On Metzler positive systems on hypergraphs

Polynomial dynamical systems [1] are powerful tools to characterize a wide range of real-world complex systems, including biological, chemical, medical, engineering, and social systems. Chemical reactions [2, 3, 4, 5] take place in energy, environmental, biological, and many other natural systems, and the inference of the reaction networks is of great significance for the understanding and design of chemical processes in engineering and life sciences [5]. In biology, the classic Lotka-Volterra model regarding the evolution of the species’ population also belongs to the class of polynomial systems [6, 7, 8, 9], where researchers treat the species pair as a fundamental unit and only capture pairwise interactions. The pairwise interactions are associated with direct and additive effects of one species on another. In the classical model [6, 7, 8, 9], one assumes interaction is always pairwise. Since the adjacency matrix of a graph usually represents the pairwise correspondence of species, the whole model can be abstracted as a system on a graph. Matrices and graphs are very useful tools for analysis of the Lotka-Volterra model [7]. Regarding epidemics and information diffusion in social networks, networked compartment models such as SIS[10, 11, 12, 13], SIR [14, 15], SIRS [16, 17] are all polynomial systems. In the classical setting, the interaction among people (one sick person infects another person) is set to be pairwise as well. Thus, matrices and graphs also play an important role in the analysis of these systems [10, 11, 13, 12, 14, 15, 16, 17].

Pairwise interactions and their representations as a graph may not always be appropriate because they exclude the possibility of group-wise interactions i.e. higher-order interactions. In a recent study in ecology [18], HOIs (Higher-order Interactions) are introduced to represent non-additive effects among species and the existence of which is supported by empirical evidence, e.g. the one on natural plant communities [18]. The paper [19] brings HOIs into the Lotka–Volterra competition model and then demonstrates the effect of HOIs, by performing simulations with empirical data, showing that HOIs appear under almost all assumptions and help to improve the accuracy of predictions. In parallel, for compartment epidemic models, [20] argues that the social contagion processes, like opinion formation, and information diffusion, may depend more on group-wise interactions, which can be captured by HOIs. Following this idea, several novel epidemic models [20, 21, 22, 23, 24] are developed by considering HOIs. While the pairwise interactions refer to a graph, the higher-order interactions usually correspond to the higher-order network [25] (sometimes referred to as simplicial complexes and hypergraphs). However, for the analysis of the higher-order Lotka-Volterra and epidemic models, there still remains an open question on how the hypergraph will influence the system evolving on it. Most current research [20, 23, 24, 19] either relies on simulations or performs model reduction. Thus, a full analysis has not been carried out.

From a network perspective, a coupled cell system is a network of dynamical systems, where the fundamental units, “cells”, are coupled together. Such systems are widely considered in control engineering [26], and mathematics [25, 27, 28]. They can be represented schematically by a directed network whose nodes correspond to cells and whose edges represent couplings [27]. A system composed of many agents interacting with each other can be naturally considered as a coupled cell system [26]. A typical example of a coupled cell system with polynomial coupling is chemical reaction networks [26]. Coupled cell systems on hypergraphs are proposed in [25] by considering higher-order interactions and introducing the concept of hypergraphs. However, there is no general way to analyze such hypergraph systems.

It is known that the behavior of a system on a (conventional) graph is closely related to its corresponding graph, which is captured, for example, by an adjacency matrix. Among the matrices frequently used to model interesting real-life behavior, one of the most important matrices is the Metzler matrix [29], whose off-diagonal elements are all non-negative. Metzler matrices are commonly associated with cooperative systems [7], such as the cooperative Lotka-Volterra [7], and single-virus systems [10, 11, 12]. For the analysis of such systems, the Perron–Frobenius theorem [29] is a fundamental tool, which indicates that an irreducible nonnegative matrix has a positive eigenvalue corresponding to a positive eigenvector. Analogously, it is shown that a hypergraph can be represented by an adjacency tensor [25, 30]. Furthermore, the tensor version of a Perron–Frobenius theorem is also available [31, 32, 33, 34], which serves as a potential suitable tool to study systems on hypergraphs. However, to the best of our knowledge, the Perron–Frobenius theorem of a tensor has never been used to study a dynamical system.

