, let's make a few observations.Notice that L (T) 0,−1 and L (B) −1,0 are the usual graph Laplacians of degree 0 for the horizontal complexes.And more generally, L (T) k,−1 and L (B) −1,l are the combinatorial higher Hodge Laplacians [15,19,31] of degree k and l , respectively, for the horizontal simplicial complexes.Furthermore, L (B)  k,−1 (resp.L (T) −1,l ) detects the k-simplices (resp.l -simplices) in the top (resp.bottom) layer complex that are not top (resp.bottom) faces of (k, 0)-crossimplices (resp.(0, l )-crossimplices).Moreover, one can see that L (B)  k,−1 is the diagonal matrix indexed over the k-simplices on the top complex and whose diagonal entries are the BO degrees.Similarly, L (T) −1,l is the diagonal matrix whose diagonal entries are the TO degrees of the l -simplices on the bottom complex.This is consistent with the interpretation of the cross-Betti numbers β (2)   0,−1 and β (1)  −1,0 given earlier in terms of connectivities between the 1-skeletons of the horizontal complexes.
Harmonic cross-hubs.Assume for the sake of simplicity that the X is equipped with the trivial weight ∼ = 1.Then, by (B4), the (0, 0)-cross-Laplacians L (T) 0,0 and L (B) 0,0 are respectively represented by the N 0,0 × N 0,0 -matrices indexed on cross-edges a i , a j ∈ X 0,0 whose entries are given by Applied to the toy example of Fig. 3, L (T) 0,0 has a zeroeigenvalue of multiplicity 3, generating the three (0, 0)-crosscycles in Table II.
Each coordinate in the eigenvectors is seen as an "intensity" along the corresponding cross-edge.Cross-edges with non-zero intensities sharing the same bottom node define certain communities in the top complex that are "controlled" by the involved bottom node.These community structures depend on both the underlying topology of the top complex and its interdependence with the other complex layer.We then refer to them as harmonic cross-clusters, and the bottom nodes controlling them are thought of as harmonic cross-hubs (HCH).The harmonic cross-hubness of a bottom node is the L 1 -norm of the intensities of all cross-edges having it in common.Here, in the eigenvectors of the eigenvalue 0, there are two subsets of cross-edges with non-zero coordinates: the cross-edges with v 2 1 in common, and the ones with v 2 4 in common.We therefore have two harmonic crosshubs (see illustration in Fig. 5), hence two harmonic cross- TABLE II: Harmonic (0, 0)-cross-forms.The 3 eigenvectors of the eigenvalue 0 of L (T) 0,0 corresponding to the synthetic CSB of Figure 3.There are 2 harmonic cross-hubs : v 2  1 and v 2 4 , their respective harmonic cross-hubness are 2.6177 and 1.4070.
clusters.The first one is responsible for the top layer crosscluster {v 1 0 , v 1 1 , v 1 2 , v 1 4 , v 1 6 }, while the second one controls the top layer cross-cluster {v 1  4 , v 1 6 }.The intensity of each involved cross-edge is the L 1 -norm of its corresponding coordinates in the 3 eigenvectors, and the harmonic cross-hubness is the sum of the intensities of the cross-edges interconnecting the corresponding cross-hub to each of the top nodes in the cross-clusters it controls.For instance, v 2  1 is the bottom node with the highest harmonic cross-hubness which is 2.6177.This reflects the fact v 2  1 not only interconnects the two connected components of the top complex (which v 2  4 does as well), but it also allows fast-track connections between the highest number of nodes that are not directly connected with intra-layer edges in the top complex.The same calculations applied to the eigenvectors of the zero-eigenvalues of L (B) 0,0 give v 1 6 as the top node with the highest harmonic cross-hubness w.r.t. the bottom complex.
