Skip to main content


Front. Complex Syst., 20 November 2023
Sec. Complex Networks
Volume 1 - 2023 |

Topology and spectral interconnectivities of higher-order multilayer networks

  • 1PERFORM Centre, Concordia University, Montreal, QC, Canada
  • 2Department of Physics, Gina Cody School of Engineering and Computer Science, Concordia University, Montreal, QC, Canada
  • 3Department of Electrical and Computer Engineering, Concordia University, Montreal, QC, Canada

Multilayer networks have permeated all areas of science as an abstraction for interdependent heterogeneous complex systems. However, describing such systems through a purely graph-theoretic formalism presupposes that the interactions that define the underlying infrastructures are only pairwise-based, a strong assumption likely leading to oversimplification. 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, and interactions might exist among groups of friends from different social networks. Although higher-order interactions have been studied for monolayer systems through the language of simplicial complexes and hypergraphs, a 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 modeling 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. Using synthetic and empirical datasets, we show that the spectra of the cross-Laplacians of a multilayer network detect different types of 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 spectra. Our framework is broad and can especially be used to study structural and functional connectomes combining connectivities of different types and orders.

1 Introduction

Multilayer networks (De Domenico et al., 2013; Boccaletti et al., 2014; Kivelä et al., 2014) have emerged over the last decade as a natural instrument in modeling myriads of heterogeneous 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 for recent progress in the understanding of many highly sophisticated structures such as heterogeneous ecological systems (Pilosof et al., 2017; Timóteo et al., 2018), spatiotemporal and multimodal human brain connectomes (Griffa et al., 2017; Mandke et al., 2018; Pedersen et al., 2018), gene–molecule–metabolite interactions (Liu et al., 2020), and interdisciplinary scientific collaborations (Vasilyeva et al., 2021). 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 flow between its different network layers. Various approaches extending concepts, properties, and centrality indices from network science (Newman, 2003; Fortunato and Hric, 2016) have been developed, leading to tremendous results in many areas of science (Solá et al., 2013; Boccaletti et al., 2014; Sánchez-García et al., 2014; Flores and Romance, 2018; Timóteo et al., 2018; Wu et al., 2019; Liu et al., 2020; Yuvaraj et al., 2021). 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-connectivities among node aggregations from different layers—as is the case for many phenomena. For example, heterogeneous 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, and collaborations and citations form a higher-order multilayer network made of teams of co-authors interconnected to articles. Many recent studies have explored higher-order interactions and structures in monolayer networks (Benson et al., 2016; Iacopini et al., 2019; Lucas et al., 2020; Schaub et al., 2020; Bianconi, 2021) using different languages, such as simplicial complexes and hypergraphs (Shi et al., 2021; Young et al., 2021; Lotito et al., 2022; Majhi et al., 2022). However, a general mathematical formalism for modeling 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 performed 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 (V1, E1) and (V2, E2), where V1, V2 are the node sets of Γ, EsVs × Vs, s = 1, 2 are the sets of intra-layer edges, and a set E1,2V1 × V2 of inter-layer edges. Intuitively, Γ might be seen as a system of interactions between two networks. It means that the node set V1 interacts not only with V2 but also with the edge set E2 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 (Mac Lane, 1963; Hatcher, 2000). 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 one hand, the clique complexes (Lim, 2020; Schaub et al., 2020) 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. Moreover, Γ can be regarded through different lenses, and each view displays different kinds 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 inter-layer 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, a 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.

2 Crossimplicial multicomplexes

2.1 General definitions

Given two finite sets, V1 and V2, and a pair of integers k, l ≥−1, a (k, l)– crossimplex a in V1 × V2 is a subset {v01,,vk1,v02,,vl2} of V1k+1×V2l+1, where visVs for s = 1, 2. The point vi1 (resp. vj2) is the vertex of a in V1 (resp. V2), and its crossfaces are its subsets of the form {v01,,vi11,vi+11,,vk1,v02,,vl2} for 0 ≤ ik and {v01,,vk1,v02,,vi12,vi+12,,vl2} for 0 ≤ il. Note that here we have used the conventions that V1n×V20=V1n and V10×V2n=V2n.

An abstract crossimplicial bicomplex X (or a CSB) on V1 and V2 is a collection of crossimplices in V1 × V2, 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. V1 and V2 are called the vertex sets of X.

Given a CSB X, for fixed integers k, l ≥ 0, we denote, by Xk,l, the subset of all its (k, l)-crossimplices. We also use the notations X0,−1 = V1, X−1,0 = V2, and X−1,−1 =∅. Recursively, Xk,−1 will denote the subset of crossimplices of the form {v01,,vk1}V1k+1, and X−1,l as the subset of crossimplices of the form {v02,,vl2}V2l+1. Such crossimplices will be referred to as intralayer simplices or horizontal simplices. We then obtain two simplicial complexes (Hatcher, 2000), X•,−1 and X−1,•, that we will refer to as the intralayer complexes and whose vertex sets are V1 and V2, respectively. In particular, X1,−1 and X−1,1 are graphs with vertex sets V1 and V2, respectively.

The dimension of a (k, l)-crossimplex is k + l + 1, and the dimension of CSB X is the dimension of its crossimplices of the highest dimension. The n-skeleton of X is the restriction of X to (k, l)-crossimplices such that k + l + 1 ≤ n. In particular, the 1-skeleton of CSB is a two-layered network, with X0,0 being the set of inter-layer links. Conversely, given a two-layered network Γ formed by two graphs Γ1 = (V1, E1) and Γ2 = (E2, V2) with the inter-layer edge set E1,2V1 × V2, define a (k, l)-clique in Γ as a pair (σ1, σ2), where σ1 is a k-clique in Γ1 and σ2 is an l-clique in Γ2 with the property that (i, j) ∈ E1,2 for every iσ1 and jσ2. We define the cross-clique bicomplex X associated with Γ by letting Xk,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 Vs,sSN and a CSB Xs,t for each pair of distinct indices s, tS. It is undirected if the sets of crossimplices in Xs,t and Xt,s are in one-to-one correspondence. In such a case, X is completely defined by the family of CSB Xs,t with s < t (see Figure 2 for a visualization of a three-layered CSM).

