A Nonlinear Perron-Frobenius Approach for Stability and Consensus of Discrete-Time Multi-Agent Systems

TL;DR

This paper introduces a nonlinear Perron-Frobenius method to analyze stability and consensus in discrete-time multi-agent systems with heterogeneous nonlinear interactions, extending linear results to a broader nonlinear class.

Contribution

It generalizes linear stability and consensus results to nonlinear, sub-homogeneous, order-preserving maps using Perron-Frobenius theory, without relying on Lyapunov functions.

Findings

Established convergence conditions for nonlinear multi-agent systems.

Demonstrated the method's effectiveness through examples.

Extended linear results to nonlinear, heterogeneous interactions.

Abstract

In this paper we propose a novel method to establish stability and, in addition, convergence to a consensus state for a class of discrete-time Multi-Agent System (MAS) evolving according to nonlinear heterogeneous local interaction rules which is not based on Lyapunov function arguments. In particular, we focus on a class of discrete-time MASs whose global dynamics can be represented by sub-homogeneous and order-preserving nonlinear maps. This paper directly generalizes results for sub-homogeneous and order-preserving linear maps which are shown to be the counterpart to stochastic matrices thanks to nonlinear Perron-Frobenius theory. We provide sufficient conditions on the structure of local interaction rules among agents to establish convergence to a fixed point and study the consensus problem in this generalized framework as a particular case. Examples to show the effectiveness of the method are provided to corroborate the theoretical analysis.

Equations

$$\begin{array}[]{c}x_{1}(k+1)=f_{1}\left(x_{1}(k),\ldots,x_{n}(k)\right)\\ \vdots\\ x_{n}(k+1)=f_{n}\left(x_{1}(k),\ldots,x_{n}(k)\right)\end{array}\quad,$$

$$x(k+1)=f(x(k))$$

$$\frac{\partial f_i(x)}{\partial x_j}\ne 0x\in {\mathbb{R}}_{\ge 0}^n\setminus S,$$

$$u\le v$$

$$u⪇v$$

$$u<v$$

$$P_I=\{x\in {\mathbb{R}}_{\ge 0}^n|x_i>0,\forall i\in I\text{ and }{x}_i=0\text{ otherwise}\},$$

$$x\le y\iff f(x)\le f(y).$$

$$x⪇y\iff f(x)⪇f(y).$$

$$x⪇y\iff f(x)<f(y).$$

$$\alpha f(x)\le f(\alpha x)$$

$$\omega (x)=\bigcap _{k\ge 0}cl\left(\{f^m(x):m\ge k\}\right),$$

$$\dot{x}=f(x)$$

$$\partial f_i/\partial x_j\ge 0\text{ for }i\ne j.$$

$$x(k+1)=f(x(k))$$

$$\partial f_i/\partial x_i>0\text{ and }\partial f_i/\partial x_j\ge 0\text{ for }i\ne j.$$

$$\frac{\partial f_i(x)}{\partial x_j}=\underset{\epsilon \to 0}{\lim }\frac{f_i(x+\epsilon e_j)-f_i(x)}{\epsilon }\ge 0,$$

$$\frac{\partial f_i(x)}{\partial x_i}=\underset{\epsilon \to 0}{\lim }\frac{f_i(x+\epsilon e_i)-f_i(x)}{\epsilon }>0,$$

$$M(x/y)=\inf \{\alpha \ge 0:y\le \alpha x\},$$

$$d_T(x,y)=\log (\max \{M(x/y),M(y/x)\})$$

$$d(f(x),f(y))\le d(x,y)$$

$${\Vert x\Vert }_{\infty }=\underset{i}{\max }\Vert x_i\Vert ,$$

$$f_i^{\ast }(z_1)=f_i(z_1,z_2),z_2=0,$$

$$g_i^{\ast }(z_1)\le g_i(z_1,z_2).$$

$$f_i^{\ast }(z_1)\le f_i(z_1,z_2).$$

$$J_f(c1)1=1\forall c\in {\mathbb{R}}_{+},$$

$$J_f(x)v=\underset{h\to 0}{\lim }\frac{f(x+hv)-f(x)}{h}.$$

$$J_f(c1)1$$

$$=\underset{h\to 0}{\lim }\frac{\cancel{c1}+h1-\cancel{c1}}{h}=1,$$

$$F_f\supseteq \{c1,c\in {\mathbb{R}}_{\ge 0}\}.$$

$$\bar{x}\ne c1,\forall c\in {\mathbb{R}}_{\ge 0}$$

$$c_1$$

Full text

††thanks: This work was supported in part by the Italian Ministry of Research and Education (MIUR) with the grant “CoNetDomeSys”, code RBSI14OF6H, under call SIR 2014 and by Region Sardinia (RAS) with project MOSIMA, RASSR05871, FSC 2014-2020, Annualita’ 2017, Area Tematica 3, Linea d’Azione 3.1.

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1 Introduction

The study of complex systems where local interactions between individuals give rise to a global collective behavior has aroused much interest in the control community. Such complex systems are often called Multi-Agent Systems (MAS), consisting of multiple interacting agents with mutual interactions among them. A topic that captured the attention of many researchers is the consensus problem [19], where the objective is to design local interaction rules among agents such that their state variables converge to the same value, the so called agreement or consensus state.

A MAS can be modeled as a dynamical system. In the discrete time linear case, classical Perron-Frobenius Theory is crucial in the convergence analysis. Indeed, in one of the most popular works in this topic ([10]), the authors established criteria for convergence to a consensus state for MAS whose global dynamics can be represented by linear time-varying systems with non-negative row-stochastic state transition matrices, which are object of study of the classical Perron-Frobenius Theory. The notable aspect of this work was to exploit such theory and graph theory instead of Lyapunov theory, allowing to study systems for which finding a common Lyapunov function to establish convergence is difficult or even impossible. Particularly, as it later became clear by the work in [20], it allowed studying switched linear systems for which there does not exist a common quadratic Lyapunov function.

Along this line of thought, in this paper we aim to exploit nonlinear Perron-Frobenius theory [14], a generalization of non-negative matrix theory, to address nonlinear interactions in MASs without Lyapunov based arguments. It follows that a MAS modeled by a non-negative row-stochastic matrix is a particular case of the proposed general theory. The literature on nonlinear consensus problems is vast. It is mostly composed by particular nonlinear consensus protocols which offer advantages such as finite-time convergence [7, 6], resilience to non-uniform time-delays [25] and many more. These protocols are usually proved to converge to the consensus state via ad hoc Lyapunov functions. Most of approaches which aim to establish convergence to consensus for some class of nonlinear MAS falls in the general convexity theory of [17], i.e., each agent’s next state is strictly inside the convex hull spanned by the state value of its neighbors. We mention the work in [27], which is the continuous-time counterpart to the result of Moreau in [17], where the authors identify a class of non-linear interactions denoted as sub-tangent and establish necessary and sufficient conditions on the network topology for convergence to consensus.

Our approach sharply differs from the previous literature. We identify a class of functions (which comprises also non-negative row-stochastic matrices) which we prove to have a special convergence properties in the positive orthant $\mathbb{R}^{n}_{\geq 0}$. In particular, we take inspiration from nonlinear Perron-Frobenius theory and considered order-preserving and sub-homogeneous nonlinear maps. Furthermore, the approach presented in this paper differs significantly from the preliminary results presented in [5] in both the statement of the theorems, lemmas, and their proof.