Besides the analysis of a networked system, one uses matrix theory to design control laws. In modern control theory, the $H_{\infty}$ controller [35] and LQR-controller [36] are two typical examples. Moreover, [37] proposes a distributed control law for a class of linear positive systems with the help of the properties of a Metzler matrix. Then, a natural question arises, namely whether we can further utilize the properties of a tensor to design a control law for a class of systems on a hypergraph.

The contributions of this paper are briefly summarized as follows. Firstly, we introduce the concept of a Metzler tensor, develop the corresponding Perron–Frobenius theorem of an irreducible Metzler tensor and explain its relationship with $\mathcal{M}$-tensors studied in e.g. [38, 39, 40]. Secondly, we use the Perron–Frobenius theorem to construct a Lyapunov-like function to study a class of positive systems on hypergraphs. To illustrate our contributions, we apply our techniques to two real systems on hypergraphs, a higher-order Lotka-Volterra and a higher-order SIS model. Thirdly, we develop a feedback control strategy to stabilize the origin of a dynamical system on a hypergraph, and further study the case of a constant control input. Finally, all the theoretical results are supported by numerical examples.

Notation: $\mathbb{C}$ and $\mathbb{R}$ denote the set of complex and real numbers, respectively. $\mathbb{R}_{+}$ denotes the set of positive real numbers. For a matrix $M\in\mathbb{R}^{n\times r}$ and a vector $a\in\mathbb{R}^{n}$, $M_{ij}$ and $a_{i}$ denote the element in the $i$th row and $j$th column and the $i$th entry, respectively. Given a square matrix $M\in\mathbb{R}^{n\times n}$, $\rho(M)$ denotes the spectral radius of $M$, which is the largest absolute value of the eigenvalues of $M$. The notation $|M|$ denotes the matrix whose entry $|M|_{ij}$ is the absolute value of $M_{ij}$. For any two vectors $a,b\in\mathbb{R}^{n}$, $a>(<)b$ represents that $a_{i}>(<)b_{i}$, for all $i=1,\ldots,n$; and $a\geq(\leq)b$ means that $a_{i}\geq(\leq)b_{i}$, for all $i=1,\ldots,n$. These component-wise comparisons are also used for matrices or tensors with the same dimension. The vector $\mathbf{1}$ ($\mathbf{0}$) represents the column vector or matrix of all ones (zeros) with appropriate dimensions. The previous notations have straightforward extensions to tensors. In the following section, we introduce the preliminaries on tensors and some further notations regarding tensors. Furthermore, the notations frequently used in this paper are listed in the table 1.

A tensor $T\in\mathbb{R}^{n_{1}\times n_{2}\times\cdots\times n_{k}}$ is a multidimensional array, where the order is the number of its dimension $k$ and each dimension $n_{i}$, $i=1,\cdots,k,$ is a mode of the tensor. A tensor is cubical if every mode has the same size, that is $T\in\mathbb{R}^{n\times n\times\cdots\times n}$. We further write a $k$-th order $n$ dimensions cubical tensor as $T\in\mathbb{R}^{n\times n\times\cdots\times n}=\mathbb{R}^{[n,k]}$. A cubical tensor $T$ is called supersymmetric if $T_{j_{1}j_{2}\ldots j_{k}}$ is invariant under any permutation of the indices. For the rest of the paper, a tensor always refers to a cubical tensor unless specified otherwise.

The identity tensor $\mathcal{I}=\left(\delta_{i_{1}\cdots i_{m}}\right)$ is defined as

The diagonal entries of a tensor are the entries with all the same index, for example, $A_{iiiiii}$. All other entries are called off-diagonal entries.

We then consider the following notation: the tensor-vector product to the power $k-1$, where $k$ is the order of the tensor, $Ax^{k-1}$ and the vector-vector product to the power $k-1$: $x^{[k-1]}$ are vectors, whose $i$-th components are

From the definition, we see that the product $Ax^{k-1}$ can capture an arbitrary homogeneous polynomial vector field since each entry of the product is a homogeneous polynomial. For example,

can be written as $\dot{x}=Ax^{k-1}$, where $A_{111}=-1,A_{112}=1,A_{211}=1,A_{212}=1,A_{222}=-1$ and all other entries of $A$ are zero. In general, the choice of the tensor $A$ is not unique. If we choose $A_{111}=-1,A_{112}=0.5,A_{121}=0.5,A_{211}=1,A_{212}=1,A_{222}=-1$ and all other entries of $A$ are zero, then we get the same vector field (1).