2.2 Orientation on crossimplices

The 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 [a]=[v01,,vk1;v02,,vl2] if k, l ≥ 0, or [v01,,vk1] (resp. [v02,,vl2]) if k ≥ 0 and l = −1 (resp. k = −1 and l ≤ 0). We shall note that an orientation of 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; a (0,0)-crossimplex is a cross-edge between layers V1 and V2; a (1, −1)-crossimplex (resp (−1, 1)-crossimplex) is a horizontal edge on V1 (resp. V2); 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 V1 or V2; a (3, −1)-crossimplex or (−1, 3)-crossimplex is a horizontal tetrahedron on V1 or V2; and a (1,1)-crossimplex, a (2,0)-crossimplex, or a (0,2)-crossimplex is a cross-tetrahedron (see Figure 1 for illustrations). On the other hand, horizontal edges, triangles, tetrahedrons, are just usual simplices on the horizontal complexes. One can consider a cross-edge as a connection between a vertex from one layer to a vertex on the other layer. In the same vein, a cross-triangle can be considered a connection between a 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.


FIGURE 1. Crossimplices. Schematic representation of (A) A (0, −1)-crossimplex (a top vertex), a (1, −1)-crossimplex (top horizontal edge), and a (−1, 2)-crossimplex (bottom horizontal triangle); (B) (0,0)-crossimplex (a cross-edge); (C) (1,0)-crossimplex (a top cross-triangle); (D) (1,1)-crossimplex (a cross-tetrahedron); and (E) (0,2)-crossimplex (also a cross-tetrahedron). Notice that cross-edges are always oriented from the vertex of the top layer to that 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 (0,1)-crossimplices (bottom cross-triangle). Moreover, there are three types of cross-tetrahedrons: the (0,2)-crossimplices, (2,0)-crossimplices, and (1,1)-crossimplices.

2.3 Weighted CSBs

A weight on CSB X is a positive function w:k,lXk,lR+ 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 aX is the number w(a).

3 Topological descriptors

3.1 Cross-boundaries

CSB X defines a bi-simplicial set (Moerdijk, 1989; Goerss and Jardine, 2009) by considering, respectively, the top and bottom crossface maps di|k,l(1):Xk,lXk1,l and di|k,l(2):Xk,lXk,l1 by


where the hat over a vertex means dropping the vertex. Moreover, for a fixed l ≥−1, X,l=(Xk,l)k1 is a simplicial complex. Similarly, Xk,=(Xk,l)l1 is a simplicial complex. We observe that if a={v01,,vk1,v02,,vl2}Xk,l, then a(1)={v01,,vk1}Xk,1 and a(2)={v02,,vl2}X1,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, v1Xk,−1 and v2X−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 v1v2. This is equivalent to requiring that if v1={v01,,vk1} and v2={v02,,vl2}, then {v01,,vk1,v02,,vl2}Xk,l. If a={v01,,vk1,v02,,vl2}Xk,l, we define its top cross-boundary (1)a as the subset of Xk−1,l consisting of all the top crossfaces of a, i.e., all the (k − 1, l)-crossimplices of the form di|k,l(1)[a] for i = 0, … , k. Analogously, its bottom cross-boundary (2)aXk,l1 is the subset of all its bottom crossfaces di|k,l(2)[a],i=0,,l.

Now, two (k, l)-crossimplices a, bXk,l are said to be as follows:

top-outer (TO) adjacent, which we write a(1)b or ac(1)b, if both are top crossfaces of a (k + 1, l)-crossimplex c; in other words, a,b(1)c;

top-inner (TI) adjacent, which we write a(1)b or a(1)db, if there exists a (k − 1, l)-crossimplex dXk−1,l, which is a top crossface of both a and b, i.e., d(1)ab(1);

bottom-outer (BO) adjacent, which we write ab(2) or ac(2)b, if both are bottom crossfaces of a (k, l + 1)-crossimplex cXk,l+1; in other words, a,b(2)c; and

bottom-inner (BI) adjacent, which we write a(2)b or a(2)db, if there exists a (k, l − 1)-crossimplex fXk,l−1, which is a bottom face of both a and b; i.e., d(2)a(2)b.

3.2 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:


• Similarly, the TI degree of a is defined as follows:


• Analogously, the BO degree of a is given as follows:


• The BO degree of a is as follows:


Observe that in the particular case where the weight function is equal to one everywhere, the TO degree of a is precisely the number of (k + 1, l)-crossimplices in X, of which a is the top crossface, while degTI(a) is the number of top crossfaces of a, which equals to k + 1. Analogous observations can be made about the BO and BI degrees.

3.3 Cross-homology groups

The space Ck,l of (k, l)-cross-chains is defined as the real vector space generated by all oriented (k, l)-crossimplexes in X. The top and bottom cross-boundary operators k,l(1):Ck,lCk1,l and k,l(2):Ck,lCk,l1 are then defined as follows by the formula


for s = 1, 2 and a generator aXk,l, where sgn(b,(s)a) is the sign of the orientation of b in (s)a; in other words, if b=di|k,l(1)[a], 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 focus more on the mixed case where both l and k are non-negative. We will often drop the indices and write (1) and (2) to avoid cumbersome notations. To see how these maps operate, let us compute, for instance, the images of the crossimplices (b), (c), (d), and (e) illustrated in Figure 1. We obtain the following:


Notice that 0,1(1)=1,0(2)=0. Moreover, by simple calculations from (6), it is easy to check that k1,l(1)k,l(1)=0 and k,l1(2)k,l(2)=0, which allows to define the top and bottom (k, l)-cross-homology groups of X as the following quotients:


For k ≥ 0 and l ≤ 0, k,1(1) and 1,l(2) are the usual boundary maps of simplicial complexes (Hatcher, 2000). Therefore, Hk,1(1)(X) and H1,l(2)(X) are the usual homology groups (Mac Lane, 1963; Hatcher, 2000) of the simplicial complexes X•,−1 and X−1,•, respectively.