The main contribution of this paper is threefold. First, we provide sufficient conditions for stability of a class of nonlinear discrete-time systems represented by positive, sub-homogeneous, type-K order preserving maps. Second, we propose a sufficient condition on the structure of heterogeneous local interaction rules among agents which guarantees that the global model of the MAS falls into the considered class of nonlinear discrete-time systems. Third, we propose a sufficient condition which links the topology of the network and the structure of the local interaction rules to guarantee the achievement of a consensus state, i.e., the network state in which all state variables have the same value. Our results are a generalization to nonlinear discrete-time system of non-negative matrix theory applied to multi-agent systems, in so doing our results do not exploit Lyapunov function arguments.

This paper is organized as follows. In Section 2 we present our notation and background material on multi-agent systems, order-preserving and sub-homogenoeus maps and recall the concept of periodic fixed-points. In Section 3 we state our main results which consists in the statement of three main theorems. In Section 4 we discuss the proof of our results, we first discuss and list the required technical lemmas and then present the proof of each theorem in separate subsections. In Section 5 we present examples of application of our theoretical results. Finally, in Section 6 we give our concluding remarks.

2 Background

In this work we propose novel tools to perform stability analysis (consensus as a special case) of MASs whose state update is represented by positive, order-preserving and sub-homogeneous maps. In this section we define a model of autonomous nonlinear MASs in discrete-time, its associated graph and present the above mentioned properties which define the class of MAS under study.

2.1 Multi-agent systems

We consider a MAS composed by a set of agents $V=\left\{1,\ldots,n\right\}$, which are modeled as autonomous discrete-time dynamical systems with scalar state in $\mathbb{R}_{\geq 0}=\{x\in\mathbb{R}:x\geq 0\}$. Agents are interconnected and update their state as follows

[TABLE]

where $k\in\{0,1,2,\ldots\}$ is a discrete-time index. Introducing the aggregate state $x=\left[x_{1},\ldots,x_{n}\right]^{T}\in\mathbb{R}^{n}$, system (1) can be written as

[TABLE]

where $f:\mathbb{R}_{\geq 0}^{n}\rightarrow\mathbb{R}_{\geq 0}^{n}$ is differentiable. Hence, in this work we consider positive systems [22]. Positivity is a term with different meanings in different contexts, here by positive system we denote a system (and the associated map) with state that evolves in $\mathbb{R}_{\geq 0}^{n}$.

Definition 1** (Positive systems and maps).**

System (2) is called positive if $f$ maps non-negative vectors into non-negative vectors, i.e., $f:\mathbb{R}_{\geq 0}^{n}\rightarrow\mathbb{R}_{\geq 0}^{n}$. Correspondingly, map $f$ is said to be positive. $\blacksquare$

We now associate to the map $f$ a graph $\mathcal{G}(f)$ which captures the pattern of interactions among agents and denote it as inference graph [16].

Let $\mathcal{G}=(\mathcal{V},\mathcal{E})$ be a graph where $\mathcal{V}=\left\{1,\ldots,n\right\}$ is the set of nodes representing the agents and $\mathcal{E}\subseteq\mathcal{V}\times\mathcal{V}$ is a set of directed edges. A directed edge $(i,j)\in\mathcal{E}$ exists if node $i$ sends information to node $j$. To each agent $i$ is associated a set of nodes called neighbors of agent $i$ defined as $\mathcal{N}_{i}=\left\{j\in\mathcal{V}:(j,i)\in\mathcal{E}\right\}$. A directed path between two nodes $p$ and $q$ in a graph is a finite sequence of $m$ edges $e_{k}=(i_{k},j_{k})\in E$ that joins node $p$ to node $q$, i.e., $i_{1}=p$, $j_{m}=q$ and $j_{k}=i_{k+1}$ for $k=1,\ldots,m-1$. A node $j$ is said to be reachable from node $i$ if there exists a directed path from node $i$ to node $j$. A node is said to be globally reachable if it is reachable from all nodes $i\in\mathcal{V}$.

Definition 2** (Inference graph).**

Given a map $f$ its inference graph $\mathcal{G}(f)=(\mathcal{V},\mathcal{E})$ is defined by a set of nodes $\mathcal{V}$ and a set of directed edges $\mathcal{E}\subseteq\left\{\mathcal{V}\times\mathcal{V}\right\}$. An edge $(i,j)\in\mathcal{E}$ from node $i$ to node $j$ exists if

[TABLE]

where $S$ is a set of measure zero in $\mathbb{R}^{n}$. $\blacksquare$

2.2 Order-preserving maps

The set of $\mathbb{R}^{n}_{\geq 0}$ is a partially ordered set with respect to the natural order relation $\leq$. For $u,v\in\mathbb{R}^{n}_{\geq 0}$, we can write the partial ordering relations as follows

[TABLE]

The partial ordering $\leq$ yields an equivalence relation $\sim$ on $\mathbb{R}^{n}_{\geq 0}$, i.e., $x$ is equivalent to $y$ ($x\sim y$) if there exist $\alpha,\beta\geq 0$ such that $x\leq\alpha y$ and $y\leq\beta x$. The equivalence classes are called parts of the cone of non-negative real vectors and the set of all parts is denoted by $\mathcal{P}$. It can be shown (see [1]) that the cone $\mathbb{R}_{\geq 0}^{n}$ has exactly $2^{n}$ parts, which are given by

[TABLE]

with $I\subseteq\{1,\ldots,n\}$. We define a partial ordering on the set of parts $\mathcal{P}$ given by $P_{I_{1}}\preceq P_{I_{2}}$ if $I_{1}\subseteq I_{2}$.

Maps which preserve such a vector order are said to be order-preserving. Next, we provide a formal definition of three kinds of order-preserving maps present in the current literature.

Definition 3** (Order-preservation).**

A positive map $f$ is said to be

• •

Order-preserving, if $\forall x,y\in\mathbb{R}^{n}_{\geq 0}$ it holds

[TABLE]

• •

Strictly order-preserving, if if $\forall x,y\in\mathbb{R}^{n}_{\geq 0}$ it holds

[TABLE]

• •

Strongly order-preserving, if $\forall x,y\in\mathbb{R}^{n}_{\geq 0}$ it holds

[TABLE]

The next remark is in order to clarify the context of the contribution of this paper.

Remark 4**.**

For linear maps, order-preservation and positivity are equivalent properties and correspond to non-negative matrices. Since this is not the case for general nonlinear maps, in this work we consider positive nonlinear maps which are also order-preserving. $\blacksquare$

Now, we are ready to introduce the definition of type-K order-preserving maps, shown next, which plays a pivotal role in the characterization of the class of nonlinear systems in which we are interested and which will be discussed at length in the proofs of our results.

