Strategy Consensus of Networked Evolutionary Games Based on Network Aggregation and Pinning Control

Abstract

The computational complexity of large-scale networked evolutionary games has become a challenging problem. Based on network aggregation and pinning control methods, this paper investigates the problem of control design for strategy consensus of large-scale networked evolutionary games. The large-size network is divided into several small subnetworks by the aggregation method, and a pinning control algorithm is proposed to achieve the strategy consensus of small subnetworks. Then, the matchable condition between the small subnetworks is realized by the input–output control. Finally, some sufficient conditions as well as an algorithm are proposed for the strategy consensus of large-scale networked evolutionary games.

1. Introduction

Evolutionary games were first proposed by biologists to study social behaviors in nature Smith and Price (1973) and have been widely applied in natural, social and economic systems in the past few decades (Fan et al., 2021; Zhang et al., 2021). Classical evolutionary games often assume that each player interacts with all other players or randomly selected players. However, due to the impact of complex network environments, the ability of each player to obtain information is limited, which is dependent of the local information among players and the network topology. Accordingly, evolutionary games on the network have been proposed and called networked evolutionary games (NEGs) Nowak and May (1992). In NEGs, nodes represent players and edges represent the interactions between players. Therefore, nodes and edges play an important role in the study of NEGs. In recent years, the state feedback control, incentive-based control and robustness of NEGs have been well studied (Jia et al., 2024; Sun et al., 2023; Wang et al., 2023; Ye, 2022) and have become popular problems in biology, sociology, system science and other disciplines (Fowler & Christakis, 2010; Iyer & Killingback, 2016).

With the development of logical dynamical systems and finite games, researchers need to adopt effective control methods to optimize the performance of systems and promote the evolution towards the expected goal in practical application scenarios such as biology, sociology and economics. However, traditional methods have difficulties dealing with complex logical relations. The semi-tensor product of matrices Cheng et al. (2011) provides an effective way to solve this problem by transforming elements of a finite set into logical vectors and then converting complex logical expressions into simple algebraic state-space representations. Based on this method, a series of important results have been obtained on the observability Y. Li et al. (2024), controllability X. Yang and Li (2023) and stability Guo et al. (2019) of Boolean networks. These studies are helpful to predict the network behavior of gene regulation and optimize the spread of social behavior. Additionally, the semi-tensor product method has been introduced in game theory (W. Liu et al., 2023; G. Zhao et al., 2024; R. Zhu et al., 2022), particularly playing a key role in the study of strategy consensus in NEGs Jia et al. (2024).

It is worth pointing out that the network topology of NEGs often consists of a large number of nodes and edges. However, the computational complexity of methods in (Jia et al., 2024; W. Liu et al., 2023; G. Zhao et al., 2024; R. Zhu et al., 2022) grows exponentially with the increase in number of nodes. As a result, it is necessary to develop some new methods to reduce the complexity of studying NEGs. Some useful techniques were proposed to study large-scale logical networks, including logical matrix factorization H. Li and Wang (2015), compositional method H. Li and Pang (2024), network aggregation Y. Zhao et al. (2013) and pinning control Zhong et al. (2021). These methods can effectively reduce the computational complexity and improve the efficiency of network analysis and control, especially for handling complex logical dynamical systems.

In recent years, the network aggregation method has been widely used in the analysis and control of logical networks, such as stabilization H. Li et al. (2021), reachability Y. Zhao et al. (2016) and observability Y. Liu et al. (2020). Particularly, the problem of controlling the strategy choices of players of a large-scale NEG in order to achieve consensus was solved using network aggregation techniques. Based on the strategy profile reachability of each control sub-game, the network aggregation was used to study the large-size networked evolutionary matrix games with a general network structure H. Li et al. (2024). The basic idea of network aggregation is to divide a network into several subnetworks and derive the behavior of the whole network by studying the properties of the subnetworks J. Yang and Cho (2016). In addition, an important method called pinning control was introduced to study logical networks Lu et al. (2016), and several interesting results were obtained (H. Li & Wang, 2015; Lin et al., 2022; S. Zhu et al., 2023, 2022). The main idea of pinning control is to achieve the control objective by controlling only a small subset of nodes Zhong et al. (2020), thereby reducing the computational complexity. The strategy consensus of large-scale NEGs was also solved through pinning control, and the strategy consensus problem of NEGs was considered in Jiang et al. (2024) via the degree-based pinning control. To our best knowledge, there are no results on combining network aggregation and pinning control to study the strategy consensus of NEGs. Therefore, it is meaningful to combine these two methods for the purpose of reducing both the computational complexity and target players of NEGs.

