Synchronous Stability in Multiplex Network Subject to Higher-Order Intralayer Interactions

Abstract

Recent research and instances have demonstrated that most real-world systems can be effectively schematized by multiplex networks. Moreover, the interactions within systems often emerge among triadic or tetradic interactions, or even interactions with more element combinations, in addition to pairwise interactions. Hypergraph coupling structures are particularly well-suited for capturing such arbitrary higher-order interactions among nodes, thereby playing a key role in accurately depicting system dynamics. Meanwhile, the functionality of numerous complex systems depends on synchronization mechanisms. Therefore, this paper focuses on investigating the synchronous stability of a multiplex hypergraph. Specifically, we examine a three-layer network where intralayer interactions are represented by hyperedges, while the interlayer interactions are modeled through pairwise couplings. By generalizing the master stability function approach to the hypergraph structure, the synchronization phenomenon of such multiplex hypergraphs is analyzed. To verify our theoretical conclusions, we apply the proposed framework to networks of FitzHugh–Nagumo neurons and Rikitake two-disk dynamos. Simulation results unveil that the presence of higher-order interactions enhances the synchronous ability within the multiplex framework.

1. Introduction

For the past three decades, network science has been successfully employed to describe numerous real-world circumstances or abstract systems [1,2,3]. However, these network models are represented as pairwise interactions and are often insufficient for describing connections involving multiple nodes in various scenarios. Examples can be found in diverse fields such as ecological systems, brain networks and protein networks [4,5,6,7], where interactions extend beyond pairwise relationships to involve arbitrary groups of units. This paradigm of higher-order coupling patterns has sparked a revolution in complex network science, fundamentally reshaping researchers’ understanding of network dynamics. Higher-order interactions not only govern the topological structure of complex systems but also profoundly shape their collective dynamics [5,6,7,8]. In particular, certain complex systems featuring higher-order interactions exhibit emergent dynamical phenomena, such as stepwise explosive phase transitions and non-trivial spreading processes [9,10,11]. Thus, to formalize these many-body interactions, it is necessary to generalize classical network structures. So far, simplicial complexes and hypergraphs have emerged as powerful frameworks to address interactions beyond pairwise [12,13]. In particular, simplicial complexes characterize the higher-order interactions with the added constraint that they include lower-order interactions, whereas hypergraphs serve as a more general framework for encoding higher-order interactions among arbitrary numbers of elements. Synchronous phenomena on complex networks represent one of the most prominent examples of collective phenomena and have garnered significant attention in a multiplex network setting [14,15,16]. Such studies have inspired researchers’ interest, with applications spanning social science, engineering and physics [17,18]. On the other hand, synchronization in networks of coupled dynamical units featuring higher-order interactions is critical for the proper operation of diverse natural and engineered systems, spanning from biological networks to social sciences frameworks. Recently, researchers have expanded the collective behaviors of networks with higher-order interactions [19,20,21,22]. In this regard, most efforts have focused on simplicial complex to represent group interactions because of their simple topological structure. Few studies have utilized hypergraphs to simulate many-body interactions. A hypergraph [23], defined as a topological structure formed by an ensemble of hyperedge links and vertices, has been explored in the context of synchronization. For instance, ref. [24] has investigated complete synchronization to hypergraphs. A new study of the condition of stability via a master stability function has been undertaken by Mulas et al. [25]. Ref. [26] has analyzed intralayer and interlayer synchronization of hypergraphs and unveiled the influence of higher-order structure on synchronous stability.

On the other hand, it has been recognized that numerous real-world scenarios are sustained by multiple layers with various interaction schemes among components, along with the mutual influence between different layers to support their efficiency and functionality [27,28,29]. So the shift from single-layer network analysis to the investigation of multilayer networks is essential for further study of complex networks. Multiplex networks have multiple interacting layers of networks, where the same node set is interconnected via more than one type of link—a feature widely adopted to model social systems, transportation networks and power-grid networks. More importantly, rich and complex collective behaviors emerge owing to interactions in multiplex networks, namely intralayer synchronization and interlayer synchronization, relay synchronization and chimera states. Nevertheless, previous studies on synchronization in multilayer networks have not taken into account the higher-order connections within layers, thereby limiting the generality of current models in describing complex systems. Based on the above discussion, in this paper, we will analyze the synchronization stability of multiplex network with higher-order intralayer interconnections and study the interplay between multiplex and higher-order structures. This work develops a mathematical model scheme for analyzing the stability of the complete synchronization state in hypergraphs with numerous interaction layers.