Definition 5** (Type-K Order-preservation).**

A positive map $f$ is said to be type-K order-preserving if for any $x,y\in\mathbb{R}_{\geq 0}^{n}$ and $x\lneq y$ it holds

1. $(i)$

$x_{i}=y_{i}\Rightarrow f_{i}(x)\leq f_{i}(y)$* , * 2. $(ii)$

$x_{i}<y_{i}\Rightarrow f_{i}(x)<f_{i}(y)$* ,*

for all $i=1,\ldots,n$, where $f_{i}$ is the $i$-th component of $f$. $\blacksquare$

As it will be shown later, such a property is sufficient but not necessary for classical order-preservation. However, since it is easily identifiable from the sign structure of the Jacobian matrix, it allows to easily establish order-preservation of a given function. Furthermore, it constraints the behavior of the system, preventing the system from evolving with periodic trajectories and thus helping in proving convergence to a steady state.

2.3 Sub-homogeneous maps

Order-preserving dynamical systems and nonlinear Perron-Frobenius theory are closely related. In the theory of order-preserving dynamical systems, the emphasis is placed on strong order-preservation. For discrete-time strongly order-preserving dynamical systems one has generic convergence to periodic trajectories under appropriate conditions [21]. An extensive overview of these results was given by Hirsch and Smith [9]. On the other hand, in nonlinear Perron-Frobenius theory one usually considers discrete-time dynamical systems that need not be strongly order-preserving, but satisfy an additional concave assumption and obtain similar results regarding periodic trajectories [13]. The concave assumption of interest in this paper is sub-homogeneity.

Definition 6** (Sub-homogeneity).**

A positive map is said to be sub-homogeneous if

[TABLE]

for all $x\in\mathbb{R}_{\geq 0}^{n}$ and $\alpha\in[0,1]$. $\blacksquare$

Order-preserving and sub-homogeneous maps arise in a variety of applications, including optimal control and game theory [2], mathematical biology [23], analysis of discrete event systems [8] and so on.

2.4 Periodic points

Concluding this section, we recall some basic concepts on periodic points which are instrumental to state our main results.

Consider the state trajectory of the system in eq. (2). A point $x\in\mathbb{R}^{n}$ is called a periodic point of map $f$ if there exist an integer $p\geq 1$ such that $f^{p}(x)=x$. The minimal such $p\geq 1$ is called the period of $x$ under $f$.

If $f(x)=x$, we call $x$ a fixed point of $f$. A fixed point is a periodic point with period $p=1$. Fixed points of a map are equilibrium points for a dynamical system. We denote $F_{f}=\{x\in X:f(x)=x\}$ the set of all fixed points of map $f$.

The trajectory of the system in eq. (2) with initial state $x$ is given by $\mathcal{T}(x,f)=\{f^{k}(x):k\in\mathbb{Z}\}$. If $f$ is clear from the context, we simply write $\mathcal{T}(x)$ to denote its trajectory, where $x$ is the initial state. If $x$ is a periodic point, we say that $\mathcal{T}(x)$ is a periodic trajectory. We denote the limit set of a point $x$ of map $f$ as $\omega(x,f)$ (or simply $\omega(x)$ if $f$ is clear from the context), which is defined as

[TABLE]

with $cl(\cdot)$ denoting the closure of a set, i.e., the set together with all of its limit points. If $x$ is a fixed point it follows that the set $\omega(x)$ is a singleton, i.e., a set containing a single point.

3 Main results

In this section we state and clarify the main results of this paper, while the following sections are dedicated to their proof.

For positive maps which are also order-preserving and sub-homogeneous, existing results (see next section for insights) do not provide any condition to ensure convergence to a fixed point, but only to periodic points [13] when the initial state is strictly positive, i.e., $x\in\mathbb{R}^{n}_{+}$. Furthermore, to the best of our knowledge, no result provides any information about trajectories whose initial state lies in the boundary of $\mathbb{R}_{\geq 0}^{n}$.

Our aim is thus to fill this void considering the previously defined class of order-preserving maps, called type-K order-preserving, for which we prove convergence to a fixed point for any initial state $x\in\mathbb{R}_{\geq 0}^{n}$ (and not only for $x\in\mathbb{R}^{n}_{+}$). This result is given in next theorem.

Theorem 7** (Convergence).**

Let a positive map $f$ be sub-homogeneous and type-K order-preserving. If $f$ has at least one positive fixed point in $\mathbb{R}^{n}_{+}$ then all periodic points are fixed points, i.e., the set $\omega(x)$ is a singleton and $\forall x\in\mathbb{R}_{\geq 0}^{n}:\lim_{k\rightarrow\infty}f^{k}(x)=\bar{x}$ where $\bar{x}$ is a fixed point of $f$. $\blacksquare$

Using this technical result, another of our contributions is a sufficient condition on the heterogeneous local interaction rules under which a MAS is stable, i.e., its state converges to a fixed point. Fixed points are synonymous for equilibrium points, while in the literature the term fixed points is widely used in the context of iterated maps, the term equilibrium point is usually preferred in the context of discrete-time dynamical systems. This result is given in Theorem 8, whose statement is shown next.

Theorem 8** (Stability).**

Consider a MAS as in (2) with at least one positive equilibrium point. If the set of differentiable local interaction rules $f_{i}$, with $i=1,\ldots,n$, satisfies the next conditions:

1. $(i)$

$f_{i}(x)\in\mathbb{R}_{\geq 0}$* for all $x\in\mathbb{R}_{\geq 0}^{n}$;* 2. $(ii)$

$\partial f_{i}/\partial x_{i}>0$* and $\partial f_{i}/\partial x_{j}\geq 0$ for $i\neq j$;* 3. $(iii)$

$\alpha f_{i}(x)\leq f_{i}(\alpha x)$* for all $\alpha\in[0,1]$ and $x\in K$;*

then the MAS converges to one of its equilibrium points for any positive initial state $x(0)\in\mathbb{R}_{\geq 0}^{n}$. $\blacksquare$

As a special case, we also study the consensus problem for the considered class of MAS. We propose a sufficient condition based on the result in Theorem 8 so that, for any initial state in $\mathbb{R}^{n}_{+}$, the MAS asymptotically reaches the consensus state, i.e., all state variable converge to same value. The proposed sufficient condition is graph theoretical and based on the inference graph $\mathcal{G}(f)$. The condition is satisfied if there exists a globally reachable node in graph $\mathcal{G}(f)$ and the consensus state is a fixed point for the considered MAS. This result is given in the next theorem.

Theorem 9** (Consensus).**

Consider a MAS as in (2). If the set of differentiable local interaction rules $f_{i}$, with $i=1,\ldots,n$, satisfies the next conditions:

1. $(i)$

$f_{i}(x)\in\mathbb{R}_{\geq 0}$* for all $x\in\mathbb{R}_{\geq 0}^{n}$;* 2. $(ii)$

$\partial f_{i}/\partial x_{i}>0$* and $\partial f_{i}/\partial x_{j}\geq 0$ for $i\neq j$;* 3. $(iii)$

$\alpha f_{i}(x)\leq f_{i}(\alpha x)$* for all $\alpha\in[0,1]$ and $x\in\mathbb{R}^{n}_{\geq 0}$;* 4. $(iv)$

$f_{i}(x)=x_{i}$* if $x_{i}=x_{j}$ for all $j\in\mathcal{N}_{i}^{in}$;* 5. $(v)$

Inference graph $\mathcal{G}(f)$ has a globally reachable node;

then, the MAS converges asymptotically to a consensus state for any initial state $x(0)\in\mathbb{R}_{\geq 0}^{n}$. $\blacksquare$

In the remainder of the paper, we discuss the proof of our main results in Theorem 7, 8 and 9.