In this paper, we study the strategy consensus of large-scale NEGs via network aggregation and pinning control. The contributions of this article are mainly summarized below. On one hand, by partitioning the whole network, we give the algebraic form of NEGs with pinning control and obtain a sufficient condition for subnetworks to achieve a strategy consensus. On the other hand, based on the network aggregation, we give a sufficient condition for a strategy consensus of large-scale NEGs by selecting a small number of control players in subnetworks.

The remainder of this paper is organized as follows. Section 2 gives some preliminaries. Section 3 presents the algebraic representations of strategy profile dynamics’ equations of the subnetworks via network aggregation. Section 4 provides the main results of this paper. Section 5 is a brief conclusion.

At the end of this section, we present some useful notations.

• $\mathbb{N}$ is the set of natural numbers, and $\mathbb{R}$ is the set of real numbers.
• $Co{l}_i\left(A\right)$ denotes the ith column of matrix A.
• $\mathcal{D}:=\{0,1\}$.
• ${\mathcal{D}}^n:=\underset{n}{\underbrace{\mathcal{D}\times \cdots \times \mathcal{D}}}$.
• ${\Delta }_n:=\{{\delta }_n^i=Co{l}_i\left(I_n\right):i=1,\cdots ,n\}$, where $I_n$ is the $n\times n$-dimensional identity matrix.
• A logical matrix $M=[{\delta }_n^{i_1}{\delta }_n^{i_2}\cdots {\delta }_n^{i_t}]$ is simply denoted as $M={\delta }_n[i_1{i}_2\cdots i_t]$.
• ${\mathbb{R}}^{n\times s}$ and ${\mathcal{L}}_{n\times s}$ denote the sets of $n\times s$-dimensional real matrices and logical matrices, respectively.

2. Preliminaries

In this section, we briefly recall some preliminaries on the semi-tensor product of matrices and the model of NEGs.

The main tool used in this paper is the semi-tensor product of matrices. This is a new multiplication that can be used for matrices with different dimensions. For more details, please refer to Cheng et al. (2011).

Definition 1.

Let $X\in {\mathbb{R}}^{m\times n}$ and $Y\in {\mathbb{R}}^{p\times q}$ be given. The semi-tensor product of two matrices X and Y, denoted by $X⋉Y$, is defined as

$$X⋉Y=\left(X\otimes I_{\alpha /m}\right)\left(Y\otimes I_{\alpha /p}\right),$$

(1)

where $\alpha =lcm(n,p)$ is the least common multiple of n and p, and ⊗ denotes the Kronecker product.

Remark 1.

When $n=p$, the semi-tensor product of matrices X and Y is consistent with the conventional matrix product. In the following text, the symbol ⋉ will be omitted.

The semi-tensor product of matrices has many useful properties, one of which is shown below. Let $f:{\mathcal{D}}_k^n\to \mathbb{R}$ be an n-dimensional pseudo-logical function. Using the semi-tensor product method, there exists a unique matrix $F\in {\mathbb{R}}^{1\times k^n}$, such that

$$f(a_1,a_2,\cdots ,a_n)=M⋉a_1⋉\cdots ⋉a_n,$$

(2)

where M is called the structural matrix of f.

In the following, we present some necessary preliminaries on NEGs.

Definition 2.