The remainder of this paper is organized as follows. First, Section 2 provides some definitions and the studied multiplex hypergraph model with multiple interaction layers. Meanwhile, synchronous stability for classes of multiplex networks based on master stability function (MSF) framework is investigated. Following that, numerical simulations on various systems of coupled oscillators are conducted to validate the theoretical findings in Section 3. Finally, we highlight our main findings as a conclusion in Section 4.

2. Preliminaries and Hypergraph

In this section, some basic concepts of hypergraphs are introduced and then the model of the multiplex hypergraph is provided.

2.1. Some Definitions

A hypergraph $H=(V, E)$ exists on a finite collection of nodes $V=\left\{v_i:i=1, 2, \dots , N\right\}$ and $E=\left\{e_j:j=1, 2, \dots , M\right\}$ denotes a family on $M$ hyperedges, where each hyperedge is a non-empty subset of $V.$ In particular, a hyperedge $e_j$ represents interactions of arbitrary order among vertices. Also, when all hyperedges are size 2, the hypergraph transforms into a traditional binary interaction graph. In a hypergraph framework, a hyperedge $e_j$ containing $n_j$ nodes is denoted as $e_j\subseteq E$ with $\left|e_j\right|=n_j$. Counterintuitively, the order of a hyperedge is defined as $n_j-1,$ where the set $e_j$ is unordered (i.e., permutations of $e_j$ represent the same hyperedge). Specifically, 1-order hyperedges correspond to traditional edges, while 2-order hyperedges refer to interactions occurring among 3 nodes, and so on. For simplicity, each node itself is defined as a 0-order hyperedge. The maximal order of its constituent hyperedges denotes the order of a hypergraph.

To further describe how the nodes are distributed among the hyperedges, we introduce the incidence matrix as follows:

$$\Phi =\left\{{\varphi }_{ij}\right\}=\left\{\begin{array}{l}1,\hspace{1em}i\in e_j\hfill \\ 0,\hspace{1em} \mathrm{otherwise}\hfill \end{array}\right.$$

(1)

On the other hand, one can obtain the adjacency matrix $A_{N\times N}=\Phi {\Phi }^T$; the elements of the matrix carries the information of hyperedges shared by two adjacent nodes. Additionally, the hyperedge matrix $B_{M\times M}={\Phi }^T\Phi$ contains the entry $b_{j_1{j}_2}$, which denotes the number of nodes in $e_{j_1}\cap e_{j_2}$. Furthermore, the corresponding Laplacian matrix is defined by $L_{N\times N}=(l_{ij})=d_i{\epsilon }_{ij}-A_{ij}$, where ${\epsilon }_{ij}$ denotes Kronecker delta and $d_i$ shows the number of hyperedges passing through the node i.

In the following step, a multiplex hypergraph $\Psi =(X, Y)$ is defined, where $X=\left\{X_k=(v_k, {\varsigma }_k):k\in \left\{1, 2, \dots , L\right\}\right\}$ is an ensemble of hypergraphs, each portraying a layer with a fixed number of nodes in each layer, and $Y$ gives the collection of links connecting each node of a layer to all its replicas in the other layers. The intralayer higher-order connections are described by the elements of ${\varsigma }_k$, and the interlayer interactions between different layers are characterized by the elements of $Y$. In this paper, we assume that intralayer connections are hyperedges of different sizes, whereas interlayer connections are pairwise couplings. That is, if we only consider binary connections between nodes within each layer, the multiplex network can be viewed as a “standard” pairwise multiplex network. A multiplex hypergraph network is depicted by the following Figure 1a; Figure 1b illustrates the intralayer network structure of the hypergraph. Moreover, the interlayer couplings are one-to-one pairwise interactions.

2.2. The Mathematical Model

This section provides the multiplex hypergraph model, with a focus on the different coupling schemes, containing higher-order intralayer and one-to-one interlayer pairwise interactions.

Let us consider a multiplex hypergraph composed of three-layer x, y and z. Also, each layer contains N nodes of identical systems connecting diffusively through M hyperedges of different sizes. The state of the $i–\mathrm{th}$ node is described by a $d–$dimensional state vector. Furthermore, the nodes across different layers are coupled by one-to-one pairwise interconnections. Then, the model of the three-layer hypergraph can be written as

$$\begin{array}{l}\left\{\begin{array}{l}{\dot{\mathit{x}}}_i=f({\mathit{x}}_i)-\alpha {\displaystyle \sum _{j=1}^N{l}_{ij}G{\mathit{x}}_j-\beta H({\mathit{y}}_i-{\mathit{x}}_i),}\hfill \\ {\dot{\mathit{y}}}_i=f({\mathit{y}}_i)-\alpha {\displaystyle \sum _{j=1}^N{l}_{ij}G{\mathit{y}}_j-\beta H({\mathit{x}}_i+{\mathit{z}}_i-2{\mathit{y}}_i),}\hfill \\ {\dot{\mathit{z}}}_i=f({\mathit{z}}_i)-\alpha {\displaystyle \sum _{j=1}^N{l}_{ij}G{\mathit{z}}_j-\beta H({\mathit{y}}_i-{\mathit{z}}_i).}\hfill \end{array}\right.\\ i=1, 2, \dots , N.\end{array}$$