Recall that $Ax^{{k}-1}\text{ and }x^{[{k}-1]}$ are both vectors. For a tensor, consider the following homogeneous polynomial equation:

where if there is a real number $\lambda$ and a nonzero real vector $x$ that are solutions of (2), then $\lambda$ is called an H-eigenvalue of $A$ and $x$ is the H-eigenvector of $A$ associated with $\lambda$ [41, 38, 31]. The number and distribution of H-eigenvalues of a real supersymmetric tensor are studied by [41]. It will be briefly introduced in the later section 12.1. Throughout this paper, the words eigenvalue and eigenvector as well as H-eigenvalue and H-eigenvector are used interchangeably. It is also worth mentioning that there are still some other kinds of definitions of the eigenvalues and eigenvectors of a tensor, e.g. Z-eigenvalues [31, 41] and U-eigenvalues [42]. However, in this paper, we will always refer to H-eigenvalues and H-eigenvectors as defined above.

It is straightforward to check that $Ax^{{k}-1}$ and $x^{[{k}-1]}$ satisfy the following : if $Ax^{k-1}=k_{1}x^{[k-1]}$ and $Bx^{k-1}=k_{2}x^{[k-1]}$, then $Ax^{k-1}+Bx^{k-1}=(A+B)x^{k-1}=k_{1}x^{[k-1]}+k_{2}x^{[k-1]}=(k_{1}+k_{2})x^{[k-1]}$.

The spectral radius of the tensor $A$ is defined as $\rho(A)=\max\{|\lambda|:\lambda\text{ is an eigenvalue of }A\}.$ A tensor $\mathcal{C}=\left(c_{{i_{1}}\ldots{i_{m}}}\right)$ of order $m$ dimension $n$ is called reducible if there exists a nonempty proper index subset $I\subset\{1,\ldots,n\}$ such that

If $\mathcal{C}$ is not reducible, then we call $\mathcal{C}$ irreducible. A tensor with all non-negative entries is called a non-negative tensor. Let $A$ be an $m$-order and $n$-dimensional tensor. The tensor $A$ is called an $\mathcal{M}$-tensor if there exist a nonnegative tensor $B$ and a positive real number $\eta\geq\rho(B)$ such that

If $\eta>\rho(B)$, then $A$ is called a non-singular $\mathcal{M}$-tensor. Moreover, a tensor ${A}$ is called diagonally dominant if

and is called strictly diagonally dominant if

We now recall the Perron–Frobenius Theorem for irreducible nonnegative tensors:

Furthermore, for nonnegative tensors, we have the following properties.

In the next, we illustrate some definitions regarding hypergraphs.