An NEG with n players is composed of the following three elements:

(i) 

An undirected network graph $\Theta =(V,E)$, where $V=\{1,2,\cdots ,n\}$ represents the set of n players, and $E\subseteq V\times V$ represents the set of edges;

(ii) 

A fundamental network game, where $S=\{1,2,\cdots ,s\}$ denotes the strategy set, $v_i\left(t\right)$ denotes the strategy of player i at time t, $v\left(t\right)={⋉}_{i=1}^n{v}_i\left(t\right)$ denotes the strategy profile at time t, and $\pi \in {\mathbb{R}}^{s\times s}$ denotes the payoff matrix;

(iii) 

A strategy updating rule (SUR)

$$v_i(t+1)=g_i\left(v_j\left(t\right),p_j\left(t\right);j\in N\left(i\right)\right),i=1,2,\cdots ,n.$$

Remark 2.

$N_l\left(i\right)$ represents the set of all neighbors of player i within l steps (including l), and N represents the set of all neighbors of player i within one step. $p_j\left(t\right)$ represents the average payoff of player j at time t, expressed as

$$p_j\left(t\right)={\displaystyle \frac{1}{\left|N\right(j\left)\right|-1}}\sum _{k\in N\left(j\right)\setminus \left\{j\right\}}{\left[\pi \right]}_{v_j\left(t\right),v_k\left(t\right)}.$$

Remark 3.

There are several common SURs, such as unconditional imitation, myopic best response adjustment, and the Fermi rule.

Remark 4.

According to the SUR in Definition 2, we can derive the fundamental evolutionary equations below:

$$v_i(t+1)=f_i\left(v_j\left(t\right);j\in N_2\left(i\right)\right),i=1,2,\cdots ,n.$$

Using the semi-tensor product of matrices, we can obtain the following strategy profile dynamics (SPDs):

$$v(t+1)=Fv\left(t\right),$$

(3)

where $F\in {\mathcal{L}}_{s^n\times s^n}$.

Definition 3.

Given an NEG $\left((V,E),\pi ,g\right)$, assume that $\{X,U\}$ is a partition of V, where $X\cap U=\varnothing ,X\cup U=V$. If the strategies of the players in U can be fixed, then $\left((X\cup U,E),\pi ,g\right)$ is called a control networked evolutionary game, where $x\in X$ is called the state player, and $u\in U$ is called the control player.

Definition 4.

Given two sets of players $A=\{i_1,i_2,\cdots ,i_r\}\subseteq V$ and $B=\{j_1,j_2,\cdots ,j_p\}\subseteq A$, where $1\le p<r$, the mapping ${\sigma }_{A,B}:{\Delta }_{s^r}\to {\Delta }_{s^p}$ is defined as

$${\sigma }_{A,B}\left({⋉}_{k=1}^r{v}_{i_k}\right)={⋉}_{k=1}^p{v}_{j_k}.$$

Definition 5.

If there exists an integer $\tau \in \mathbb{R}$, such that $v_i(t;v_0)=\theta$, $i=1,2,\cdots ,n$ is satisfied for any integer $t\ge \tau$ and any initial strategy profile $v_0\in {\Delta }_{s^n}$, then an NEG $\left((V,E),\pi ,g\right)$ is said to achieve strategy consensus at $\theta \in {\Delta }_s$.

3. Problem Formulation

We divide the set V of a large-scale NEG into k subsets:

$$V=V_1\cup V_2\cup \cdots \cup V_k,$$

where $V_i\cap V_j=\varnothing$ holds for any two different $i,j\in \{1,\cdots ,k\}$, and each set $V_i$ represents a set of players in a subnetworked evolutionary game ${\Xi }_i$. Define $Z_i$ and $Y_i$ as the sets of input players and output players corresponding to $V_i$, respectively, where

$$Z_i=\left(\bigcup _{j\in V_i}{N}_2\left(j\right)\right)\setminus V_i:=\{{\check{\epsilon }}_{i,1},{\check{\epsilon }}_{i,2},\cdots ,{\check{\epsilon }}_{i,q_i}\},$$

$$Y_i=\left(\bigcup _{j\in V\setminus V_i}{N}_2\left(j\right)\right)\cap V_i:=\{{\widehat{\epsilon }}_{i,1},{\widehat{\epsilon }}_{i,2},\cdots ,{\widehat{\epsilon }}_{i,p_i}\},$$

and ${\widehat{\epsilon }}_{i,1}<\cdots <{\widehat{\epsilon }}_{i,p_i}$.