(2)

where $f({\mathit{x}}_i), f({\mathit{y}}_i), f({\mathit{z}}_i):$${ℝ}^d\to {ℝ}^d$ represents the local dynamics of the $i\mathrm{th}$ uncoupled nodes in the first, second and third layers, respectively. $G(·):{ℝ}^d\times {ℝ}^d\to {ℝ}^d$ and $H(·):{ℝ}^d\times {ℝ}^d\to {ℝ}^d$ give the intralayer and interlayer coupling functions, respectively. The intralayer coupling and interlayer coupling strengths are represented by $\alpha$ and $\beta$. Here, we assume that $G=diag(g_1, g_2, g_3)$ and $H=diag(h_1, h_2, h_3)$ are diagonal matrices. Furthermore, the Laplacian matrices $L={(l_{ij})}_{N\times N}$ of the hypernetwork in each layer are identical and satisfy the dissipation condition ${\displaystyle \sum _{j=1}^N{l}_{ij}}=0\begin{array}{cc}& (i=1, 2, \end{array}\dots , N).$ According to the topological structure of Figure 1, the interlayer general Laplacian matrix is given by

$$D=\left[\begin{array}{ccc}1& -1& 0\\ -1& 2& -1\\ 0& -1& 1\end{array}\right].$$

In the following, the interlayer supra-Laplacian matrix $L^D=D\otimes I_N$ is given, $I_N$ denotes the identity matrix, $\otimes$ represents the Kronecker product, and the intralayer supra-Laplacian matrix $L^L$

$$L^L=\underset{l=1}{\overset{3}{\oplus }}{L}^{(l)}=\left[\begin{array}{ccc}{L}^{(1)}& 0& 0\\ 0& L^{(2)}& 0\\ 0& 0& L^{(3)}\end{array}\right].$$

where $\oplus$ means direct sum and $L^{(l)}$ is the Laplacian matrix of the hypernetwork in layer l.

2.3. Stability Condition for Multiplex Hypergraph

In this subsection, we aim to develop a general framework for investigating the synchronous stability of multiplex hypergraphs using the master stability function method [30]. This framework simplifies large-scale networked systems to node-scale systems via diagonalization and decoupling. Consequently, the problem of determining network synchronization reduces to verifying whether all characteristic modes of the network lie within the corresponding synchronized regions. For simplicity, some new variables are defined as follows:

$$\mathit{x}=\left[\begin{array}{c}{\mathit{x}}_1\\ {\mathit{x}}_2\\ ⋮\\ {\mathit{x}}_N\end{array}\right],\hspace{1em}\begin{array}{c}\mathit{y}=\left[\begin{array}{c}{\mathit{y}}_1\\ {\mathit{y}}_2\\ ⋮\\ {\mathit{y}}_N\end{array}\right]\end{array},\hspace{1em}\begin{array}{c}\mathit{z}=\left[\begin{array}{c}{\mathit{z}}_1\\ {\mathit{z}}_2\\ ⋮\\ {\mathit{z}}_N\end{array}\right]\end{array},\hspace{1em}\tilde{f}(\mathit{x})=\left[\begin{array}{c}f({\mathit{x}}_1)\\ f({\mathit{x}}_2)\\ ⋮\\ f({\mathit{x}}_N)\end{array}\right],\begin{array}{c}\tilde{f}(\mathit{y})=\left[\begin{array}{c}f({\mathit{y}}_1)\\ f({\mathit{y}}_2)\\ ⋮\\ f({\mathit{y}}_N)\end{array}\right]\end{array},\hspace{1em}\begin{array}{c}\tilde{f}(\mathit{z})=\left[\begin{array}{c}f({\mathit{z}}_1)\\ f({\mathit{z}}_2)\\ ⋮\\ f({\mathit{z}}_N)\end{array}\right]\end{array},\hspace{1em}x=\left[\begin{array}{c}\mathit{x}\\ \mathit{y}\\ \mathit{z}\end{array}\right],\hspace{1em}\begin{array}{c}F(x)=\left[\begin{array}{c}\tilde{f}(\mathit{x})\\ \tilde{f}(\mathit{y})\\ \tilde{f}(\mathit{z})\end{array}\right].\end{array}$$