3.4 Cross-Betti vectors

The cross-homology groups are completely determined by their dimensions, the top and bottom (k, l)-cross-Betti numbers βk,l(s)(X)=dimHk,l(s)(X), s = 1, 2. In particular, βk,1(1) and β1,l(2) are the usual Betti numbers for the horizontal simplicial complexes (Hatcher, 2000). The couple βk,l=(βk,l(1),βk,l(2)) 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, β0,−1 and β−1,0 encode the connectivities within and in-between the 1-skeletons of the horizontal complexes associated with X. Precisely, β0,1(1) is the number of connected components of the graph X1,−1, and β0,1(2) is the number of nodes in V1 with no interconnections with any nodes in V2. Similarly, β1,0(1) is the number of nodes in V2 with no interconnections with any nodes in V1, while β1,0(2) is the number of connected components of the bottom horizontal graph X−1,1. Furthermore, β1,−1 simultaneously counts the number of loops in X1,−1 and the number of its intra-layer links that do not belong to cross-triangles formed with the graph X−1,1. Analogous topological information is provided by β−1,1. In addition, β0,0 measures the extent to which individual nodes of one complex layer serve as communication channels between different hubs from the other layers. More precisely, an element in H0,0(1)(X) represents either an inter-layer one-dimensional loop formed by a path in X1,−1, whose end-nodes interconnect with the same node in V2, or two connected components in the top complex communicating with each other through a node in the bottom complex. β0,0 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 cross-boundaries of cross-triangles; we call such paths cones. In other words, β0,0 quantifies node clusters in one layer that are “controlled” by nodes in the other layer. Detailed proof of this description is provided in Methods 7.1.

Now, given a CSM X, its cross-Betti table βk,l is obtained by computing all the cross-Betti vectors of all its underlying CSBs. Computation of the cross-Betti table of the CSM of Figure 2 is presented in Table 1.


FIGURE 2. Schematic representation of a two-dimensional crossimplicial multicomplex X with 3 layers and 30 nodes in total; X consists of the vertex sets V1, V2, and V3 and the three CSBs X1,2,X1,3,X2,3 defined, respectively, on the products V1 × V2, V1 × V3, and V2 × V3.


TABLE 1. Cross-Betti table. The cross-Betti table for CSM of Figure 2. 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, and the number of paths of length 2 connecting the nodes in one layer and passing through a node from another layer.

To illustrate what the cross-Betti vectors represent, we consider the simple two-dimensional CSB X of Figure 3. We get β0,1(1)=2,β1,1(1)=1 and β1,0(2)=1,β1,1(2)=0. This reflects the fact the top layer has two connected components and one cycle, while the bottom one has one component and no cycles. Moreover, three top nodes are not interconnected to the bottom complex, six top edges are not top faces of cross-triangles, two bottom nodes are not interconnected to the top layer, and five bottom edges are not bottom faces of cross-triangles. This information is encoded in β0,−1 = (2, 3), β1,−1 = (1, 6), β−1,0 = (2, 1), and β−1,1 = (5, 0). There are three generating inter-layer cycles, two of which are formed by an intra-layer path in the bottom layer and a node in the top layer (v41 and v61), and the other one is formed by an intra-layer path in the top layer and a node (v12) in the bottom layer. Moreover, the two nodes v12 and v42 of V2 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 inter-layer cycles are exactly what β0,0 encodes. Specifically, by computing the cross-homology of X, we get β0,0(1)=3, which counts the cycle v21v31v41v12v21 and the nodes v42 and v12 that interconnect v41 to v61, and v21 to v61, β0,0(2)=2 counting the inter-layer cycles v41v12v22v32v42v41 and v61v22v32v42v61. In each of these cycles, the top node allows a shortest (inter-layer) path between the end-points of the involved intra-layer path.


FIGURE 3. Cross-Betti vectors. Schematic representation of a two-dimensional CSB with 14 nodes in total, whose oriented maximal crossimplices are the intra-layer triangle [v51,v61,v71] in X2,−1, the intra-layer edges [v01,v11],[v11,v21],[v11,v31],[v21,v31] in X1,−1, the bottom intra-layer triangles [v02,v12,v22],[v32,v42,v52] in X−1,2, the intra-layer edge [v22,v32] in X−1,1, the cross-triangles [v01,v11;v12],[v11,v21;v12] in X1,0, [v61;v12,v22],[v61;v42,v52] in X0,1, and the cross-edges [v41;v12],[v41;v42] in X0,0.

Using algebraic topological methods to calculate the cross-Betti vectors for larger multicomplexes can quickly become computationally heavy. We provide powerful linear-algebraic tools that not only allow to easily compute βk,ls but also tell exactly where the topological structures being counted are located within the multicomplex.

4 Spectral descriptors

4.1 Cross-forms

Ck,lCk,l(X,R) denotes the dual space HomR(Ck,l,R) of the real vector space Ck,l. In other words, Ck,l is the vector space of real linear functional ϕ:Ck,lR. We will refer to such functionals as (k, l)-forms or cross-forms on X. In particular, (k, −1)-forms correspond to k-forms on the simplicial complex X•,−1, and (−1, l)-forms are l-forms on the complex X−1,•. We have C−1,−1 = 0, and by convention, we set Ck,l (∅) = 0.

Notice that a natural basis of Ck,l is given by the following set of linear forms:


called elementary cross-forms, where

eab=1,if  a=b,0,otherwise,

which naturally identify Ck,l with Ck,l. Now, the maps δk,l(1):Ck,lCk+1,l and δk,l(2):Ck,lCk,l+1 are defined by the following equations:


for ϕCk,l, aXk+1,l and cXk,l+1. Next, given a weight w on X, we get an inner product on cross-forms by the following setting:

ϕ,ψk,laXk,lwaϕaψa,for  ϕ,ψCk,l.(8)

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 the following equation:


for ϕCk+1,l, aXk,l. We also obtain a similar formula for the dual (δk,l(2))*.

4.2 The cross-Laplacian operators

Identifying Ck,l with Ck,l and equipping it with an inner product, as (21), we define the following self-adjoint linear operators on Ck,l for all k, l ≥−1:

- the top (k, l)-cross-Laplacian is as follows:


- and the bottom (k, l)-cross-Laplacian is as follows:


Being defined on finite dimensional spaces, these operators can be represented as square matrices indexed over crossimplices. Specifically, denoting Nk,l = |Xk,l|, Lk,l(T) can be represented by positive definite Nk,l × Nk,l matrices (see Methods 7.3).

Moreover, the null spaces, the elements of which we call harmonic cross-forms, are easily seen to be in one-to-one correspondence with cross-cycles on X. In other words, we have the following isomorphisms (see Methods 7.2 for the proof):


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. However, before we see how this works, let us make a few observations. Notice that L0,1(T) and L1,0(B) are the usual graph Laplacians of 0 degree for the horizontal complexes. More generally, Lk,1(T) and L1,l(B) are the combinatorial higher Hodge Laplacians (Horak and Jost, 2013; Lim, 2020; Schaub et al., 2020) of degrees k and l, respectively, for the horizontal simplicial complexes. Furthermore, Lk,1(B) (resp. L1,l(T)) 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 Lk,1(B) is the diagonal matrix indexed over the k-simplices on the top complex and whose diagonal entries are the BO degrees. Similarly, L1,l(T) 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 β0,1(2) and β1,0(1) given earlier in terms of connectivities between the 1-skeletons of the horizontal complexes.