The concept of a hypergraph is defined in [30]. Here, we utilize a set of tensors to represent the information of the weights of a hypergraph. A weighted and directed hypergraph is a triplet $\mathscr{H}=(\mathcal{V},\mathcal{E},\tilde{A})$. The set $\mathcal{V}$ denotes a set of vertices and $\mathcal{E}=\{E_{1},E_{2},\cdots,E_{n}\}$ is the set of hyperedges. A hyperedge is an ordered pair $E=(\mathcal{X},\mathcal{Y})$ of disjoint subsets of vertices; $\mathcal{X}$ is the tail of $E$ and $\mathcal{Y}$ is the head. As a special case, a weighted and undirected hypergraph is a triplet $\mathscr{H}=(\mathcal{V},\mathcal{E},\tilde{A})$, where $\mathcal{E}$ is now a finite collection of non-empty subsets of $\mathcal{V}$. If all hyperedges of the hypergraph contain the same number of (tails, heads) nodes, then the hypergraph is uniform. For details, see [43] for undirected uniform hypergraphs and see [44] for directed uniform hypergraphs. The connection between undirected hyperedges and directed hyperedges and their physical meanings are introduced in [45]. Briefly speaking, for example, an undirected hyperedge of 3 nodes $i,j,k$ which can capture the overall interaction effect when $i,j,k$ are interacting can be decomposed by three kinds of directed hyperedges, namely, $j,k$s’ influence on $i$ ($A_{ijk},A_{ikj}$), $i,j$s’ influence on $k$ ($A_{kij},A_{kji}$), $i,k$s’ influence on $j$ ($A_{jik},A_{jki}$). For modeling purposes, regarding the nodal dynamics, one directed hyperedge usually denotes the joint influence of a group of agents on one agent. Thus, it suffices to deal with hyperedges with one single tail, and we assume that each hyperedge has only one tail but one or multiple ($\geq 1$) heads. This setting is consistent with [44] and has the advantage that a directed uniform hypergraph can be represented by an adjacency tensor. In section 4, we will show why this setting is sufficient for modeling nodal dynamics (evolution of nodes). For now, we mention that from a modeling perspective, the hyperedges with multiple tails may be helpful when one wants to capture the influence of one group of agents on another one. If the interactions of the model are supersymmetric, then an undirected hypergraph can be adopted instead of a directed one. For example, the influence of $i,j$ on $k$ is the same as $j,k$ on $i$ and $i,k$ on $j$. Generally, an undirected uniform hypergraph can be represented by a supersymmetric adjacency tensor. We now introduce the set of tensors ${\tilde{A}}=\{A_{2},A_{3},\cdots\}$ to collect the weights of all hyperedges, where $A_{2}=[A_{ij}]$ denotes the weights of all second-order hyperedges, $A_{3}=[A_{ijk}]$ denotes the weights of all third-order hyperedges, and so on. For instance, $A_{ijkl}$ denotes the weight of the hyperedge where $i$ is the tail and $j,k,l$ are the heads. For simplicity, in this paper, we also use the weight (for example, $A_{\bullet}$) to denote the corresponding hyperedge. If all hyperedges only have one tail and one head, then the network is a standard directed and weighted graph.

For convenience, throughout this paper, we define a multi-index notation $I=i_{2},\cdots,i_{k}$, where $k$ is the order of the associated tensor.

Similar to the definition of a Metzler matrix [46], we define a Metzler tensor as a tensor whose off-diagonal entries are non-negative. Then, it is straightforward to see that any Metzler tensor $A$ can be represented as $A=B-s\mathcal{I}$, where $s$ is a real scalar and $B$ is a nonnegative tensor.

Next, we show the Perron-Frobenius theorem for an irreducible Metzler tensor.

We have the following results.

In this section, we give a brief introduction to the modeling framework of nodal dynamics on a hypergraph. Proposed by [25], coupled cell systems [27] on a hypergraph are in the form of

where $F$ is a function that describes the intrinsic coupling of the node, the adjacency matrix $A_{2}$ together with the coupling functions $G_{i}$ represent the pairwise network interactions, and the coefficients of $A_{s}$ (which are adjacency tensors) and coupling function $G_{i}^{(s)}$ (with $s\geq 3$ ) are called higher-order network interactions. For example, $(A_{3})_{ijl}$ and $G_{i}^{(3)}\left(x_{k},x_{j},x_{i}\right)$ describe the joint influence of nodes $l,j$ on node $i$, which can be captured by a directed hyperedge with a single tail. Since we focus on nodal dynamics, using a hypergraph with all edges with one single tail is sufficient. Then, we further assume that the intrinsic coupling $F$ can be represented by some self-arc $(A_{s})_{i\cdots i}x_{i}^{s-1}$ and the coupling function $G_{k}^{(s)}(x_{i},x_{j},x_{l},\cdots)=x_{i}x_{j}x_{l}\cdots$. Note that this multiplicative interaction has some important physical interpretations. For example, the probability of some independent event occurring at the same time is captured by the multiplicative interaction. Then, it is straightforward to see that (3) can be written in a tensor form $\dot{x}=A_{k}x^{k-1}+A_{k-1}x^{k-2}+\cdots+A_{2}x$. It is worthwhile mentioning that any polynomial system, where the powers are positive integers, can be written in this form. Especially, any homogeneous polynomial system is in the form of $\dot{x}=Ax^{k-1}$, which is described by a uniform hypergraph.