4 Proof of main results

We begin by clarifying the relationships among the different kinds of order-preservation.

Remark 10**.**

Strong order-preservation $\Rightarrow$ Type-K order-preservation $\Rightarrow$ strict order-preservation $\Rightarrow$ order-preservation. $\blacksquare$

Every converse relationship in Remark 10 does not hold. Given $x,y\in\mathbb{R}$, let $f:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$, we have the following counter-examples:

• •

$f(x,y)=[1,1]^{T}$ is order-preserving but not strictly;

• •

$f(x,y)=[y,x]^{T}$ is strictly order-preserving but not type-K;

• •

$f(x,y)=[\sqrt{x}+y,y]^{T}$ is type-K order-preserving but not strongly.

Usually, to verify order-preservation is not an easy task. For differentiable continuous-time systems $\dot{x}=f(x)$ a sufficient condition to ensure order-preservation is given by Kamke [24, 12]. The Kamke condition usually exploited in the analysis of continuous time systems is shown next.

Lemma 11** (Kamke Condition).**

[24, 12]**  The map $f$ of a continuous-time system

[TABLE]

is order-preserving if its Jacobian matrix is Metzler, i.e.,

[TABLE]

As a counterpart to Lemma 11, for discrete-time systems we propose a sufficient condition to ensure type-K order-preservation of a map $f$, instrumental to the analysis of discrete-time systems, which we denote Kamke-like condition.

Proposition 12** (Kamke-like condition).**

The map $f$ of a discrete-time system

[TABLE]

is type-K order-preserving if its Jacobian matrix is Metzler with strictly positive diagonal elements, i.e., if

[TABLE]

{pf*}

Proof. Let $x\in\mathbb{R}^{n}$ and, without lack of generality, $y=x+\varepsilon e_{j}$ where $\varepsilon>0$ and $e_{j}$ denotes a canonical vector with all zero values but the $j$-th which is $1$, thus $x\lneq y$. If (3) holds, then

If $i\neq j$ then $y_{i}=x_{i}+\varepsilon 0=x_{i}$ and

[TABLE]

which implies that $f_{i}(x)\leq f_{i}(x+\varepsilon e_{j})=f_{i}(y)$, i.e., condition $(i)$ of Definition 5. 2. 2.

If $i=j$ then $y_{i}=x_{i}+\varepsilon 1>x_{i}$ and

[TABLE]

which implies that $f_{i}(x)<f_{i}(x+\varepsilon e_{i})=f_{i}(y)$, i.e., condition $(ii)$ of Definition 5.

Since $1)\Rightarrow\eqref{condition}$ and $2)\Rightarrow\eqref{condition}$, the proof is complete. $\square$

Having clarified how to verify the type-K order-preserving property for a discrete-time system, we move on in the next subsection to discuss a significant property of order-preserving maps which are also sub-homogeneous, i.e., non-expansivness with respect to the so-called Thompson’s metric.

4.1 Non-expansive maps

Dynamical systems defined by order-preserving and sub-homogeneous maps are non-expansive under the Thompson’s metric. Here, we introduce the concepts of non-expansiveness and Thompson’s metric and give a few useful lemmas.

Definition 13** (Thompson’s metric [26]).**

For $x,y\in\mathbb{R}^{n}_{\geq 0}$ define

[TABLE]

with $M(x/y)=\infty$ if the set is empty. By mean of function $M(y/x)$, Thompson’s metric $d_{T}:\mathbb{R}^{n}\times\mathbb{R}^{n}\rightarrow[0,\infty]$ is defined for all $(x,y)\in(\mathbb{R}^{n}\times\mathbb{R}^{n})\setminus(0,0)$ as follows

[TABLE]

*with $d_{T}(0,0)=0$. $\blacksquare$ *

Definition 14** (Non-expansiveness).**

A positive map $f$ is called non-expansive with respect to a metric $d:\mathbb{R}^{n}\times\mathbb{R}^{n}\rightarrow\mathbb{R}_{\geq 0}$, if

[TABLE]

for all $x,y\in\mathbb{R}_{\geq 0}^{n}$. $\blacksquare$

The next result taken from [1] but stated in reference to positive cones $K$. In this paper we always consider as a particular case the cone of non-negative vectors, i.e., $K=\mathbb{R}^{n}_{\geq 0}$, which is a solid, closed and convex cone.

Lemma 15**.**

[1]**  Let $K$ be a solid closed convex cone111A set $K\in\mathbb{R}^{n}$ is called a convex cone if $\alpha K\subseteq K$ for all $\alpha\geq 0$ and $K\cap(-K)=\{0\}$. The convex cone $K$ is closed if it is a closed set in $\mathbb{R}^{n}$ and it is solid if it has a non-empty interior. in $\mathbb{R}^{n}$. If $f:K\rightarrow K$ is an order-preserving map, then it is sub-homogeneous if and only if it is non-expansive with respect to Thompson’s metric $d_{T}$. $\blacksquare$

Such a property allows one to prove detailed results concerning the behavior of dynamical systems, see [1, 15, 18] and also Chapter 8 in [14] and reference therein.

Thompson’s metric $d_{T}$ and the sup-norm $\left\lVert{\cdot}\right\rVert_{\infty}$ defined by

[TABLE]

are closely related thanks to the following lemma.

Lemma 16**.**

[26]**  The coordinate-wise logarithmic function $L:\mathbb{R}^{n}_{+}\rightarrow\mathbb{R}^{n}$ is an isometry from $(\mathbb{R}^{n}_{+},d_{T})$ to $(\mathbb{R}^{n},\left\lVert\cdot\right\rVert_{\infty})$, with $\mathbb{R}_{+}=\{x\in\mathbb{R}:x>0\}$. $\blacksquare$

By Lemma 16, if a positive map $f$ can be restricted to $\mathbb{R}_{+}^{n}=int(\mathbb{R}_{\geq 0}^{n})$ and if it is order preserving and sub-homogenous, then $g:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ given by $g=log\circ f\circ exp$, is a sup-norm non-expansive map that has the same dynamical properties as $f$. The dynamics of sup-norm non-expansive maps is widely known. In fact, there exists the following result, which is the simplified version of Theorem 4.2.1 in [14].

Lemma 17**.**

[26]**  If $f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ is a sup-norm non-expansive then only one of the next cases can occur:

1. $(i)$

$\forall x\in\mathbb{R}^{n}$* trajectories $\mathcal{T}(x)$ are unbounded;* 2. $(ii)$

$\forall x\in\mathbb{R}^{n}$* trajectories $\mathcal{T}(x)$ are bounded. $\blacksquare$*

4.2 Proof of Theorem 7

As pointed out in the previous section, for positive maps $f$ which are also order-preserving and sub-homogeneous it is possible to establish the boundedness of any trajectory with initial states $x(0)\in\mathbb{R}_{+}^{n}$ and entirely enclosed in $\mathbb{R}_{+}^{n}$. On the contrary, there are still no results for trajectories with initial state $x(0)$ in the boundary of $\mathbb{R}_{\geq 0}^{n}$.