4.3 Harmonic cross-hubs

For the sake of simplicity, it is assumed that X is equipped with the trivial weight 1. Then, by (26), the (0,0) cross-Laplacians L0,0(T) and L0,0(B) are, respectively, represented by the N0,0 × N0,0 matrices indexed on cross-edges ai, ajX0,0, whose entries are given by the following equations:

L0,0(T)ai,aj=degTOai+1,if i=j,1,if ij,ai=vi1;vk2,aj=vj1;vk2and vi1,vj1,vk2X1,0,0,otherwise,(10)


L0,0(B)ai,aj=degBOai+1,if i=j,1,if ai=vi01;vi2,aj=vi01;vj2,and vi01,vi2,vj2X0,1,0,otherwise.(11)

Applied to the toy example of Figure 3, L0,0(T) has a zero-eigenvalue of multiplicity 3, generating the three (0,0) cross-cycles in Table 2.


TABLE 2. Harmonic (0,0) cross-forms. The three eigenvectors of the eigenvalue 0 of L0,0(T) corresponding to the synthetic CSB of Figure 3. There are two harmonic cross-hubs, v12 and v42, and their respective harmonic cross-hubness are 2.6177 and 1.4070.

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 considered harmonic cross-hubs (HCHs). The harmonic cross-hubness of a bottom node is the L1-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 v12 in common and those with v42 in common. We, therefore, have two harmonic cross-hubs (see illustration in Figure 4), hence two harmonic cross-clusters. The first harmonic cross-hub is responsible for the top layer cross-cluster {v01,v11,v21,v41,v61}, while the second harmonic cross-hub controls the top layer cross-cluster {v41,v61}. The intensity of each involved cross-edge is the L1-norm of its corresponding coordinates in the three 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, v12 is the bottom node with the highest harmonic cross-hubness, which is 2.6177. This reflects the fact that v12 not only interconnects the two connected components of the top complex (which v42 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 L0,0(B) yield v61 as the top node with the highest harmonic cross-hubness with respect to the bottom complex.


FIGURE 4. Cross-Laplacians, harmonic, and principal cross-hubs. (A) and (D) Heat-maps of the top and bottom (0,0) cross-Laplacian matrices for the example in Figure 3. Both matrices are indexed over the cross-edges of CSB, and the diagonal entries correspond to one added to the number of cross-triangles containing the corresponding cross-edge. L0,0(T) has a zero eigenvalue of multiplicity 3, while L0,0(B) has a zero eigenvalue of multiplicity 2. (B) and (E) Harmonic cross-hubs with respect to the top (resp. the bottom) horizontal complex of X; the intensity of a cross-edge is given by the L1-norm of the corresponding coordinates in the eigenvectors of the eigenvalue 0. (C) and (F) Principal cross-hubs in the bottom (resp. top) layer with respect to the top (resp. bottom) layer; by definition, they are the spectral cross-hubs obtained from the largest eigenvalues of the top and bottom (0,0) cross-Laplacians, respectively.

4.4 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=(x1,,xN0,0) is an eigenvector for an eigenvalue λT of L0,0(T). Then, denoting the cross-edges by ai, i = 1, … , N0,0, we have the following relations:


where χ is such that χ(ai, aj) = 1 if i = j or if ai and aj are adjacent but do not belong to a top cross-triangle, and χ(ai, aj) = 0 otherwise. It follows that the cross-edge intensity |xi| grows larger as degTO (ai) → λT. In particular, for λT = 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, λT=λmaxT, to be the largest eigenvalue of L0,0(T). Then, the intensity |xi| 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 two-layered network, for λT = 0, |xi| is larger for a cross-edge pointing to a bottom node interconnecting the largest number of top nodes that are not directly connected with intra-layer edges; for λT = łmax, |xi| 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, 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 λ1Tλ2TλmaxT of L0,0(T). At every stage, we regroup the cross-edges with non-zero coordinates in the associated eigenvectors and pointing to the same nodes, and 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 L0,0(B). In particular, we define the principal cross-hubs (PCHs) in the bottom layer with respect to the top layer as the SCHs obtained from λmaxT. 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 with respect to the bottom layer using the largest eigenvalue λmaxB of L0,0(B). Going back to the bicomplex of Figure 3, the largest eigenvalue of L0,0(T) is λmaxT=5, and the corresponding eigenvector is represented by Table 3.


TABLE 3. Principal eigenvector of L0,0(T) for the CSB of Figure 3. By definition, this is the eigenvector associated with the largest eigenvalue.

There is only one PCH in the bottom layer with respect to the top layer, which is the bottom node v12, and its principal cross-hubness is 2.2360.

Interestingly, the number of SCHs that appear for a given eigenvalue tend to vary dramatically with respect to the smallest eigenvalues before it eventually decreases or stabilizes at a very low number (see Figure 5; Figure 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 layers. The further we move away from the smallest non-zero eigenvalue, the more powerful are the 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 line—spectral persistence bar—running through the indices of the corresponding eigenvalues (Figure 5). The spectral persistence bars corresponding to all SCHs (the spectral bar codes) obtained from L0,0(T) (resp. L0,0(B)) constitute a signature for all the clustering structures imposed by the bottom (resp. top) layer to the top (resp. bottom) layer.


FIGURE 5. Spectral persistence of cross-hubs. Schematic illustrations of the variations in spectral cross-hubs along the eigenvalues and the spectral persistence bars codes for the toy CSB of Figure 3: (A) shows the number of bottom nodes that emerge as spectral cross-hubs with respect to the top layer as a function of the eigenvalues of L0,0(T), and (B) represents the number of top nodes revealed as spectral cross-hubs with respect to the bottom layer as a function of the eigenvalues of L0,0(B). (C) and (D) represent the spectral persistence bar codes for L0,0(T) and L0,0(B), respectively. For both the top and bottom (0,0) cross-Laplacians, most of the spectral cross-hubs, hence, spectral cross-clusters, emerge during the first stages (smallest eigenvalues), and very few of them survive at later stages; here, only one cross-hub emerges or survives at the largest eigenvalue (v12 for L0,0(T) and v61 for L0,0(B)).