We further emphasize that polynomial coupling functions are widely used in many real-world systems on hypergraphs. For example, an epidemic model on a hypergraph and a generalized Lotka-Volterra model adopt polynomial coupling functions. We will introduce both systems later in sections 10 and 11 in detail. We further observe that system (3) is based on a network with a coupled cell structure. That is a network that describes how nodes are influenced and interact with each other. Sometimes, this hypergraph has a direct physical meaning, such as social networks (epidemic model) [22] and ecological networks (Lotka-Volterra) [19, 47]. In addition, notice that a system may be represented by different networks. For example, for a chemical reaction system with $X+Y\rightarrow Z+R$, the evolution of $X$ is described by $\dot{x}_{1}=-k_{1}x_{1}x_{2}$. According to the chemical reaction $X+Y\rightarrow Z+R$, it is straightforward to define a hyperedge where $X,Y$ are heads and $Z,R$ are tails. However, if one uses the concept of a coupled cell network, the evolution of $X$ is only related to itself and $Y$. Thus, we just need one hyperedge with a single tail $X$ and heads $X,Y$. Since the system dynamics are just the same, these are two different representations of the given model.

In the section 6, we focus on the particular case where the tensor has a Metzler structure and will utilize Theorem 1 to analyze the stability of some positive systems on a uniform hypergraph.

In this section, we introduce the concept of hypergraph spectrum, strong connectivity, and H-eigenvector centrality. We know that a uniform hypergraph with hyperedges of $k$ nodes can be captured by an adjacency tensor of order $k$. In the literature [48, 49], all these concepts (spectrum, centrality) are for a signless hypergraph (all weights of its edges are non-negative). To the best of our knowledge, there is no literature about these concepts on a signed hypergraph with both positive and negative weights. In this paper, we deal with a tensor or a hypergraph with a Metzler structure. We know that $A=B-s\mathcal{I}$, where $s\in\mathbb{R}$, $A$ is a Metzler tensor and $B$ is non-negative. Instead of investigating $\mathscr{H}(A)$, we can study a signless hypergraph $\mathscr{H}(B)$. Later, we will show that this manipulation has very little influence on the properties of a hypergraph, i.e., $\mathscr{H}(A)$ and $\mathscr{H}(B)$ have some similar properties, where $\mathscr{H}(A)$ denotes a hypergraph with the adjacency tensor $A$.

First of all, throughout the section, to make the choice of $B,s$ unique given a Metzler tensor $A$, we choose the smallest $s$ such that $B$ is non-negative. If $A$ is already non-negative, then $A=B,s=0$.

In [49], the spectrum of a signless uniform hypergraph $\mathscr{H}(B)$ is defined as the spectrum (spectral radius) of the corresponding adjacency tensor $\rho(B)$. According to Theorem 1, the Perron-eigenvalue of $A$ is a shift on the spectrum of $B,\mathscr{H}(B)$, i.e., $\lambda(A)=\rho(B)-s$. In fact, in the next section, we will show that the stability of a Metzler dynamical system on a uniform hypergraph is indeed related to $\rho(B)$ and $s$.

Next, a $k$-uniform, $n$-node hypergraph with adjacency tensor ${{T}}$ is strongly connected if the graph induced by the $n\times n$ matrix ${M}$ obtained by summing the modes of ${{T}},M_{ij}=\sum_{j_{3}\ldots\ldots,j_{k}}{T}_{i,j,j_{3},\ldots,j_{k}}$, is strongly connected. As a consequence, a strongly connected hypergraph has an irreducible adjacency tensor [48, 50].