When $f$ satisfies the additional property of type-K order preservation, Theorem 7 gives a sufficient condition for the boundedness of any trajectory with initial state $x(0)\in\mathbb{R}_{\geq 0}^{n}$ and convergence of such trajectories to a fixed point of map $f$.

The proof of Theorem 7 requires the preliminary discussion of several lemmas which are needed to prove that type-K order-preservation as opposed to simple order-preservation, together with other properties, is sufficient to extend result to trajectories starting at any point in $\mathbb{R}^{n}_{\geq 0}$ and exclude the existence of periodic trajectories.

Lemma 18**.**

Let a positive map $f$ be type-K order-preserving. For all $x\in\mathbb{R}_{\geq 0}^{n}$ it holds that $f_{i}^{k}(x)>0$ for all $i$ such that $x_{i}>0$ and $k\geq 1$.

{pf*}

Proof. For any $x\in\mathbb{R}_{\geq 0}^{n}$ let $I(x)\subset\{1,\ldots,n\}$ be such that $x_{i}>0$ for $i\in I(x)$ and $x_{i}=0$ otherwise. Since $\mathbf{0}\leq x$, by type-K order-preservation of $f$ follows $f(\mathbf{0})\leq f(x)$. More precisely it holds $f_{i}(x)>f_{i}(\mathbf{0})\geq 0$ for $i\in I(x)$ and $f_{i}(x)\geq f_{i}(\mathbf{0})\geq 0$ otherwise, implying $I(x)\subseteq I(f(x))$. By induction, $I(x)\subseteq I(f^{k}(x))$, i.e., $f^{k}_{i}(x)>0$ for all $i\in I(x)$, completing the proof. $\square$

Lemma 19**.**

Let a positive map $f$ be sub-homogeneous and type-K order-preserving. For all $x\in\mathbb{R}_{\geq 0}^{n}$ there exists a part $P\in\mathcal{P}(\mathbb{R}_{\geq 0}^{n})$ and an integer $k_{0}\in\mathbb{Z}$ such that $f^{k}(x)\in P$ for all $k\geq k_{0}$.

{pf*}

Proof. Since by Lemma 15 $f$ is non-expansive under the Thompson’s metric $d_{T}$, then $x\sim y$ implies $f(x)\sim f(y)$. This can be easily proved by noticing that $d_{T}(f(x),f(y))\leq d_{T}(x,y)<\infty$ since $x\sim y$. This means that $f$ maps parts into parts, i.e., for all $x\in\mathbb{R}_{\geq 0}^{n}$ and $x^{\prime}\in[x]=P_{I_{0}}$ it holds $f(x^{\prime})\in[f(x)]=P_{I_{1}}$. By Lemma 18 it follows $P_{I_{0}}\preceq P_{I_{1}}$ and therefore $[x]\preceq[f(x)]$. Generalizing, we say that $f^{k}(x)\in P_{I_{k}}$ with $k\in\mathbb{Z}$ and $I_{k}\subseteq I_{k+1}\subseteq\{1,\ldots,n\}$. There exists $k_{0}\in\mathbb{Z}$ such that $I_{k}=I_{k_{0}}$ for all $k>k_{0}$ and thus $P_{k}=P_{k_{0}}$. This completes the proof. $\square$

Lemma 20**.**

Let a positive map $f$ be sub-homogeneous and type-K order-preserving. If $f$ has a positive fixed point $\bar{x}\in\mathbb{R}_{+}^{n}$, then for all $x\in\mathbb{R}_{\geq 0}^{n}$ the trajectory $\mathcal{T}(x)$ is bounded.

{pf*}

Proof. By Lemma 16 it follows that $g=\log\circ f\circ\exp$ is a sup-norm non-expansive map that has the same dynamical properties as $f$ for all $x\in\mathbb{R}_{+}^{n}$. By Lemma 17 we know that one of the two cases can occur:

1. $(i)$

all trajectories $\mathcal{T}(\log(x),g)$ are unbounded; 2. $(ii)$

all trajectories $\mathcal{T}(\log(x),g)$ are bounded.

Since $f$ has a fixed point $x_{f}\in\mathbb{R}_{+}^{n}$, such that $f(x_{f})=x_{f}$, then $x_{g}=\log(x_{f})$ is a fixed point of $g$, i.e., $g(x_{g})=x_{g}$. The trajectory $\mathcal{T}(\log(x_{f}),g)$ is obviously bounded and therefore case $(ii)$ holds.

By Lemma 19, we can partition $\mathbb{R}_{\geq 0}^{n}$ in two disjoint sets $S_{1}$, $S_{2}$ such that if for $x$ there exists $k_{0}\in\mathbb{Z}$ such that $f^{k_{0}}(x)\in\mathbb{R}_{+}^{n}$, then $x\in S_{1}$, otherwise $x\in S_{2}$. We analyze these two cases.

1. For all $x\in S_{1}$, by Lemma 19, it holds that $f^{k}(x)\in\mathbb{R}_{+}^{n}$ for all $k\geq k_{0}$. Let $x_{0}=f^{k_{0}}(x)$. Since case $(ii)$ holds $\mathcal{T}(\log(x_{0}),g)$ is bounded, because of the isometry also $\mathcal{T}(x_{0},f)$ is bounded, and therefore also $\mathcal{T}(x,f)$. We conclude that for all $x\in S_{1}$ trajectories $\mathcal{T}(x,f)$ are bounded.

2. For all $x\in S_{2}$, by Lemma 19, there exists $k_{0}\in\mathbb{Z}$ such that $f^{k}(x)\in P_{I}$ with $I(x)\subset N=\{1,\ldots,n\}$ for all $k\geq k_{0}$. Without loss of generality, here we assume $I=\{1,\ldots,m\}$, where $m<n$. Let $x=[z_{1}^{T},z_{2}^{T}]^{T}$ with $z_{1}\in\mathbb{R}^{m}_{\geq 0}$ and consider the following $m$-dimensional map $f^{*}:\mathbb{R}_{+}^{m}\rightarrow\mathbb{R}_{+}^{m}$ defined by

[TABLE]

with $i\in I(x)$. It is not difficult to check that $f^{*}$ is still sub-homogeneous and type-K order preserving. Accordingly, $g^{*}=\log\circ f^{*}\circ\exp$ is a sup-norm non-expansive map that has the same dynamical properties as $f^{*}$ for all $x\in\mathbb{R}_{+}^{m}$. The main point now is to prove that if $(ii)$ occurs then all trajectories $\mathcal{T}(\log(z_{1}),g^{*})$ are also bounded. To this aim, we first need to show that for all $i\in I(x)$ it holds

[TABLE]

Since both the exponential and the logarithmic functions are strictly increasing, (4) is equivalent to

[TABLE]