In the following, our main goal is to obtain the necessary conditions for the stable complete synchronization state of the hypergraph (2). Firstly, we look into the conditions for invariance in the synchronization state. It is well known that a complete synchronization is achieved in the hypergraph (2) when each unit follows in unison with the rest nodes. Therefore, there exists a solution ${\mathit{s}}_0(t)\in {ℝ}^p$ such that

$$t\to \infty ,‖{\mathit{x}}_i(t)-{\mathit{s}}_0(t)‖\to 0,\begin{array}{c}‖{\mathit{y}}_i(t)-{\mathit{s}}_0(t)‖\to 0,‖{\mathit{z}}_i(t)-{\mathit{s}}_0(t)‖\to 0,\hspace{1em}i=1, 2, \dots , N.\end{array}$$

(3)

Then, the synchronized manifold can be described by

$$S=\left\{{\mathit{s}}_0(t)\in {ℝ}^P:{\mathit{x}}_i(t)={\mathit{y}}_i(t)={\mathit{z}}_i(t)={\mathit{s}}_0(t), i=1, 2, \dots , N, t\in {ℝ}^{+}\right\}.$$

(4)

To further analyze the stability of the complete synchronization solution, small perturbation around the synchronization states is considered, i.e., $\delta {\mathit{x}}_i={\mathit{x}}_i-{\mathit{s}}_0, \delta {\mathit{y}}_i={\mathit{y}}_i-{\mathit{s}}_0, \delta {\mathit{z}}_i={\mathit{z}}_i-{\mathit{s}}_0$, and the linear stability analysis of Equation (2) is performed. One obtains

$$\left\{\begin{array}{l}\delta {\dot{x}}_i=JF({\mathit{s}}_0)\delta x_i-\alpha {\displaystyle \sum _{j=1}^N{L}_{ij}^{H}G({\mathit{s}}_0)\delta x_j+\beta [JH({\mathit{s}}_0)\delta y_i-JH({\mathit{s}}_0)\delta x_i], }\hfill \\ \delta {\dot{y}}_i=JF({\mathit{s}}_0)\delta y_i-\alpha {\displaystyle \sum _{j=1}^N{L}_{ij}^{H}G({\mathit{s}}_0)\delta y_j+\beta [JH({\mathit{s}}_0)\delta x_i+JH({\mathit{s}}_0)\delta z_i-2JH({\mathit{s}}_0)\delta y_i], }\hfill \\ \delta {\dot{z}}_i=JF({\mathit{s}}_0)\delta z_i-\alpha {\displaystyle \sum _{j=1}^N{L}_{ij}^{H}G({\mathit{s}}_0)\delta z_j+\beta [JH({\mathit{s}}_0)\delta y_i-JH({\mathit{s}}_0)\delta z_i].}\hfill \end{array}\right.$$

(5)

where ${\mathit{s}}_0$ is the synchronization solution of a network satisfying ${\dot{\mathit{s}}}_0=f({\mathit{s}}_0)$ and $JF({\mathit{s}}_0)$ is the Jacobian of $F$ calculated at the synchronization solution ${\mathit{s}}_0$. To get the block matrix form, we define the new vector $\delta x_i={[\delta x_1^{tr}, \delta x_2^{tr},\dots ,\delta x_N^{tr}]}^{tr},\hspace{1em}\begin{array}{c}\delta y_i={[\delta y_1^{tr}, \delta y_2^{tr},\dots ,\delta y_N^{tr}]}^{tr}\end{array}$ and $\delta z_i={[\delta z_1^{tr}, \delta z_2^{tr},\dots ,\delta z_N^{tr}]}^{tr}$. Equation (5) can be simplified as

$$\left\{\begin{array}{l}\begin{array}{l}\delta {\dot{x}}_i=[I_N\otimes JF({\mathit{s}}_0)-\alpha L^H\otimes JG]\delta x_i+\beta [I_N\otimes JH({\mathit{s}}_0)\delta y_i-I_N\otimes JH({\mathit{s}}_0)\delta x_i],\\ \end{array}\hfill \\ \begin{array}{l}\delta {\dot{y}}_i=[I_N\otimes JF({\mathit{s}}_0)-\alpha L^H\otimes JG]\delta y_i+\beta [I_N\otimes JH({\mathit{s}}_0)\delta x_i+I_N\otimes JH({\mathit{s}}_0)\delta y_i-2I_N\otimes JH({\mathit{s}}_0)\delta z_i],\\ \end{array}\hfill \\ \delta {\dot{z}}_i=[I_N\otimes JF({\mathit{s}}_0)-\alpha L^H\otimes JG]\delta z_i+\beta [I_N\otimes JH({\mathit{s}}_0)\delta y_i-I_N\otimes JH({\mathit{s}}_0)\delta z_i].\hfill \end{array}\right.$$