Eigenvector centrality is a standard network analysis tool for determining the importance of entities in a strongly connected system that is represented by a network. In [48], an H-eigenvector centrality (HEC) of a uniform signless hypergraph is proposed. Notice that although the main results in [48] are based on the setting of an unweighted hypergraph, they further claim that the unweighted setting is not necessary and that all of the proposed methods work seamlessly if the hypergraph is weighted. In the next, we give further details regarding the weighted case. Let us start with an example of a $3$-uniform hypergraph. Generally, a centrality of an $n$-node $k$-uniform hypergraph $\mathscr{H}(B)$ is defined in the following way:

1. Some function $f$ of the centrality of node $u,f\left(c_{u}\right)$, should be proportional to the weighted sum of some function $g$ of the centrality score of its neighbors, the weights are the same as the weights of an adjacency tensor. In a 3-uniform hypergraph, this means that for some positive constant $\lambda$,

2. The centrality scores should be positive, i.e., $c=(c_{1},\cdots,c_{n})^{\top}>\mathbf{0}$.

Then, we may choose $f\left(c_{u}\right)=c_{u}^{2}$ and $g\left(c_{v},c_{w}\right)=c_{v}c_{w}$, which gives $c_{u}^{2}=\frac{1}{\lambda}\sum_{u,v,w}B_{uvw}c_{v}c_{w},\Longleftrightarrow B{c}^{2}=\lambda{c}^{[2]}$. This coincides with the H-eigenproblem (2). As $c>\mathbf{0}$, the centrality vector $c$ is the Perron-H-eigenvector of an adjacency tensor $B$. The physical meaning of $f\left(c_{u}\right),g\left(c_{v},c_{w}\right)$ is that $f,g$ should be at the same degree and the contribution $g$ of the centralities of two nodes in a 3-node hyperedge is multiplicative for the third. Generally, let $\mathscr{H}$ be a strongly connected $k$-uniform hypergraph with adjacency tensor $B$. Then the H-eigenvector centrality vector for $\mathscr{H}$ is the positive real vector ${c}$ satisfying

for some eigenvalue $\lambda>0$. Clearly, $c$ is the normalized ($\|{c}\|_{1}=1$) Perron-H-eigenvector. Recall that if $A=B-s\mathcal{I}$ is some shift on the $B$, then according to Theorem 1, they have the same centrality vector.

In [48], a topology named sunflower hypergraph is studied under an unweighted setting. Here, we perform a further analysis under a weighted setting. A sunflower hypergraph has a hyperedge set $E$ with a common pairwise intersection. Formally, for any hyperedges (called petals) $E_{i},E_{j}\in E,E_{i}\cap E_{j}=\tilde{E}$. The common intersection $\tilde{E}$ is called the core. The sunflower hypergraph is similar to the star graph, see figure 1. In [48], the centrality of a $k$-uniform $r$-petal undirected sunflower hypergraph is calculated. Let the central node be $u$ and some other arbitrary node be called $v$. For an unweighted case, $\frac{c_{u}}{c_{v}}=r^{\frac{1}{k}}$. Now, let us consider an extreme case of a weighted hypergraph. We set one petal $R$ with an extremely large weight; all other weights remain $0$ or $1$ as in the unweighted case. We can observe that the value of the centrality $c$ will be largely influenced by $R$ since it dominates the H-eigenproblem (2). In the unweighted case, all the non-central nodes have the same centrality. In the extreme case we consider, the centrality of non-central nodes also has different values, mainly determined by whether the node is in the petal $R$. Thus, we can see that in an unweighted (or equally weighted case), the centrality is mainly determined by the topology of a hypergraph, while in the extremely weighted case (the weights of each edge are very different), the centrality is largely determined by the weights.

For a non-uniform hypergraph, there is very limited related literature addressing the concepts of spectrum and centrality. However, a non-uniform hypergraph can be regarded as a multi-layer network, where each layer is a uniform hypergraph [52], illustrated in Figure 2. We suggest that we utilize the spectrum and centrality of each layer to analyze the whole hypergraph. We will use this idea later to study a system on a non-uniform hypergraph. Alternatively, we can also project a hypergraph into a graph, for graph projection see [52] for details. It is important to notice that graph projection brings a higher-order hypergraph into a graph (of order two). This will lead to a loss of information. Indeed, by utilizing the graph projection, [52] is only able to capture the local stability of a system on a hypergraph. Our work will fill this gap and talk about the global one.

Here, we consider a positive Metzler-tensor-based system on a uniform hypergraph of $n$ nodes. The dynamical system is given by

where $A$ is an irreducible Metzler order $k$ dimension $n$ tensor, and $x\in\mathbb{R}^{n}$ is the state variable. Component-wise, (6) reads as

Firstly, we show that (6) is a positive system.