Spectral persistence of cross-hubs.To better grasp the idea of cross-hubness, let us have a closer look at the coordinates of the eigenvectors of the (0, 0)-cross-Laplacians (( 10) and ( 11)) whose eigenvalues are all non-negative real numbers.Suppose φ = (x 1 , ..., x N 0,0 ) is an eigenvector for an eigenvalue λ T of L (T) 0,0 .Then, denoting the cross-edges by a i , i = 1, ..., N 0,0 , we have the relations where χ is such that χ(a i , a j ) = 1 if i = j or if a i and a j are adjacent but do not belong to a top cross-triangle, and χ(a i , a j ) = 0 otherwise.It follows that the cross-edge intensity |x i | grows larger as deg T O (a i ) → λ T .In particular, for λ T = 0, the intensity is larger for cross-edges that belong to a large number of cones and to the smallest number of top cross-triangles.Now, consider the other extreme of the spectrum, namely λ T = λ T max to be the largest eigenvalue of L (T) 0,0 .Then, the intensity |x i | is larger for cross-edges belonging to the largest number of top cross-triangles and a large number of top cones at the same time.
Taking the case of a 2-layered network, for λ T = 0, |x i | is larger for a cross-edge pointing to a bottom node interconnecting a largest number of top nodes that are not directly FIG. 4: Spectral persistence of cross-hubs.Schematic illustrations of the variations of spectral cross-hubs along the eigenvalues and the spectral persistence bars codes for the toy CSB of Fig. 3: (A) shows the number of bottom nodes that emerge as spectral crosshubs w.r.t. the top layer as a function of the eigenvalues of L (T) 0,0 , and (B) represents the number of top nodes revealed as spectral cross-hubs w.r.t. the bottom layer as a function of the eigenvalues of L (B) 0,0 .(C) and (D) represent the spectral persistence bar codes for L (T)  0,0 and L (B) 0,0 , respectively.For both the top and bottom (0, 0)cross-Laplacians, most of the spectral cross-hubs, hence of spectral cross-clusters, emerge during the first stages (smallest eigenvalues), very few of them survive at later stages, and here only one cross-hub emerge or survive at the largest eigenvalue (v 2  1 for L (T) 0,0 and v 1 6 for L (B)  0,0 ).
connected with intra-layer edges; and for λ T = λ max , |x i | is larger for a cross-edge pointing to a bottom node interconnecting a large number of top intra-layer communities both with each other and with a large number of top nodes that are not directly connected to each other via intra-layer edges.More generally, by applying the same process to each distinct eigenvalue, we obtain clustering structures in the top layer that are controlled by the bottom nodes and that vary along the spectrum At every stage, we regroup the cross-edges with non-zero coordinates in the associated eigenvectors and pointing to the same nodes, then sum up their respective intensities to obtain a ranking among a number of cross-hubs that we call spectral cross-hubs (SCHs).Intuitively, the intensities held by cross-edges gather to confer a 'restructuring power' onto the common bottom node -the cross-hub, allowing it to control a cluster on the top layer.It is clear that, by permuting the top layer with the bottom layer, the same reasoning applies to L (B) 0,0 .In particular, we define the principal cross-hubs (PCH) in the bottom layer w.r.t. the top layer as the SCHs obtained from λ T max .The principal cross-hubness of a bottom PCH is defined as its restructuring power.In a similar fashion, we define the principal cross-hubness in the top layer w.r.t. the bottom layer using the largest eigenvalue λ B max of L (B) 0,0 .Going back to the bicomplex of Fig. 3, the largest eigenvalue of L (T) 0,0 is λ T max = 5, the corresponding eigenvector is represented by Table III.
TABLE III: Principal eigenvector of L (T) 0,0 for the CSB of Figure 3.By definition, this is the eigenvector associated to the largest eigenvalue.
There is only one PCH in the bottom layer w.r.t. the top layer, which is the bottom node v 2  1 , and its principal crosshubness is 2.2360.
Interestingly, the number of SCHs that appear for a given eigenvalue tend to vary dramatically w.r.t. the smallest eigenvalues before it eventually decreases or stabilizes at a very low number (see Fig. 4 and Fig. 6).Some cross-hubs may appear at one stage along the spectrum and then disappear at a future stage.This suggests the notion of spectral persistence of cross-hubs.Nodes that emerge the most often or live longer as cross-hubs along the spectrum might be seen as the most central in restructuring the topology of the other complex layer.The more we move far away from the smallest non-zero eigenvalue, the most powerful are he nodes that emerge as hubs facilitating communications between aggregations of nodes in the other layer.The emergence of spectral cross-hubs is represented by a horizontal linespectral persistence bar -running through the indices of the corresponding eigenvalues (Fig. 4).The spectral persistence bars corresponding to all SCHs (the spectral bar codes) obtained from L (T) 0,0 (resp.L (B) 0,0 ) constitute a signature for all the clustering structures imposed by the bottom (resp.top) layer to the top (resp.bottom) layer.