(6)

It is noticed that $L^H$ is a real symmetric matrix with zero row-sum, it can be diagonalized, and all its eigenvalues are non-zero real numbers with a smallest eigenvalue of zero. In the following, we investigate network synchronization via the idea of MSF. Thus, the dynamics of the variational equation can be described as follows:

$$\left\{\begin{array}{l}\begin{array}{l}{{\dot{\xi }}_i}^{(1)}=[I_N\otimes JF({\mathit{s}}_0)-\alpha L^H\otimes JG]{{\xi }_i}^{(1)}+\beta [(I_N\otimes JH({\mathit{s}}_0)){{\xi }_i}^{(2)}-(I_N\otimes JH({\mathit{s}}_0)){{\xi }_i}^{(1)}],\\ \end{array}\hfill \\ \begin{array}{l}{{\dot{\xi }}_i}^{(2)}=[I_N\otimes JF({\mathit{s}}_0)-\alpha L^H\otimes JG]{{\xi }_i}^{(2)}+\beta [I_N\otimes JH({\mathit{s}}_0){{\xi }_i}^{(1)}+I_N\otimes JH({\mathit{s}}_0){{\xi }_i}^{(2)}-2I_N\otimes JH({\mathit{s}}_0){{\xi }_i}^{(3)}],\\ \end{array}\hfill \\ {{\dot{\xi }}_i}^{(3)}=[I_N\otimes JF({\mathit{s}}_0)-\alpha L^H\otimes JG]{{\xi }_i}^{(3)}+\beta [I_N\otimes JH({\mathit{s}}_0){{\xi }_i}^{(2)}-I_N\otimes JH({\mathit{s}}_0){{\xi }_i}^{(3)}].\hfill \end{array}\right.$$

(7)

For simplicity, we assume that supra-Laplacian matrices $L^L$ and $L^D$ are symmetric matrices, and satisfy $L^L{L}^D=L^D{L}^L$; therefore, there exists an invertible matrix $R$ such that

$$V_1=R^{-1}{L}^{L}R=diag\left[\begin{array}{ccc}{\lambda }_1& & \\ & \ddots & \\ & & {\lambda }_N\end{array}\right],\hspace{1em}\begin{array}{c}{V}_2=R^{-1}{L}^{D}R=diag\left[\begin{array}{ccc}{\mu }_1& & \\ & \ddots & \\ & & {\mu }_N\end{array}\right]\end{array}.$$

which satisfies $0={\lambda }_1<{\lambda }_2\le {\lambda }_3\dots \le {\lambda }_N,0={\mu }_1<{\mu }_2\le {\mu }_3\dots \le {\mu }_N.$ Then, Equation (7) can be transformed into

$$\left\{\begin{array}{l}\begin{array}{l}{\dot{\xi }}_i^{(1)}=JF(s_0){\xi }_i^{(1)}-\alpha {\lambda }_{i}JG{\xi }_i^{(1)}+\beta [JH(s_0)){\xi }_i^{(2)}-JH(s_0)){\xi }_i^{(1)}],\\ \end{array}\hfill \\ \begin{array}{l}{\dot{\xi }}_i^{(2)}=JF(s_0){\xi }_i^{(2)}-\alpha {\lambda }_{i}JG{\xi }_i^{(2)}+\beta [JH(s_0)){\xi }_i^{(2)}+JH(s_0)){\xi }_i^{(1)}-2JH(s_0){\xi }_i^{(3)}],\\ \end{array}\hfill \\ {\dot{\xi }}_i^{(3)}=JF(s_0){\xi }_i^{(3)}-\alpha {\lambda }_{i}JG{\xi }_i^{(3)}+\beta [JH(s_0)){\xi }_i^{(2)}-2JH(s_0){\xi }_i^{(3)}].\hfill \end{array}\right.$$

(8)

Further, we write Equation (8) in matrix form denoting a new vector $\eta ={(R\otimes I_N)}^{-1}\xi$, then

$$\dot{\eta }=(I_N\otimes JF(s_0)\xi -\alpha \Gamma \otimes G)\eta -\beta \alpha (diag\left\{{\mu }_1, {\mu }_2, \dots ,{\mu }_N\right\}\otimes H))\eta .$$

(9)

Since matrices $V_1$ and $V_2$ are diagonal matrices, $I_N\otimes JF(s_0)$ has a block also in diagonal matrix form. Furthermore, each diagonal block has the same form. Thus, the multiplex master stability equation of network (2) can be described by

$${\dot{\eta }}_k=JF({\mathit{s}}_0){\eta }_k-\alpha {\lambda }_{k}JG{\eta }_k-\beta u_{k}JH{\eta }_k\hspace{1em}\begin{array}{c}(k=1, 2, \dots , N)\end{array}.$$