FIGURE 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 labeled with their ICAO codes (see The most persistent cross-hubs correspond to the airports that provide the most efficient correspondences from the first airline network to the second.

5 Experiments on multiplex networks

5.1 Diffusion CSBs

Let M be a multiplex formed by M graphs Γs = (Es, V), s = 1, … , M. Denoting the vertex set V as an ordered set {1, 2, … , N}, we will write vis to represent the node i in the graph Γs, following the same notations we have used for multicomplexes.

For every pair of distinct indices s, t, we define the two-dimensional CSB Xst on V × V such that Xk,1st= for k ≥ 1, and X1,kst is the 2-clique complex of the layer indexed by t in the multiplex M; a pair (vis,vjt)V×V, forms a cross-edge if i < j, and nodes i and j are connected in Γs; and a (0,1) crossimplex is a triple (vis,vjt,vkt)V3 such that i is connected to j and k in Γs, and j and k are connected in Γt, while X1,0st=. We call Xst the diffusion bicomplex of (layer) s onto t. Notice that by construction, the (0,0) cross-Laplacians of Xst are indexed over Es, while the (0,0) cross-Laplacians of Xts are indexed over Et. This shows that Xst and Xts are not the same. The diffusion bicomplex Xst is a way to look at the topology of Γs through the topology of Γt; in other words, it diffuses the topology of the former into the topology of the latter.

5.2 Cross-hubs in air transportation networks

We used a subset of the European air transportation network (ATN) dataset (from Cardillo et al., 2013) to construct a three-layered multiplex M on 450 nodes, each representing a European airport (Wu et al., 2019). The three layer networks Γ1, Γ2, and Γ3 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 X1→2, X1→3, X2→1, X3→1, X2→3, and X3→2, we obtain the spectral persistence bar codes describing the emergence of SCHs for each airline with respect to the others (see Figure 6). The induced SCH rankings are presented in Table 4.


TABLE 4. Ranking of the 10 most persistent SCHs for the diffusion bicomplexes associated with the European air transportation multiplex network.

6 Discussion and conclusion

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 higher-order multilayer complex systems. Nevertheless, through synthetic examples of CSM and applications in multiplex networks, we have shown that our framework provides powerful tools to reveal important topological features in multilayer networks and address questions that would not arise from the standard pairwise-based formalism of multilayer networks. We specially focused on the (0,0) cross-Laplacians to show how their spectra quantify the extent to which nodes in one layer control topological structures in other layers in a multilayer network. Specifically, we saw that the spectra of these matrices allow detection of nodes from one layer that serve as inter-layer connecting hubs for clusters in the other layer; we referred to such nodes as SCHs. Such hubs vary in function of the eigenvalues, and they can be ranked according to their spectral persistence along the spectra of the cross-Laplacians. SCHs obtained from the largest eigenvalues (principal cross-hubs or PCHs) are those that interconnect 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. It might be the case that two SCHs persist equally long enough to be considered 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 PCH.

One can observe that the topological and geometric interpretations given for L0,0(T) and L0,0(B) can theoretically be generalized to the higher-order (k, l) cross-Laplacians as well. In other words, 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. However, in practice, dealing with the higher-order cross-Laplacians could quickly become computationally heavy or infeasible. It would, however, be interesting to see how the (0,1) and (1,1) cross-Laplacians translate in real datasets involving intrinsic intra- and inter-layer triangles and tetrahedrons.

Finally, many recent advances in structural and functional neuroimaging and genomics are based on the analysis of complex network representations of the human brain. The proposed framework will enable us to analyze the intimate relationships between the brain structure, genomics, and functional representations within unified multilayered structures.

7 Methods

7.1 Cones, kites, and the (0,0) cross-Betti numbers

Let vj2 be a fixed vertex in V2. A kite from V1 to vj2 is an ordered tuple (vi11,,vip1) of vertices in V1 such that {vir1,vir+11,vj2}X1,0 for r = 1, … , p − 1. Such an object is denoted as (vi01,,vip1vj2). Beware that the vertices vi11,,vip1 do not need to be pair-wise connected in V1. We observe the cross-triangles all pointing to vj2 that are pieced together in the form of an actual kite, as shown in Figure 7. In particular, if vj2 is the bottom face of a (1,0) cross-triangle [vi1,vk1;vj2], then (vi1,vk1vj2) is a kite. If (vi11,,vip1vj2) is a kite, its boundary is the triple (vi11,vip1,vj2)V12×V2. Similarly, given a fixed vertex vi1V1, one can define a kite from V2 to vi1 by a tuple (vj12,,vjp2) of vertices in V2 satisfying the analogous conditions. Such a kite will be denoted as (vi1vj12,,vjp2).


FIGURE 7. Two-dimensional crossimplicial bicomplex containing kites and cones. (v01,v11,v21) is a kite from V1 to v12V2 with boundary (v01,v21,v12)V12×V2, and (v12,v22,v32,v42) is a kite from V2 to v61V1 with boundary (v61,v12,v42)V1×V22. The tuples (v12,v22,v32),(v12,v22) and (v22,v32,v42) are also kites from V2 to v61. Furthermore, there are three cones with bases in V1: (v21,v41,v12) is a closed cone with base (v21,v41)V1 and vertex v12V2, and (v41,v61,v12) is an open cone with base (v41,v61)V12 and vertex v12V2. In addition, (v41,v61,v42) is an open cone with base (v41,v61)V12 and vertex v42V2, and (v12,v42,v41)V22×V1 is a closed cone with base (v12,v42)V22 and vertex v41V1. It follows from Theorem 7.1 that β0,0 = (3, 1).

It is worth noting that if (vi11,,vip1) is a kite from V1 to vj2, then so is each tuple (vir1,vir+11,,vir+q1) with 1 ≤ r and r + qp.

By a cross-chain on a kite, we mean one that is a linear combination of the triangles composing the kite; in other words, a cross-chain on the kite (vi11,,vip1vj2) is an element aC1,0(X) of the form


where γ1,,γp1R. In a similar fashion, cross-chains on a kite of the form (vi1vj12,,vjp2) are defined.

Now, given a pair (vi1,vk1) of vertices in the layer V1 and the vertex vj2V2, we say that the triple (vi1,vk1,vj2)V12×V2 is a cone with base (vi1,vk1) and vertex vj2 if it satisfies the following conditions:

vi1vj2 and vk1vj2, i.e., [vi1;vj2],[vk1;vj2]X0,0;

• the triple (vi1,vk1,vj2)V12×V2 is not the boundary of a kite from V1 to vj2.