V. EXPERIMENTS ON MULTIPLEX NETWORKS
Diffusion CSBs.Let M be a multiplex formed by M graphs Γ s = (E s ,V ), s = 1, . . ., M .Denoting the vertex set V as an ordered set {1, 2, . . ., N }, we will write v s i to represent the node i in the graph Γ s , following the same notations we have used for multicomplexes.
For every pair of distinct indices s, t , we define the 2dimensional CSB X s→t on V × V such that X s→t k,−1 = for k ≥ 1, X s→t −1,k is the 2-clique complex of the layer indexed by t in the multiplex M ; a pair (v s i , v t j ) ∈ V ×V , forms a crossedge if i < j , and nodes i and j are connected in Γ s ; and a (0, 1)-crossimplex is a triple (v s i , v t j , v t k ) ∈ V 3 such that i is connected to j and k in Γ s , and j and k are connected in Γ t , while X s→t 1,0 = .We call X s→t the diffusion bicom- 6: Spectral persistent cross-hubs.The spectral persistence bar codes of the six diffusion bicomplexes of the European ATN multiplex.The nodes represent European airports labelled with their ICAO codes.The most persistent cross-hubs correspond to the airports that provide the most efficient correspondences from the first airline network to the second.plex of (layer) s onto t .Notice that by construction, the (0, 0)-cross-Laplacians of X s→t are indexed over E s , while the (0, 0)-cross-Laplacians of X t →s are indexed over E t .This shows that X s→t and X t →s are not the same.In fact, the diffusion bicomplex X s→t is a way to look at the the topology of Γ s through the topology of Γ t ; or put differently, it diffuses the topology of the former into the topology of the latter.
Cross-hubs in air transportation networks.We use a subset of the European Air Transportation Network (ATN) dataset from [5] to construct a 3-layered multiplex M on 450 nodes each representing a European airport [36].The 3 layer networks Γ 1 , Γ 2 , and Γ 3 of M represent the direct flights served by Lufthansa, Ryanair, and Easyjet airlines, respectively; that is, intra-layer edges correspond to direct flights between airports served by the corresponding airline.Considering the respective bottom (0, 0)-cross-Laplacians of the six diffusion bicomplexes X 1→2 , X 1→3 , X 2→1 , X 3→1 , X 2→3 , and X 3→2 , we obtain the spectral persistence bar codes describing the emergence of SCH's for each airline w.r.t. the others (see Fig. 6).The induced SCH rankings are presented in Table V, while the corresponding PCHs are illustrated in Fig. 7.

VI. DISCUSSION AND CONCLUSIONS
We have introduced CSM as a generalization of both the notions of simplicial complexes and multilayer networks.We further introduced cross-homology to study their topology and defined the cross-Laplacian operators to detect more structures that are not detected by homology.Our goal here was to set up a mathematical foundation for studying higherorder multilayer complex systems.Nevertheless, through synthetic examples of CSM and applications to multiplex networks, we have shown that our framework provides powerful tools to revealing important topological features in a multilayer networks and address questions that would not arise from the standard pairwise-based formalism of multilayer networks.We put a special focus on the (0, 0)-cross-Laplacians to show how their spectra quantify the extent to which nodes in one layer restructure the topology of other layers in a multilayer network.Indeed, given a CSB X or even a 2-layered network, we defined L (T) 0,0 and L (B) 0,0 as two self-adjoint positive operators operators that allow to look at the topology of one layer through the lens of the other layer.Specifically, we saw that their spectra allow to detect nodes from one layer that serve as interlayer connecting hubs for clusters in the other layer; we referred to such nodes as spectral cross-hubs (SCHs).Such hubs vary in function of the eigenvalues of the cross-Laplacians, the notion of spectral persistent cross-hubs was used to rank them according to their frequency along the spectra.The SCHs obtained from the largest eigenvalues were referred here as principal crosshubs (PCHs) as they are the ones that interconnects the most important structures of the other layer.We should note that a PCH is not necessarily spectrally persistent, and two SCHs can be equally persistent but at different ranges of the spectrum.This means that, depending on the applications, some choices need to be made when ranking SCHs based on their spectral persistence.Indeed, it might be the case that two SCHs persist equally longer enough to be considered as the most persistent ones, but that one persists through the first quarter of the spectrum while the other persists through the second quarter of the spectrum, so that none of them is a PCH.For instance, in the example of the European ATN multiplex networks, when two nodes were equally persistent, we ranked higher the one that came later along the spectrum.Finally, one can observe that the topological and geometric interpretations given for these operators can be generalized to the higher-order (k, l )-cross-Laplacians as well.That is, the spectra of these operators encode the extent to which higher-order topological structures (edges, triangles, tetrahedrons, and so on) control the emergence of higher-order clustering structures in the other layers.