(10)

where ${\lambda }_k$, $u_k$ represent the smallest eigenvalues of the super-Laplace matrices $L^L$ and $L^D$. Therefore, the MSF of the multiplex hypergraph can be described as follows:

$$\dot{\eta }=(JF({\mathit{s}}_0)-\alpha \lambda G-\beta uH)\eta .$$

(11)

According to the MSF approach, the stability of system (2) can be transformed into solving Equation (11) along with ${\dot{\mathit{s}}}_0=f({\mathit{s}}_0)$ for obtaining the maximum transverse Lyapunov exponents, which are a function of $\alpha$ and $\beta$, denoted as $\Lambda (\alpha , \beta )$. When the topological structures are fixed, one can determine the eigenvalues $\lambda$ and $\mu$, and the synchronous regions with respect to intralayer coupling $\alpha$ and interlayer coupling $\beta$ will be obtained.

When $\mu =0$ and $\lambda \ne 0$, there exists no interlayer coupling within the network. Analogously, when $\lambda =0$ and $\mu \ne 0$, there exists no intralayer coupling within the network.

3. Numerical Simulations

In this section, based on the stability framework developed, two specific examples of three-layer hypergraphs will be studied. This provides a more intuitive representation of the dynamic behavior and measure the synchronization degree of coupled neurons. In the following step, the average synchronization error is considered to quantify complete synchronization as

$$E(t)={\displaystyle \frac{1}{NM}}{\displaystyle \sum _{k=1}^3{\displaystyle \sum _{i=1}^N‖x_i^{(k)}(t)-\overline{x}(t)‖}},$$

(12)

where ‖ ‖ denotes the Euclidean norm and $\overline{x}(t)$ is the average state of all the nodes in the multiplex network. The synchronization error reaching zero indicates the emergence of synchrony. The hypergraph (2) is integrated using the fourth-order Runge–Kutta method with an integration step size $dt=0.01$.

In what follows, the influence of higher-order intralayer interactions on synchronous stability will be investigated. The system parameters of individual nodes are selected in their chaotic regimes. This work concentrates on the changes in coupling strength alongside the dynamics of different node dynamics.

3.1. Synchronization on Three-Layer Hypergraph of Neuron Systems

Firstly, neuronal dynamics on the basis of the presented framework are analyzed. Many researchers have revealed that neurons interact through many-body interactions [31]. It is shown that cells contact with thousands of synapses and ensure their functions. In specific, one neuron can transfer useful information to other neurons via gap junctions and chemical synapses. Therefore, the presented hypergraph framework may suitably describe these many-body interactions for the neuron network. Furthermore, it is well known that synchronization in neural behavior is very important. Here, the individual node dynamics of each layer are chosen as identical FitzHugh–Nagumo (FHN) oscillators [32], which can be described as follows:

$$\left\{\begin{array}{l}{\displaystyle \frac{dx}{dt}}=x-{\displaystyle \frac{1}{3}}{z}^3-y+e\sin (z),\hfill \\ {\displaystyle \frac{dy}{dt}}=a_1(x+a_2-a_{3}y),\hfill \\ {\displaystyle \frac{dz}{dt}}=fx-g\sin (z).\hfill \end{array}\right.$$

(13)

where $x$ denotes the membrane potential, $y$ denotes represents the recovery variable and $z$ is the adaptive current. $a_1, a_2, a_3$ are system parameters and $e, f, g$ means the dimensionless parameters of the neural circuit formed by Josephson junctions. When the system parameters $a_1=1.3, a_2=0.5, a_3=0.01, e=0.53, f=0.08, g=0.01$, the coupled neural system gives rise to discharging behavior. The following Figure 2 depicts the dynamics of the coupled system.

For inner coupling matrices $G, H$, this paper considers the family of choices that fit the simplest form $I_{ij}\in {ℝ}^{3\times 3}, i, j=1, 2, 3$, which denotes a matrix whose element $(i, j)=1$, and all other elements are 0.

In addition, a three-layer network is constructed based on the FHN neuron model and, to investigate synchronization performance, the schematic diagram is depicted in Figure 1. That is, three layers share identical intralayer topologies, with each node in one layer connected to its corresponding replica nodes in other layers. It is assumed that each layer is a hypergraph comprising 11 nodes. Here, we select the intralayer and interlayer coupling functions $G=diag(1, 0, 0), H=diag(0, 0, 1)$, respectively. The topological structure is also described in Figure 1.