We also say that (vi1,vk1,vj2) is a cone with the base in V1 and the vertex in V2. In a similar fashion, one defines a cone with the base in V2 and the vertex in V1. We refer to Figure 7 for examples of cones.

An immediate consequence of a triple (vi1,vk1,vj2)V12×V2 being a cone is that the vertices {vi1,vk1,vj2} are not (1,0) crossimplex. The vertices vi1 and vk1 might, however, be connected by a horizontal path of some length; when we mean that there might be a sequence of vertices vi01,,vip1 in V1, not all of which form cross-triangles with vj2, and such that


in which case, the cone is said to be closed; it is called open 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 H0,0(1)(X), H0,0(2)(X), and hence, the (0,0) cross-Betti numbers.

Theorem 7.1. The (0,0) cross-homology group H0,0(1)(X) (resp. H0,0(2)(X)) is generated by the cross-homology classes of cones with bases in V1 and vertices in V2 (resp. with bases in V2 and vertices in V1). Therefore, the (0,0) cross-Betti number β0,0(t) counts the cones with bases in Vt, t = 1, 2.

Here, by the cross-homology class of the cone (vi1,vk1,vj2)V12×V2, for instance, we mean the top cross-homology of the (0,0) cross-chain [vk1;vj2][vi1;vj2]C0,0(X).

Proof. We prove the theorem for H0,0(1)(X) since the same arguments apply to H0,0(2)(X). Every cone (vi1,vk1,vj2) defines a non-trivial (0,0) cross-cycle; in other words, the difference of the corresponding cross-edges [vi1;vj2][vk1;vj2]ker0,0(1). More generally, suppose we are given p cones (vi11,vi21,vj2),(vi2,vi31,vj2),,(vip11,vip1,vj2) with bases in V1 and all with the same vertex vj2V2. Then, for all real numbers α1, … , αp such that r=1pαr=0, the cross-chain


is clearly a (0,0) cross-cycle with a non-trivial cross-homology class, i.e., bker0,0(1) and bim1,0(1). Conversely, let bker0,0(1). We can write


so that 0,0(1)(b)=m=1Mαm[vim2]=0. Then, either all the vim2 values are pair-wise different, in which case b′ is the trivial cross-cycle; or there exist p + 1 subsets ({mr,1,,mr,Mr})r=1p+1 of {1, … , M} such that

vimr,12=vimr,22==vimr,Mr2,for 1rp


vimp+1,j2vimp+1,j2,for all jj,1j,jMp+1.

It follows that

j=1Mrαmr,j=0,for each r=1,,p(15)

and αmp+1,j=0for all j=1,,Mp+1. Hence, we get the following general expression of a (0,0) cross-cycle:


where the coefficients satisfy (15). Furthermore, it is straightforward to see that bim1,0(1) if, and only if, for each r = 1, … , p, there exists a permutation τr of {1, … , Mr} such that


is a kite. In that case, we get b=1,0(1)(a), where


and where, for r = 1, … , p, the coefficients γmr,j are given by


This shows that trivial cross-homology classes in H0,0(1)(X) are given by cross-cycles obtained from cross-chains on kites, i.e., images of sums of cross-chains in the form of (13).

7.2 Inner product, cross-Laplacians, and harmonic cross-forms

We define the following maps on cross-forms:

by the following equations:

δk,l(1)ϕa=ba(1)sgnb,a(1)ϕb,and  δk,l(2)ϕc=dc(2)sgnd,c(2)ϕd,(19)

for ϕCk,l(X), aXk+1,l and cXk,l+1.

Now, for each pair of integers (k, l), we choose inner products ⟨⋅,⋅⟩k,l, ⟨⋅,⋅⟩k+1,l and ⟨⋅,⋅⟩k,l+1 on the real vector spaces Ck,l(X), Ck+1,l(X) and Ck,l+1(X), respectively. The adjoint operators (δk,l(1))*:Ck+1,l(X)Ck,l(X) and (δk,l(2))*:Ck,l+1(X)Ck,l(X) are determined by the following relations:


for ϕCk,l(X), ψCk+1,l(X), and ψ′ ∈ Ck,l+1(X).

Any weight w on X defines such inner products on cross-forms by the following setting:

ϕ,ψk,laXk,lwaϕaψa,for  ϕ,ψCk,lX.(21)

It can be seen that, with respect to such an inner product, elementary cross-forms form an orthogonal basis. Moreover, given a weight function w and such an inner product, we get the following by simple calculations using (20):


for ϕCk+1,l(X), aXk,l. In addition, we obtain a similar formula for (δk,l(2))*.

Definition 7.2. We define the following self-adjoint linear operators on Ck,l(X) for all k, l ≥−1:

• the top-outer (k, l)-cross-Laplacian (or TO-Laplacian of order (k, l)


• the top-inner (k, l)-cross-Laplacian (or TI-Laplacian of order (k, l)


• the top (k, l) cross-Laplacian


• and similarly, the bottom-outer (k, l) cross-Laplacian (BO-Laplacian of order (k, l))


• the bottom-inner (k, l)-cross-Laplacian (BI-Laplacian)


• and the bottom (k, l)-cross-Laplacian


The null spaces of these operators, defined as the following sub-groups


are called the spaces of harmonic top (resp. bottom) cross-forms on X.

There is a one-to-one correspondence between cross-cycle classes and harmonic cross-forms on CSB X. In other words, we have the following group isomorphisms generalizing (Eckmann, 1944; Horak and Jost, 2013).

Lemma 7.3. For s = 1, 2 and for all k, l ≥−1, we have


where we have used the notations Lk,l(1)=Lk,l(T) and Lk,l(2)=Lk,l(B).

Proof. Let us prove this result for s = 1 (similar arguments apply to s = 2). First, notice that from the identification Ck,l(X) = Ck,l(X), we obtain the following:


and the analog holds for H(2)k,l(X). Moreover, recall from linear algebra that if EfF is a linear operator on two vector spaces equipped with inner products, then ker (f*f) = ker f. Indeed, we clearly have ker f ⊂ ker (f*f). Next, if x ∈ ker (f*f), then f*fx,yE=fx,fyF=0 for all yE, which implies that x ∈ ker f. In our case, we have δk,l(1)δk1,l(1)=0 and (δk1,l(1))*(δk,l(1))*=0; hence,




and the isomorphism (23) follows from (24).