We assume $|V_i|=n_i$ and define

$$V_i=\{{\epsilon }_i^1,{\epsilon }_i^2,\cdots ,{\epsilon }_i^{p_i},{\epsilon }_i^{p_i+1},{\epsilon }_i^{p_i+2},\cdots ,{\epsilon }_i^{n_i}\},$$

where ${\epsilon }_i^l\in Y_i$, ${\epsilon }_i^m\in V_i\setminus Y_i$, $l=1,2,\cdots ,p_i$ and $m=p_i+1,p_i+2,\cdots ,n_i$. The degree of players in $V_i$ satisfies the following conditions:

$$deg\left({\epsilon }_i^1\right)\ge deg\left({\epsilon }_i^2\right)\ge \cdots \ge deg\left({\epsilon }_i^{p_i}\right),$$

$$deg\left({\epsilon }_i^{p_i+1}\right)\ge deg\left({\epsilon }_i^{p_i+2}\right)\ge \cdots \ge deg\left({\epsilon }_i^{n_i}\right),$$

and if there exist two nodes that have equal degree, then we keep the order of these two nodes in V.

We assume that the network graph of the considered NEG is connected, and after partitioning the set V into k subsets, the network graph ${\Theta }_i$ of each subnetwork is also connected, where $i=1,\cdots ,k$. Consider the ith subnetworked evolutionary game ${\Xi }_i$ below:

$$\begin{array}{c}\hfill v_{{\epsilon }_i^{\alpha }}(t+1)=f_{{\epsilon }_i^{\alpha }}\left(\{v_k\left(t\right);k\in Z_i\cup V_i\}\right),\end{array}$$

(4)

where $\alpha =1,\cdots ,n_i$ and $i=1,\cdots ,k$. Then, the SPDs are constructed from (4) as follows:

$$\begin{array}{c}\hfill {\phi }_i(t+1)=F_i{\gamma }_i\left(t\right){\phi }_i\left(t\right),\end{array}$$

(5)

where ${\phi }_i\left(t\right)={⋉}_{l=1}^{n_i}{v}_{{\epsilon }_i^l}\left(t\right)$, ${\gamma }_i\left(t\right)={⋉}_{l=1}^{q_i}{v}_{{\check{\epsilon }}_{i,l}}\left(t\right)$ and $F_i\in {\mathcal{L}}_{s^{n_i}\times s^{n_i+q_i}}$.

This paper aims to achieve the strategy consensus in subnetworked evolutionary games and eventually achieve the strategy consensus in whole NEGs by network aggregation. In the next section, we focus on how to choose control players in order to achieve the strategy consensus in the subnetworks.

4. Main Results

Firstly, we establish a criterion for achieving the strategy consensus in subnetworked evolutionary games and present a control design procedure based on this criterion.

In order to facilitate the analysis, we need to give some basic assumptions. Let the target strategy profile of an NEG with n players be $v_e={\theta }^n={\delta }_{s^n}^{\xi }\in {\Delta }_{s^n}$. Then, define the target strategy profile of the ith subnetwork as ${\phi }_{i,e}={\sigma }_{V,V_i}\left(v_e\right)={\theta }^{n_i}={\delta }_{s^{n_i}}^{{\xi }_i}\in {\Delta }_{s^{n_i}}$, the target strategy profile of the input players as ${\gamma }_{i,e}={\sigma }_{V,Z_i}\left(v_e\right)={\theta }^{q_i}={\delta }_{s^{q_i}}^{{\overline{w}}_i}\in {\Delta }_{s^{q_i}}$, and the target strategy profile of the output players as $P_{i,e}={\sigma }_{V,Y_i}\left(v_e\right)={\theta }^{p_i}={\delta }_{s^{p_i}}^{{\overline{m}}_i}\in {\Delta }_{s^{p_i}}$.