Then, the dynamical descriptions of the three layers are given by, respectively,

$$1-\mathrm{layer}:\begin{array}{cc}& \end{array}\left\{\begin{array}{l}{\dot{x}}_1=x_1-{\displaystyle \frac{1}{3}}{x}_3^3-x_2+e\sin (x_3),\hfill \\ {\dot{x}}_2=a_1(x_1+a_2-a_3{x}_2)-\alpha {\displaystyle \sum _{j=1}^N{l}_{ij}{x}_2-\beta (x_2-y_2),}\hfill \\ {\dot{x}}_3=f{x}_1-g\sin (x_3).\hfill \end{array}\right.$$

$$2-\mathrm{layer}:\begin{array}{cc}& \end{array}\left\{\begin{array}{l}{\dot{y}}_1=y_1-{\displaystyle \frac{1}{3}}{y}_3^3-y_2+e\sin (y_3),\hfill \\ {\dot{y}}_2=a_1(y_1+a_2-a_3{y}_2)-\alpha {\displaystyle \sum _{j=1}^N{l}_{ij}{y}_2-\beta (x_2-2y_2+z_2),}\hfill \\ {\dot{y}}_3=f{y}_1-g\sin (y_3).\hfill \end{array}\right.$$

$$3-\mathrm{layer}:\begin{array}{cc}& \end{array}\left\{\begin{array}{l}{\dot{z}}_1=z_1-{\displaystyle \frac{1}{3}}{z}_3^3-y_2+e\sin (z_3),\hfill \\ {\dot{z}}_2=a_1(z_1+a_2-a_3{z}_2)-\alpha {\displaystyle \sum _{j=1}^N{l}_{ij}{z}_2-\beta (y_2-z_2),}\hfill \\ {\dot{z}}_3=f{z}_1-g\sin (z_3).\hfill \end{array}\right.$$

Numerical integrations of the chaotic units are performed by means of an Runge–Kutta method, with an integration step equal to $dt=0.001$, in a window of time equal to 500.

For the calculation of the maximum Lyapunov exponent in Equation (2), we used the algorithm reported in ref. [33] with the following parameters: integration step size equal to $dt=0.001$, length of the simulation L = 500.

Firstly, we examine the evolution of the largest Lyapunov exponent for adjusting the coupling strengths to understand the synchronous behaviors in our hypergraph (2) model. Figure 3a displays the largest Lyapunov exponent for various values of intralayer and interlayer coupling strengths. Figure 3b shows the variation in the largest Lyapunov exponent as a function of the interlayer coupling $\beta$ for various values of the intralayer coupling strengths $\alpha$. The figure delineates different curves with respect to different values of intralayer coupling strengths $\alpha$. Furthermore, in Figure 4, it can be observed that the error converges to zero with the evolution of time, indicating that the multiplex hypergraph achieves a complete synchronous state.

Thereafter, when the higher-order interactions are not taken into account, Figure 5a,b illustrate the results associated with the synchronization region by adjusting the values of coupling strengths in dimensional parameter planes with reference to the largest Lyapunov exponent. As shown in Figure 5b, it can be observed that the critical values of interlayer coupling reduce as the higher-order intralayer coupling strengths increase. Hence, simulation results imply that higher-order intralayer interactions can enhance the synchronizability.

3.2. Synchronization on Three-Layer Hypergraph of Rikitake Two-Disk System

In order to further elucidate the synchronization performance of hypergraphs and demonstrate the vital role of higher-order interactions in synchronous ability, we study the multiplex hypergraph defined in Equation (2), which comprises identical node dynamics, as Rikitake two-disk dynamo systems [34]. The specific node system is given by

$$\left\{\begin{array}{l}{\displaystyle \frac{d{x}_1}{dt}}=-a{x}_1+x_2{x}_3,\hfill \\ {\displaystyle \frac{d{x}_2}{dt}}=-a{x}_2+x_1(x_3-b),\hfill \\ {\displaystyle \frac{d{x}_3}{dt}}=1-x_1{x}_2.\hfill \end{array}\right.$$

(14)

where $x_1, x_2, x_3$ are the state variables, $a$ is resistive dissipation and $b$ denotes the difference in the angular speeds of the two disks. This is the function $f(·)$ in the multiplex network Equation (2). The state of each individual node in the multiplex network is characterized by a three-dimensional vector with each component evolving according to the dynamics in Equation (13). Here, the parameter values are taken as $a=1, b=1$. Under these parameters, a two-scroll attractor exists in system (13), as shown in Figure 6a. Meanwhile, the time series of state variable $x_1$ is described as Figure 6b.