It follows that the eigenvectors corresponding to the zero eigenvalue of the (k, l) cross-Laplacian Lk,l(s) are representative cross-cycles in the homology group Hk,l(s)(X). Henceforth, we see that in order to get the dimensions of the cross-homology groups Hk,l(s)(X), it suffices to find the eigenspaces corresponding to the zero eigenvalues of Lk,l(s). In other words,

βk,l(1)=dimkerLk,l(1),and βk,l(2)=dimkerLk,l(2).(25)

7.3 Matrix representations

Since the sets Xk,l are finite, the vector spaces Ck,l(X) and Ck,l(X) are finite dimensional, and as we have seen, the latter has the elementary cross-forms ea, aXk,l as orthogonal basis with respect to inner products defined from weight functions. So, the cross-Laplacian operators Lk,l(s),s=T,B can be represented as real square matrices of dimension |Xk,l|×|Xk,l| whose entries are indexed by elementary cross-forms ea. In order to compute these matrix representations, we first need to give their formal expressions as linear operators. Thanks to (22), we get the following for ϕCk,l, aXk,l:




In particular, when ϕ is an elementary cross-form eb, bXk,l, we obtain the following:

Lk,l(TO)eba=1wadegTOa,if a=b,wcwa,if  aband  ac(1)b,and have same orientation,wcwa,if  aband  ac(1)b,and have opposite orientations,0,otherwise ,


Lk,l(TI)eba=wadegTIa,if a=b,wbwd,if aband a(1)db,and have same orientation,wbwd,if aband a(1)dband have opposite  orientations,0,otherwise .

It follows that the (ea, eb)-th entry of the matrix representation of the top (k, l) cross-Laplacian Lk,l(T) with respect to the inner product defined from the weight function w is given by the following:

Lk,l(T)ea,eb=1wadegTOa+wadegTIa,if a=b,wbwdwcwa,if  ab,ac(1)band a(1)db,and have same orientationwcwawbwd,if  ab,ac(1)band a(1)db,and have opposite orientationswbwd,if ab,a(1)db,same orientation,and not  top-outer  adjacent,wbwd,if ab,a(1)db,opposite orientations,and not  top-outer  adjacent,0,otherwise .(26)

It is clear that we get similar matrix representation for the bottom (k, l) cross-Laplacian Lk,l(B).

Data availability statement

Publicly available datasets were analyzed in this study. These data can be found at:∼atnmultiplex/.

Author contributions

EM: conceptualization, data curation, formal analysis, investigation, methodology, software, validation, visualization, writing–original draft, and writing–review and editing. OA: writing–review and editing. HB: Funding acquisition, investigation, project administration, resources, supervision, validation, visualization, and writing–review and editing.


The author(s) declare financial support was received for the research, authorship, and/or publication of this article. This work was supported by the Natural Sciences and Engineering Research Council of Canada through the CRC grant NC0981.


The code used for visualizations and computations will be made available upon request.

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.


Benson, A. R., Gleich, D. F., and Leskovec, J. (2016). Higher-order organization of complex networks. Science 353 (6295), 163–166. doi:10.1126/science.aad9029

PubMed Abstract | CrossRef Full Text | Google Scholar

Bianconi, G. (2021). Higher-order networks. Cambridge University Press.

Google Scholar

Boccaletti, S., Bianconi, G., Criado, R., Del Genio, C. I., Gómez-Gardenes, J., Romance, M., et al. (2014). The structure and dynamics of multilayer networks. Phys. Rep. 544 (1), 1–122. doi:10.1016/j.physrep.2014.07.001

PubMed Abstract | CrossRef Full Text | Google Scholar

Cardillo, A., Gómez-Gardenes, J., Zanin, M., Romance, M., Papo, D., Pozo, F. d., et al. (2013). Emergence of network features from multiplexity. Sci. Rep. 3 (1), 1344–1346. doi:10.1038/srep01344

PubMed Abstract | CrossRef Full Text | Google Scholar

De Domenico, M., Solé-Ribalta, A., Cozzo, E., Kivelä, M., Moreno, Y., Porter, M. A., et al. (2013). Mathematical formulation of multilayer networks. Phys. Rev. X 3 (4), 041022. doi:10.1103/physrevx.3.041022

CrossRef Full Text | Google Scholar

Eckmann, B. (1944). Harmonische funktionen und randwertaufgaben in einem komplex. Comment. Math. Helvetici 17, 240–255. doi:10.1007/bf02566245

CrossRef Full Text | Google Scholar

Flores, J., and Romance, M. (2018). On eigenvector-like centralities for temporal networks: discrete vs. continuous time scales. J. Comput. Appl. Math. 330, 1041–1051. doi:10.1016/

CrossRef Full Text | Google Scholar

Fortunato, S., and Hric, D. (2016). Community detection in networks: a user guide. Phys. Rep. 659, 1–44. doi:10.1016/j.physrep.2016.09.002

CrossRef Full Text | Google Scholar

Goerss, P. G., and Jardine, J. F. (2009). Simplicial homotopy theory. Springer Science and Business Media.

Google Scholar

Griffa, A., Ricaud, B., Benzi, K., Bresson, X., Daducci, A., Vandergheynst, P., et al. (2017). Transient networks of spatio-temporal connectivity map communication pathways in brain functional systems. NeuroImage 155, 490–502. doi:10.1016/j.neuroimage.2017.04.015

PubMed Abstract | CrossRef Full Text | Google Scholar

Hatcher, A. (2000). Algebraic topology. Cambridge: Cambridge Univ. Press.