In the SPDs (5) of subnetworked evolutionary game ${\Xi }_i$, we fix the strategies of input and output players in $Z_i\cup Y_i$ as the target strategy $\theta ={\delta }_s^r$. Then, (5) can be written as

$$\begin{array}{ccc}\hfill & & {\phi }_i(t+1)=F_i{\gamma }_{i,e}{P}_{i,e}{⋉}_{j=p_i+1}^{n_i}{v}_{{\epsilon }_i^j}\left(t\right)\hfill \\ \hfill & =& F_i{\delta }_{s^{q_i+p_i}}^{({\overline{w}}_i-1)p_i+{\overline{m}}_i}{\tilde{\phi }}_i\left(t\right)\hfill \\ \hfill & =& Bl{k}_{Q_i}\left(F_i\right){\tilde{\phi }}_i\left(t\right)\hfill \\ \hfill & =& \widetilde{F_i}{\tilde{\phi }}_i\left(t\right),\hfill \end{array}$$

where $\widetilde{F_i}=Bl{k}_{Q_i}\left(F_i\right)$, $Q_i=({\overline{w}}_i-1)p_i+{\overline{m}}_i$ and ${\tilde{\phi }}_i\left(t\right)={⋉}_{j=p_i+1}^{n_i}{v}_{{\epsilon }_i^j}\left(t\right)$. Thus, we have

$${\left({\delta }_s^r\right)}^{p_i}{\tilde{\phi }}_i(t+1)=\widetilde{F_i}{\tilde{\phi }}_i\left(t\right).$$

Since ${\left({\left({\delta }_s^r\right)}^{p_i}\right)}^{\top }{\left({\delta }_s^r\right)}^{p_i}=1$, we derive that

$$\begin{array}{c}\hfill {\tilde{\phi }}_i(t+1)={\left({\left({\delta }_s^r\right)}^{p_i}\right)}^{\top }{\tilde{F}}_i{\tilde{\phi }}_i\left(t\right).\end{array}$$

(6)

Then, we have the following result.

Theorem 1.

Let $Z_i\cup Y_i$ be the set of input and output control players of ${\Xi }_i$. Then, the subnetworked evolutionary game ${\Xi }_i$ achieves strategy consensus to ${\phi }_{i,e}={\sigma }_{V,V_i}\left(v_e\right)={\theta }^{n_i}={\left({\delta }_s^r\right)}^{n_i}:={\delta }_{s^{n_i}}^{{\xi }_i}$, if and only if

$$\begin{array}{c}\hfill Ro{w}_{{\psi }_i}{\left[{\left({\left({\delta }_s^r\right)}^{p_i}\right)}^{\top }{\tilde{F}}_i\right]}^{s^{n_i-p_i}}={\mathbf{1}}_{s^{n_i-p_i}}^{\top },\end{array}$$

(7)

where ${\psi }_i=1+(r-1){\displaystyle \frac{s^{n_i-p_i}-1}{s-1}}$.

Proof. 

(Necessity) Suppose that the subnetworked evolutionary game ${\Xi }_i$ achieves the strategy consensus to ${\phi }_{i,e}={\sigma }_{V,V_i}\left(v_e\right)={\theta }^{n_i}$. By Definition 5, there exists $T_i\in \mathbb{N}$ such that ${\phi }_i\left(t;{\phi }_i\left(0\right)\right)={\phi }_{i,e}={\theta }^{n_i}$ is satisfied for any integer $t\ge T_i$ and any initial strategy profile ${\phi }_i\left(0\right)\in {\mathcal{D}}_s^{n_i}$, that is, $v_l\left(t;{\phi }_i\left(0\right)\right)={\delta }_s^r$ holds for any $l\in V_i$.