Similarly, we select the inner coupling matrices $G, H$ from the family of $I_{ij}\in {ℝ}^{3\times 3},i, j=1, 2, 3$, which shows the $(i, j)$-element of matrix is one and other elements are zero. Additionally, the interlayer topology is taken as one-to-one connections; that is, each node in one layer is connected to a counterpart node in the other layers. The coupling of these nodes occurs by the second variable. Then, the dynamical descriptions of three layers are given by, respectively,

$$1-\mathrm{layer}:\begin{array}{cc}& \end{array}\left\{\begin{array}{l}{\dot{x}}_1=-a{x}_1+x_2{x}_3,\hfill \\ {\dot{x}}_2=-a{x}_2+x_1(x_3-b)-\alpha {\displaystyle \sum _{j=1}^N{l}_{ij}{x}_2-\beta (x_2-y_2),}\hfill \\ {\dot{x}}_3=1-x_1{x}_2-\alpha .\hfill \end{array}\right.$$

$$2-\mathrm{layer}:\begin{array}{cc}& \end{array}\left\{\begin{array}{l}{\dot{y}}_1=-a{y}_1+y_2{y}_3,\hfill \\ {\dot{y}}_2=-a{y}_2+y_1(y_3-b)-\alpha {\displaystyle \sum _{j=1}^N{l}_{ij}{y}_2-\beta (x_2-2y_2+z_2),}\hfill \\ {\dot{y}}_3=1-y_1{y}_2-\alpha .\hfill \end{array}\right.$$

$$3-\mathrm{layer}:\begin{array}{cc}& \end{array}\left\{\begin{array}{l}{\dot{z}}_1=-a{z}_1+z_2{z}_3,\hfill \\ {\dot{z}}_2=-a{z}_2+z_1(z_3-b)-\alpha {\displaystyle \sum _{j=1}^N{l}_{ij}{z}_2-\beta (y_2-z_2),}\hfill \\ {\dot{z}}_3=1-z_1{z}_2-\alpha .\hfill \end{array}\right.$$

Here, the topological structure is also described in Figure 1, and the corresponding intralayer and interlayer coupling matrices are the same as those in the first example.

Figure 7a,b depict the corresponding results in terms of the largest Lyapunov exponent and synchronous error of the three-layer hypergraph with high-order intralayer coupling and one-to-one interlayer coupling schemes. From Figure 7b, it can be observed that the synchronization error tends to zero. This implies that the multiplex hypergraph can achieve synchronization. In addition, some numerical simulations are presented in accordance with the theoretical result derived from Equation (11) on the basis of the maximum Lyapunov exponent, similar to the first example of the neuronal system.

4. Discussion

In summary, this work develops a master stability function framework which captures an essential feature of multiplex hypergraphs, that the intralayer and interlayer coupling strategies are distinct. The proposed model takes into account that intralayer interactions are represented by hyperedges, while the interlayer interactions are modeled through pairwise couplings. Here, we consider that edges are undirected and define supra-Laplacian matrices for intralayer connections and interlayer connections.

Using the multiplex MSF framework requires specifying f (·), and the inter- and intralayer coupling functions. This paper considers a three-layer network of FHN oscillators and Rikitake two-disk systems. Furthermore, the framework can be extended to real systems. Results indicate that dynamical behaviors observed in the network are determined by the topological structure and coupling strengths describing complete synchronization, intralayer synchronization and interlayer synchronization. In particular, given a specified network topology and nodal dynamics, the complete synchronization is mainly determined by both inner coupling matrices. Similarly, the intralayer synchronization is mainly determined by the intralayer coupling matrix, and the interlayer synchronization by the interlayer coupling matrix. Hence, in addition to nodal dynamics, the coupling function is an essential factor in determining which kind of synchronization the network will achieve.

5. Conclusions

To conclude, we have investigated the synchronization of dynamical systems coupled in hypergraph architectures, which are composed of three-layer network topologies. Moreover, the stability conditions for the synchronous motion are derived, involving the use of generalized Laplacian matrices that characterize the effects of higher-order interactions. The presented approach relies on linear stability, thus offering a local investigation of synchronization. For fixed network nodal dynamics, the complete synchronized region is mainly determined by intralayer and interlayer coupling matrices. In addition, our findings reveal that higher-order representations significantly influence collective dynamics, highlighting the importance of selecting appropriate system representations when studying non-pairwise interaction systems.

Here we have numerically studied specific three-layer networks of two chaotic systems where different layers have the same topological structures. The proposed framework can be extended to multiplex networks with more complex intralayer topologies and various nodal dynamics. In addition, fixed-time synchronization is an important research direction in the synchronization control of dynamic systems. The system can reach a synchronized state within a predetermined finite time. Thus, this kind of synchronization type has important applications in engineering scenarios requiring fast responses, such as robot coordination and power system stability. In the future, it would be interesting to generalize our results to fixed-time synchronization for higher-order networks.