By definition, (5) holds if $z_{2}=0$. If $z_{2}\neq 0$, for any $x=[z_{1}^{T},z_{2}^{T}]^{T}$ consider $\bar{x}=[z_{1}^{T},\bar{z}_{2}^{T}]^{T}$ such that $\bar{z}_{2}=\mathbf{0}$. Since $f$ is order-preserving, for all $i\in I$ it holds that $f_{i}(\bar{x})\leq f_{i}(x)$, which is equivalent to write $f_{i}(z_{1},\bar{z}_{2})\leq f_{i}(z_{1},z_{2})$ . By definition, $f_{i}^{*}(z_{1})=f_{i}(z_{1},\bar{z}_{2})$. Therefore, $f_{i}^{*}(z_{1})\leq f_{i}(z_{1},z_{2})$ for all $z_{2}\neq 0$, i.e., (5) and (4) hold. Suppose that $(ii)$ occurs and there exist $\hat{z}_{1}\in\mathbb{R}_{+}^{m}$ such that $\mathcal{T}(\log(\hat{z}_{1}),g^{*})$ is unbounded. By (4) it is clear that given $\hat{x}=[\hat{z}_{1}^{T},z_{2}^{T}]^{T}$ the trajectory $\mathcal{T}(\log(\hat{x}),g)$ is also unbounded, contradicting $(ii)$. Let $x_{0}=f^{k_{0}}(x)$. Since all trajectories $\mathcal{T}(\log(x_{0}),g)$ are bounded, because of the isometry also $\mathcal{T}(x_{0},f)$ is bounded, and therefore also $\mathcal{T}(x,f)$. We conclude that for all $x\in S_{2}$ trajectories $\mathcal{T}(x,f)$ are bounded. $\square$

Lemma 21**.**

[11]**  Let a positive map $f$ be sub-homogeneous and type-K order-preserving. If for all $x\in K$ the trajectory $\mathcal{T}(x)$ is bounded, then for all $x\in\mathbb{R}_{\geq 0}^{n}$, $\omega(x)$ is a singleton. $\blacksquare$

Finally, after restating the Theorem 7 for convenience of the reader, we present a compact proof based on the results presented in this section and Lemma 3.1.3 in [14].

Theorem 7 (Convergence) Let a positive map $f$ be sub-homogeneous and type-K order-preserving. If $f$ has at least one positive fixed point in $\mathbb{R}^{n}_{+}$ then all periodic points are fixed points, i.e., the set $\omega(x)$ is a singleton and $\forall x\in\mathbb{R}_{\geq 0}^{n}:\lim_{k\rightarrow\infty}f^{k}(x)=\bar{x}$ where $\bar{x}$ is a fixed point of $f$.

{pf*}

Proof of Theorem 7.   By Lemma 20 it follows that all trajectories $\mathcal{T}(x)$ are bounded for all $x\in\mathbb{R}_{+}^{n}$. By Lemma 21 it follows that all periodic points are fixed points, i.e., the set $\omega(x)$ is a singleton. By Lemma 3.1.3 in [14], since $f$ is continuous all trajectories are bounded and all periodic points are fixed points it holds $\lim_{k\rightarrow\infty}f^{k}(x)=\bar{x}$ where $\bar{x}$ is a fixed point of $f$. $\square$

4.3 Proof of Theorem 8

By means of the technical result in Theorem 7 we prove our second main result, a sufficient condition on the structure of the heterogeneous local interaction rules of the MAS under consideration so that the global map (possibly unknown due to an unknown network topology) is positive, type-K order preserving and sub-homogeneous map, thus falling within the class of systems considered in Theorem 12. We also restate the theorem for convenience of the reader.

Theorem 8 (Stability) Consider a MAS as in (2) with at least one positive equilibrium point. If the set of differentiable local interaction rules $f_{i}$, with $i=1,\ldots,n$, satisfies the next conditions:

1. $(i)$

$f_{i}(x)\in\mathbb{R}_{\geq 0}$* for all $x\in\mathbb{R}_{\geq 0}^{n}$;* 2. $(ii)$

$\partial f_{i}/\partial x_{i}>0$* and $\partial f_{i}/\partial x_{j}\geq 0$ for $i\neq j$;* 3. $(iii)$

$\alpha f_{i}(x)\leq f_{i}(\alpha x)$* for all $\alpha\in[0,1]$ and $x\in K$;*

then the MAS converges to one of its equilibrium points for any positive initial state $x(0)\in\mathbb{R}_{\geq 0}^{n}$.

{pf*}

Proof of Theorem 8. We start the proof by establishing equivalence relationships between the properties $(i)-(iii)$ of the local interaction rules of the MAS listed in the statement of Theorem 8 and properties (a)-(c) shown next:

1. $(a)$

$f$ is positive; 2. $(b)$

$f$ is type-K order-preserving; 3. $(c)$

$f$ is sub-homogeneous;

We now prove all equivalences one by one.

• •

$[(i)\Leftrightarrow(a)]$ Condition $(i)$ implies that $f$ maps a point of $\mathbb{R}_{\geq 0}^{n}$ into $\mathbb{R}_{\geq 0}^{n}$ and is, therefore, a positive map (see Definition 1).

• •

$[(ii)\Rightarrow(b)]$ due to Proposition 12 (Kamke-like condition).

• •

$[(iii)\Leftrightarrow(c)]$ by Definition 6 of a sub-homogeneous map, sub-homogeneity can be verified element-wise for map $f$, thus the equivalence follows.

Thus, if conditions $(i)$ to $(iii)$ hold true for all local interaction rules $f_{i}$ with $i=1,\ldots,n$, since by assumption map $f$ has at least one positive fixed point, we can exploit the result in Theorem 7 to establish that for all positive initial conditions, the state trajectories of the MAS converge one of its positive equilibrium points. $\square$

4.4 Proof of Theorem 9

To prove our third main result, namely Theorem 9, we need to introduce two technical lemmas. The first lemma, shown next, states sufficient conditions under which the elements along the rows of the Jacobian matrix of a map $f$ computed at a consensus point $c\mathbf{1}$ sum to one.

Lemma 22**.**

Let a map $f$ be positive and differentiable. If the set of fixed points $F_{f}$ of map $f$ satisfies $F_{f}\supseteq\{c\mathbf{1},\quad c\in\mathbb{R}_{\geq 0}\}$, i.e., the set of fixed points contains at least all positive consensus states, then

[TABLE]

where $J_{f}(c\mathbf{1})$ denotes the Jacobian of $f$ evaluated in $c\mathbf{1}$.

{pf*}

Proof. Since $f$ is differentiable, we can apply directly the definition of directional derivative in a point $x\in\mathbb{R}^{n}_{\geq 0}$ along a vector $v\in\mathbb{R}^{n}$ obtaining

[TABLE]

Now we evaluate this expression in a consensus point $x=c\mathbf{1}\in F_{f}$, and along the direction $v=\mathbf{1}$ which is an invariant direction of $f$. We obtain

[TABLE]

thus proving the statement. $\square$

Next, we introduce a critical lemma needed to prove our third main result. In particular, we show that if there exists a fixed point of map $f$ different from a consensus point, then there exists a consensus point such that the Jacobian of map $f$ computed at that consensus point has a unitary eigenvalue with multiplicity strictly greater than one.