) of vertices in What we have here are cross-triangles all pointing to v 2 j that are pieced together in the form of an actual kite as in Figure 8.In particular, if v 2 j is the bottom face of a (1, 0)-cross-triangle ) of vertices in V 2 satisfying analogous conditions.Such a kite will be denoted as ) with 1 ≤ r and r + q ≤ p.
By a cross-chain on a kite we mean one that is a linear combination of the triangles composing the kite; that is, a cross-chain on the kite (v where γ 1 , . . ., γ p−1 ∈ R. In a similar fashion, cross-chains on a kite of the form (v We also say that (v In a similar fashion one defines a cone with base in V 2 and vertex in V 1 .We refer to Figure 8 for examples of cones.
An immediate consequence of a triple (v i and v 1 k might however be connected by a horizontal path of some length; by which we mean that there might be a sequence of vertices v 1 i 0 , . . ., v 1 i p in V 1 not all of which form cross-triangles with v 2 j and such that otherwise. Cones in a crossimplicial bicomplex are classified by the top and bottom (0, 0)-cross-homology groups of the bicomplex.Specifically, we have the following topological interpretation of H (1)  0,0 (X ), H (2)  0,0 (X ), and hence, the (0, 0)-cross-Betti numbers.
It follows that the eigenvectors corresponding to the zero eigenvalue of the (k, l )-cross-Laplacian L (s)  k,l are representative cross-cycles in the homology group H (s)  k,l (X ).Hence-forth, we see that in order to get the dimensions of the crosshomology groups H (s) k,l (X ), it suffices to find the eigenspaces corresponding to the zero eigenvalues of L (s) k,l .That is, β (1)  k,l = dim ker L (1)  k,l , and β (2)  k,l = dim ker L (2)  k,l .(B3)
In particular, when φ is an elementary cross-form e b , b ∈ X k,l , we get

5 FIG. 5 :
FIG.5: Cross-Laplacians, harmonic and principal cross-hubs.(A) and (D): Heat-maps of the the top and bottom (0, 0)-cross-Laplacian matrices for the example of Fig.3.Both matrices are indexed over the cross-edges of the CSB, and the diagonal entries correspond to one added to the number of cross-triangles containing the corresponding cross-edge.L(T)  0,0 has a zero eigenvalue of multiplicity 3, while L (B) 0,0 has a zero eigenvalue of multiplicity 2. (B) and (E): The harmonic cross-hubs w.r.t. to the top (resp.the bottom) horizontal complex of X ; the intensity of a cross-edge is given by the L 1 -norm of the corresponding coordinates in the eigenvectors of the eigenvalue 0. (C) and (F): the principal cross-hubs in the bottom (resp.top) layer w.r.t. the top (resp.bottom) layer; by definition, they are the spectral cross-hubs obtained from by the largest eigenvalues of the top and bottom (0, 0)-cross-Laplacians, respectively.

15 FIG. 7 :
FIG. 7: PCHs of the diffusion bicomplexes for the European ATN multiplex.The nodes represent airports labelled with their ICAO codes.

TABLE I :
Cross-Betti table.The cross-Betti table for the CSM of Figure

TABLE IV :
Ranking of the ten most persistent SCHs for the diffusion bicomplexes associated to the European air transportation multiplex network.
which case the cone is said to be closed; it is called open