Now, we are ready to discuss the stability of the origin. Recall that a Metzler tensor can be written as $A=B-s\mathcal{I}$, with $B$ a nonnegative tensor.

In this section, we develop feedback control strategies for a positive Metzler-tensor-based system on a uniform hypergraph. The general control goal is to stabilize the origin. Consider system (6) with some control inputs, namely

Firstly, we design a feedback control as $u_{i}={q}x_{i}^{k-1}$ ($u={q}x^{[k-1]}$). Then, the closed loop system reads as $\dot{x}=Ax^{k-1}+{q}x^{[k-1]}=(A+{q}\mathcal{I})x^{k-1}$. According to Theorems 1 and 2, $\lambda(A+{q}\mathcal{I})=\lambda(A)+{q}$ and if $\lambda(A)+{q}<0$, then we drive the solution of (12) asymptotically to the origin. From a perspective of distributed control of a networked system, this control law only requires self-information, which is truly an advantage. Furthermore, the controller does not change the centrality measure of the nodes. However, the cost to stabilize the system, $q$ (the scalar one needs to manipulate), may be large.

Next, let us consider a more general feedback controller $u={D}x^{k-1}$ with ${D}$ a non-positive tensor. Then, the closed-loop system reads as $\dot{x}=Ax^{k-1}+{D}x^{k-1}=(A+{D})x^{k-1}$. We know that a Metzler tensor can be written as $A={B}+s\mathcal{I}$, where ${B}$ is a nonnegative tensor and we assume $s<0$ such that $A$ may have negative eigenvalue. Therefore, $A+{D}={B}+{D}+s\mathcal{I}$. Now, suppose $|{D}|\leq{B}$ such that ${D}+{B}$ is still a non-negative tensor and ${D}+{B}\leq{B}$. Thus, we have $\lambda(A+{D})=\rho({D}+{B})+s\mathcal{I}\leq\lambda(A)=\rho(A)+s\mathcal{I}$ according to Lemma 2. We can see that the Perron-eigenvalue of the closed loop is actually smaller than that of the open loop. Since ${D}+{B}$ is still a non-negative tensor, $\lambda({D}+{B})\geq 0$. Then, according to Lemma 2, if we increase the entries of $|{D}|$, then $\rho({D}+{B})$ is decreasing but greater than zero. Thus, with a large enough $|{D}|$, we can have $\lambda({D}+{B})+s<0$ and thus $\lambda(A+{D})<0$, indicating that the origin of the closed loop (12) is globally asymptotically stable. Notice, in contrast to the previous case, that using the controller potentially changes the centrality measure of the nodes.

Finally, we consider the feedback controller $u={D}x^{p-1}$, and $p\neq k$. This leads to the closed loop system $\dot{x}=Ax^{k-1}+{D}x^{p-1}$. The system is now based on a general hypergraph consisting of one layer of a $k$-th order uniform hypergraph and another layer of a $p$-th order uniform hypergraph. Since the closed-loop system is no longer a uniform hypergraph, our analytical results may no longer be applicable but the general problem remains an interesting open problem for future work. The main challenge is how to deal with the combination of tensors with different orders and usually, these two tensors could have different eigenvalues and eigenvectors. However, in the following, we discuss some special cases when the problem can be transformed into a problem on a uniform hypergraph.

In this section, we consider system (12) with a constant control input $u=b$ with $b$ a positive vector. This leads to the affine system

This is the simplest in-homogeneous polynomial case. It is easy to check that (13) is still a positive system since $b$ is a positive vector. If $b$ is no longer a positive vector, the system may not be a positive system. Recall that if a Metzler tensor $A$ of order $k$ satisfies $\lambda(A)<0$, then $-A$ is a non-singular $\mathcal{M}$-tensor.

Let $\dot{x}=0$ in (13), then we get that $-Ax^{k-1}=b$. Thus, Lemma 8.1 directly indicates that the system (13) has a unique positive equilibrium if $-A$ is a non-singular $\mathcal{M}$-tensor.