Lemma 23**.**

Let $f$ be positive, sub-homogeneous, type-K order-preserving and have a set of fixed points $F_{f}$ such that

[TABLE]

If there exists a fixed point $\bar{x}\in\mathbb{R}^{n}_{\geq 0}$ such that

[TABLE]

then there exists $\bar{c}(\bar{x})>0$ such that the Jacobian matrix $J_{f}(\bar{c}(\bar{x})\mathbf{1})$ of map $f$ computed at $\bar{c}(\bar{x})\mathbf{1}$ has a unitary eigenvalue with multiplicity strictly greater than one.

{pf*}

Proof. Let $\bar{x}=\left[\bar{x}_{1},\ldots,\bar{x}_{n}\right]^{T}\in\mathbb{R}^{n}_{\geq 0}$ be a fixed point of map $f$ and let $c_{1},c_{2}\in\mathbb{R}_{\geq 0}$ be such that

[TABLE]

We define three sets

[TABLE]

Consider a point $y$ such that the $i$-th component is defined by

[TABLE]

and such that

[TABLE]

By (6) and (7) it follows that

[TABLE]

$\bullet$ Now, we prove that

[TABLE]

Since map $f$ is type-K order preserving, from (7) it follows $c_{1}\leq f_{i}(y)\leq\bar{x}_{i}$ and from (8) $f_{i}(y)\leq c_{3}$ for $i=1,\ldots,n$. For $i\in I_{min}(\bar{x})$, by definition $\bar{x}_{i}=c_{1}$ and thus $f_{i}(y)=c_{1}$, otherwise for $i\in I(\bar{x})\cup I_{max}(\bar{x})$ by (7) $\bar{x}_{i}\geq y_{i}=c_{3}$ and it follows $c_{1}\leq f_{i}(y)\leq c_{3}$. Thus, (9) holds.

$\bullet$ Now, we prove that

[TABLE]

Since $f$ is order-preserving and sub-homogeneous, then $f$ is non-expansive under the Thompson’s metric (see Definition 13) by Lemma 15. Now, by exploiting the definition of non-expansive map, we compute an upper bound to $d_{T}(\bar{x},f(y))$. It holds

[TABLE]

where

[TABLE]

Since $c_{2}\geq c_{3}$, it holds

[TABLE]

Now, we compute a lower bound to $d_{T}(\bar{x},f(y))$, where

[TABLE]

and

[TABLE]

Thus $\max\left\{M(\bar{x}/f(y)),M(f(y)/\bar{x})\right\}\geq\frac{c_{2}}{c_{3}},$ therefore

[TABLE]

By inequalities (11) and (12) it follows that

[TABLE]

thus proving that the inequality in (10) holds true.

$\bullet$ Due to Theorem 7 it holds $\displaystyle\lim_{k\rightarrow\infty}f^{k}_{i}(y)=\bar{y}_{i}$. Now, we prove that

[TABLE]

In (13) three cases may occur:

If $i\in I_{min}(\bar{x})$ then $\bar{x}_{i}=c_{1}$ and by (7) it follows $f_{i}(y)=c_{1}$. 2. 2.

If $i\in I_{max}(\bar{x})$ then $\bar{x}_{i}=c_{2}$ and by (10) it follows $f_{i}(y)=c_{3}$. 3. 3.

If $i\in I(\bar{x})$, by (9) two cases may occur:

1. (a)

There exists $k^{*}>0$ such that $f^{k^{*}}_{i}(y)<y_{i}$. In this case, by type-K order-preservation it holds that $f^{k}_{i}(y)<f_{i}^{k-1}(y)\quad\forall k\geq k^{*}+1$ and therefore

[TABLE] 2. (b)

Otherwise $f^{k}_{i}(y)=f^{k-1}(y)\quad\forall k>0$ and therefore

[TABLE]

Thus, by (13) we proved that for any fixed point $\bar{x}$ different from a consensus point $c\mathbf{1}$ there exists a fixed point $\bar{y}$ with elements corresponding to either $c_{1}$ or $c_{3}$ and such that $I(\bar{y})=\emptyset$.

Now, consider a point $z$ such that its $i$-th component is defined as follows

[TABLE]

with $c_{4}\in\left[c_{1},c_{3}\right]$. By (13) and (14), we can conclude that $z$ is fixed point, i.e., $f(z)=z$, for all values of $c_{4}$ in the interval $c_{4}\in\left[c_{1},c_{3}\right]$. Now, let $v(\bar{x})$ be a vector such that

[TABLE]

Thus, by (15) the point $c_{1}\mathbf{1}+hv(\bar{x})$ is a fixed point of map $f$ for all $h\in[0,c_{3}-c_{1}]$. Thus, it follows that

[TABLE]

Since $v(\bar{x})\neq\mathbf{1}$, it holds (by reasoning along the lines of Lemma 22) that the Jacobian of map $f$ computed at $c_{1}\mathbf{1}$ has a right eigenvector equal to $v(\bar{x})$, i.e., $J_{f}(c_{1}\mathbf{1})v(\bar{x})=v(\bar{x})$. By Lemma 22 it holds that the Jacobian of $f$ satisfies $J_{f}(c\mathbf{1})\mathbf{1}=\mathbf{1}$ for all $c>0$. Thus, if there exists $\bar{x}\neq c\mathbf{1}$ then there exists $\bar{c}(\bar{x})=\displaystyle\min_{i=1,\ldots,n}\bar{x}_{i}=c_{1}$ such that matrix $J_{f}(\bar{c}(\bar{x})\mathbf{1})$ has a untiray eigenvalue with multiplicity strictly greater than one, thus proving the statement of this lemma. $\square$

Finally, we recall for convenience of the reader and detail a proof of our third and last main result, i.e., Theorem 9, which provides sufficient conditions for asymptotic convergence to the consensus state.

Theorem 9 (Consensus) Consider a MAS as in (2). If the set of differentiable local interaction rules $f_{i}$, with $i=1,\ldots,n$, satisfies the next conditions:

1. $(i)$

$f_{i}(x)\in\mathbb{R}_{\geq 0}$* for all $x\in\mathbb{R}_{\geq 0}^{n}$;* 2. $(ii)$

$\partial f_{i}/\partial x_{i}>0$* and $\partial f_{i}/\partial x_{j}\geq 0$ for $i\neq j$;* 3. $(iii)$

$\alpha f_{i}(x)\leq f_{i}(\alpha x)$* for all $\alpha\in[0,1]$ and $x\in\mathbb{R}^{n}_{\geq 0}$;* 4. $(iv)$

$f_{i}(x)=x_{i}$* if $x_{i}=x_{j}$ for all $j\in\mathcal{N}_{i}^{in}$;* 5. $(v)$

Inference graph $\mathcal{G}(f)$ has a globally reachable node;

then, the MAS converges asymptotically to a consensus state for any initial state $x(0)\in\mathbb{R}_{\geq 0}^{n}$.

{pf*}

Proof of Theorem 9 We start the proof by establishing the relations between properties $(i)-(v)$ and the following:

1. $(a)$

$f$ is positive; 2. $(b)$

$f$ is type-K order-preserving; 3. $(c)$

$f$ is sub-homogeneous; 4. $(d)$

$F_{f}=\{c\mathbf{1}:c\in\mathbb{R}_{\geq 0}\}$.

We go through all equivalences one by one.