Google Scholar

Horak, D., and Jost, J. (2013). Spectra of combinatorial laplace operators on simplicial complexes. Adv. Math. 244, 303–336. doi:10.1016/j.aim.2013.05.007

CrossRef Full Text | Google Scholar

Iacopini, I., Petri, G., Barrat, A., and Latora, V. (2019). Simplicial models of social contagion. Nat. Commun. 10 (1), 2485. doi:10.1038/s41467-019-10431-6

PubMed Abstract | CrossRef Full Text | Google Scholar

Kivelä, M., Arenas, A., Barthelemy, M., Gleeson, J. P., Moreno, Y., and Porter, M. A. (2014). Multilayer networks. J. complex Netw. 2 (3), 203–271. doi:10.1093/comnet/cnu016

CrossRef Full Text | Google Scholar

Lim, L.-H. (2020). Hodge laplacians on graphs. SIAM Rev. 62, 685–715. doi:10.1137/18m1223101

CrossRef Full Text | Google Scholar

Liu, X., Maiorino, E., Halu, A., Glass, K., Prasad, R. B., Loscalzo, J., et al. (2020). Robustness and lethality in multilayer biological molecular networks. Nat. Commun. 11 (1), 6043. doi:10.1038/s41467-020-19841-3

PubMed Abstract | CrossRef Full Text | Google Scholar

Lotito, Q. F., Musciotto, F., Montresor, A., and Battiston, F. (2022). Higher-order motif analysis in hypergraphs. Commun. Phys. 5 (1), 79. doi:10.1038/s42005-022-00858-7

CrossRef Full Text | Google Scholar

Lucas, M., Cencetti, G., and Battiston, F. (2020). Multiorder laplacian for synchronization in higher-order networks. Phys. Rev. Res. 2 (3), 033410. doi:10.1103/physrevresearch.2.033410

CrossRef Full Text | Google Scholar

Mac Lane, S. (1963). Homology. Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete. Springer Berlin, Heidelberg: Academic Press. doi:10.1007/978-3-642-62029-4

CrossRef Full Text | Google Scholar

Majhi, S., Perc, M., and Ghosh, D. (2022). Dynamics on higher-order networks: a review. J. R. Soc. Interface 19 (188), 20220043. doi:10.1098/rsif.2022.0043

PubMed Abstract | CrossRef Full Text | Google Scholar

Mandke, K., Meier, J., Brookes, M. J., O’Dea, R. D., Van Mieghem, P., Stam, C. J., et al. (2018). Comparing multilayer brain networks between groups: introducing graph metrics and recommendations. NeuroImage 166, 371–384. doi:10.1016/j.neuroimage.2017.11.016

PubMed Abstract | CrossRef Full Text | Google Scholar

Moerdijk, I. (1989). Bisimplicial sets and the group-completion theorem. Dordrecht: Springer Netherlands, 225–240.

CrossRef Full Text | Google Scholar

Newman, M. E. (2003). The structure and function of complex networks. SIAM Rev. 45 (2), 167–256. doi:10.1137/s003614450342480

CrossRef Full Text | Google Scholar

Pedersen, M., Zalesky, A., Omidvarnia, A., and Jackson, G. D. (2018). Multilayer network switching rate predicts brain performance. Proc. Natl. Acad. Sci. 115 (52), 13376–13381. doi:10.1073/pnas.1814785115

PubMed Abstract | CrossRef Full Text | Google Scholar

Pilosof, S., Porter, M. A., Pascual, M., and Kéfi, S. (2017). The multilayer nature of ecological networks. Nat. Ecol. Evol. 1 (4), 0101. doi:10.1038/s41559-017-0101

PubMed Abstract | CrossRef Full Text | Google Scholar

Sánchez-García, R. J., Cozzo, E., and Moreno, Y. (2014). Dimensionality reduction and spectral properties of multilayer networks. Phys. Rev. E 89 (5), 052815. doi:10.1103/physreve.89.052815

CrossRef Full Text | Google Scholar

Schaub, M. T., Benson, A. R., Horn, P., Lippner, G., and Jadbabaie, A. (2020). Random walks on simplicial complexes and the normalized hodge 1-laplacian. SIAM Rev. 62 (2), 353–391. doi:10.1137/18m1201019

CrossRef Full Text | Google Scholar

Shi, D., Chen, Z., Sun, X., Chen, Q., Ma, C., Lou, Y., et al. (2021). Computing cliques and cavities in networks. Commun. Phys. 4, 249. doi:10.1038/s42005-021-00748-4

CrossRef Full Text | Google Scholar

Solá, L., Romance, M., Criado, R., Flores, J., García del Amo, A., and Boccaletti, S. (2013). Eigenvector centrality of nodes in multiplex networks. Chaos (Woodbury, N.Y.) 23, 033131. doi:10.1063/1.4818544

PubMed Abstract | CrossRef Full Text | Google Scholar

Timóteo, S., Correia, M., Rodríguez-Echeverría, S., Freitas, H., and Heleno, R. (2018). Multilayer networks reveal the spatial structure of seed-dispersal interactions across the great rift landscapes. Nat. Commun. 9 (1), 140. doi:10.1038/s41467-017-02658-y

PubMed Abstract | CrossRef Full Text | Google Scholar

Vasilyeva, E., Kozlov, A., Alfaro-Bittner, K., Musatov, D., Raigorodskii, A., Perc, M., et al. (2021). Multilayer representation of collaboration networks with higher-order interactions. Sci. Rep. 11, 5666. doi:10.1038/s41598-021-85133-5

PubMed Abstract | CrossRef Full Text | Google Scholar

Wu, M., He, S., Zhang, Y., Chen, J., Sun, Y., Liu, Y.-Y., et al. (2019). A tensor-based framework for studying eigenvector multicentrality in multilayer networks. Proc. Natl. Acad. Sci. 116, 15407–15413. doi:10.1073/pnas.1801378116

PubMed Abstract | CrossRef Full Text | Google Scholar

Young, J.-G., Petri, G., and Peixoto, T. P. (2021). Hypergraph reconstruction from network data. Commun. Phys. 4 (1), 135. doi:10.1038/s42005-021-00637-w

CrossRef Full Text | Google Scholar

Yuvaraj, M., Dey, A. K., Lyubchich, V., Gel, Y. R., and Poor, H. V. (2021). Topological clustering of multilayer networks. Proc. Natl. Acad. Sci. U. S. A. 118 (21), e2019994118. doi:10.1073/pnas.2019994118

PubMed Abstract | CrossRef Full Text | Google Scholar

Keywords: multilayer networks, cross-homology, multicomplexes, Laplacian, cross-hubs, spectral persistence

Citation: Moutuou EM, Ali OBK and Benali H (2023) Topology and spectral interconnectivities of higher-order multilayer networks. Front. Complex Syst. 1:1281714. doi: 10.3389/fcpxs.2023.1281714

Received: 22 August 2023; Accepted: 19 October 2023;
Published: 20 November 2023.

Edited by:

Byungjoon Min, Chungbuk National University, Republic of Korea

Reviewed by:

Flaviano Morone, New York University, United States
Wei Wang, Chongqing Medical University, China

Copyright © 2023 Moutuou, Ali and Benali. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Elkaïoum M. Moutuou,