$[(i)\Leftrightarrow(a)]$ See Proof of Theorem 8. 2. 2.

$[(ii)\Leftrightarrow(b)]$ See Proposition 12. 3. 3.

$[(iii)\Leftrightarrow(c)]$ See Proof of Theorem 8. 4. 4.

$[(i-v)\Rightarrow(d)]$ The proof of this implication is given below.

Condition $(iv)$ implies that the consensus space $c\mathbf{1}$ is a subset of the set of fixed points $F_{f}$ of map $f$, i.e.,

[TABLE]

By Theorem 22 the Jacobian matrix $J_{f}(c\mathbf{1})$ evaluated at a consensus point is row-stochastic, i.e., $J_{f}(c\mathbf{1})\mathbf{1}=\mathbf{1}$. By the definition of inference graph (see Definition 2), it holds that $\mathcal{G}(f)=\mathcal{G}(J_{f}(c\mathbf{1}))$. Thus $\mathcal{G}(J_{f}(c\mathbf{1}))$ has a globally reachable node by hypothesis and is aperiodic because condition $(ii)$ ensures a self-loop at each node.

Now, we are ready to prove by contradiction that $[(i-v)\Rightarrow(d)]$. In particular, if there exists a fixed point $\bar{x}\neq c\mathbf{1}$, then by Lemma 23 the Jacobian matrix $J_{f}(c\mathbf{1})$ has a unitray eigenvalue with multiplicity strictly greater than one. On the other hand, by the widely known Theorem 5.1 in [4], if $\mathcal{G}(J_{f}(c\mathbf{1}))$ has a globally reachable node and is aperiodic then $J_{f}(c\mathbf{1})$ has a simple unitary eigenvalue with corresponding eigenvector equal to $\mathbf{1}$, unique up to a scaling factor $c$.

This is a contradiction, therefore it does not exist a fixed point $\bar{x}$ such that $\bar{x}\neq c\mathbf{1}$ with $c>0$. Thus, we conclude that the set of fixed points of map $f$ satisfies

[TABLE]

Finally, if conditions $(a)$ to $(c)$ are satisfied, then Theorem 8 the MAS converges to its set of fixed points $F_{f}$. If $(d)$ is satisfied, the $F_{f}$ contains only consensus points and thus the MAS in (2) converges to a consensus state for all $x\in\mathbb{R}^{n}_{\geq 0}$. $\square$

5 Examples

In this section we provide examples to corroborate our theoretical analysis of the convergences properties of discrete-time, nonlinear, positive, type-K order-preserving and sub-homogeneous MAS.

Example 24**.**

As a first example we consider a susceptible-infected-susceptible (SIS) epidemic model [3] described by the following

[TABLE]

Such model was originally derived to describe the propagation of an infectious diseases over a group of individuals. Each group is subdivided according to susceptible and infectious. Individuals can be cured and reinfected many times, there is not an immune group. Given $n$ groups, let $x_{i}(k)$, $y_{i}(k)$ be the portion of, respectively, susceptibles and infectious of group $i$ at time $k$, it is clear that $x_{i}(k),y_{i}(k)\geq 0$ and $x_{i}(k)+y_{i}(k)=1$ for any $k$. Thus, it is sufficient to consider the dynamics of one of them to completely describe the system. In model (16) variables have the following meaning:

• •

$\beta_{ij}\geq 0$ are the infectious rates;

• •

$\delta_{i}\geq 0$ is the healing rate;

• •

$h\geq 0$ is the sampling rate.

We now evaluate conditions $(i)-(iv)$ of Theorem 8 to establish the convergence of the associated MAS to a positive fixed point. Due to space limitations, we omit all steps and give directly conditions under which the theorem holds.

• •

First we notice that $x_{i}(k)$ belongs to $[0,1]$ for all $k$. It is guaranteed that for all $x_{i}(k)\in[0,1]$ also $x_{i}(k+1)+\in[0,1]$ if and only if (17) and (18) hold,

[TABLE]

We conclude that for any $x\in[0,1]^{n}\subset\mathbb{R}_{\geq 0}^{n}$ then $f_{i}(x)\in[0,1]\subset\mathbb{R}_{\geq 0}$, thus proving that condition $(i)$ holds.

• •

Condition $(ii)$ holds if and only if the next inequality holds

[TABLE]

• •

Condition $(iii)$ holds if and only if the next inequality holds

[TABLE]

• •

Condition $(iv)$ is satisfied since $\bar{x}=\mathbf{1}\in\mathbb{R}^{n}_{\geq 0}$ is a positive fixed point.

One can prove that (17), (18), (19), (20) are equivalent to (21). Let $\beta=\sum_{j\neq i}\beta_{ij}$,

[TABLE]

If (21) holds, then conditions of Theorem 8 are satisfied, and we conclude that the MAS converges to a fixed point for all $x\in[0,1]^{n}$. For a MAS described by graph $\mathcal{G}_{1}$ in Figure 1(a), a numerical simulation is given in Figure 2(a).

Example 25**.**

Consider a MAS described by graph $\mathcal{G}_{2}$ in Figure 1(b) and nonlinear local interaction rule

[TABLE]

We now evaluate conditions $(i)-(v)$ of Theorem 9 to establish the convergence of the associated MAS to consensus state.

• •

Condition $(i)$ holds $\forall x\in\mathbb{R}^{n}_{\geq 0}$.

• •

Condition $(ii)$ holds if and only if $\varepsilon_{i}\in\left(0,\left|\mathcal{N}_{i}\right|^{-1}\right]$ for $i=1,\ldots,n$.

• •

Condition $(iii)$ holds $\forall x\in\mathbb{R}^{n}_{\geq 0}$.

• •

Condition $(iv)$ is satisfied since $\bar{x}=c\mathbf{1}$ with $c\in\mathbb{R}_{\geq 0}$ is a solution for $x=f(x)$.

• •

Condition $(v)$ is satisfied since graph $\mathcal{G}$ has a globally reachable node.

Thus, the conditions of Theorem 9 are satisfied, and we conclude that the MAS in (22) converges to a consensus state. A numerical simulation is given in Figure 2(b).

6 Conclusions and future works

In this paper we presented three main results related to a class of nonlinear discrete-time multi-agent systems represented by a state transition map which is positive, sub-homogeneous and type-K order preserving.

The first result establishes that a general discrete-time dynamical system converges to one of its equilibrium points asymptotically if its corresponding state transition map is positive, sub-homogeneous and type-K order preserving. The second result provides sufficient conditions for a set of nonlinear, discrete-time heterogeneous local interaction rules which define the MAS to establish stability of the MAS, independently of its graph topology (which is considered unknown) by exploiting our first main result. Finally, the third result provides sufficient conditions for a set of nonlinear, discrete-time heterogeneous local interaction rules which define the MAS to establish asymptotic convergence to a consensus state if the inference graph of the MAS has a globally reachable node.

This paper generalizes results for discrete-time linear MAS whose state transition matrix is stochastic to the nonlinear case thanks to nonlinear Perron-Frobenius theory. Examples are provided to show the effectiveness of the stability analysis of a MAS based on our method.

Future work will consider MAS represented by a time-varying set of heterogeneous local interaction rules.