Since $Z_i\cup Y_i$ is the set of input and output players of ${\Xi }_i$, we can derive that

$$\begin{array}{c}\hfill v_m\left(t\right)={\delta }_s^r\end{array}$$

(8)

is satisfied for any $m\in Z_i\cup Y_i$ and any $t\in \mathbb{N}$. Then, according to (6) and (8), we obtain that

$$\begin{array}{c}\hfill {\left({\delta }_s^r\right)}^{n_i-p_i}={\tilde{\phi }}_i\left(t\right)={\left({\left({\delta }_s^r\right)}^{p_i}\right)}^{\top }{\tilde{F}}_i{\tilde{\phi }}_i(t-1)={\left[{\left({\left({\delta }_s^r\right)}^{p_i}\right)}^{\top }{\tilde{F}}_i\right]}^t{\tilde{\phi }}_i\left(0\right)\end{array}$$

holds for any integer $t\ge T_i$ and any ${\tilde{\phi }}_i\left(0\right)\in {\Delta }_{s^{n_i-p_i}}$. Based on ${\left({\delta }_s^r\right)}^{n_i-p_i}={\delta }_s^{1+(r-1){\displaystyle \frac{s^{n_i-p_i}-1}{s-1}}}$ and the arbitrariness of ${\tilde{\phi }}_i\left(0\right)\in {\Delta }_{s^{n_i-p_i}}$, we have

$$\begin{array}{c}\hfill Ro{w}_{{\psi }_i}{\left[{\left({\left({\delta }_s^r\right)}^{p_i}\right)}^{\top }{\tilde{F}}_i\right]}^t={\mathbf{1}}_{s^{n_i-p_i}}^{\top }.\end{array}$$

(9)

Next, we prove that there exists a positive integer $t\le s^{n_i-p_i}$ satisfying (9). Let $t^{*}$ be the smallest positive integer satisfying (9), and assume that $t^{*}\ge s^{n_i-p_i}$. Then, there must exist a ${\phi }_i\left(0\right)\in {\Delta }_{s^{n_i}}$ such that ${\phi }_i(t;{\phi }_i\left(0\right))\ne {\left({\delta }_s^r\right)}^{n_i}$, $0\le t\le t^{*}-1$ and ${\phi }_i\left(t^{*},{\phi }_i\left(0\right)\right)={\left({\delta }_s^r\right)}^{n_i}$.

When the strategy of the input and output players of ${\Xi }_i$ are fixed to the target strategy $\theta$, there exists $0\le t_1\le t_2\le t^{*}-1$ such that ${\phi }_i\left(t_1;{\phi }_i\left(0\right)\right)={\phi }_i\left(t_2;{\phi }_i\left(0\right)\right)\ne {\left({\delta }_s^r\right)}^{n_i}$. Then, the trajectory of ${\Xi }_i$ from ${\phi }_i{\left(0\right)}^{\prime }={\phi }_i\left(t_1;{\phi }_i\left(0\right)\right)$ constitutes a cycle, which has the form of ${\phi }_i\left(t_1;{\phi }_i\left(0\right)\right)\to {\phi }_i\left(t_1+1;{\phi }_i\left(0\right)\right)\to \cdots \to {\phi }_i\left(t_2;{\phi }_i\left(0\right)\right)$. This contradicts the definition of a strategy consensus. Thus, (7) is satisfied.

(Sufficiency) Assume that (7) is satisfied. Then, we have

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\left\{{\left[{\left({\left({\delta }_s^r\right)}^{p_i}\right)}^{\top }{\tilde{F}}_i\right]}^{s^{n_i-p_i}}\right\}}_{i,j}=0,i\ne {\psi }_i,j=1,\cdots ,s^{n_i-p_i},\hfill \\ ⋮\hfill \\ {\left\{{\left[{\left({\left({\delta }_s^r\right)}^{p_i}\right)}^{\top }{\tilde{F}}_i\right]}^{s^{n_i-p_i}}\right\}}_{i,j}=1,i={\psi }_i,j=1,\cdots ,s^{n_i-p_i}.\hfill \end{array}\right.\end{array}$$

(10)

Based on (6), for any integer $t>s^{n_i-p_i}$ and any ${\tilde{\phi }}_i\left(0\right)\in {\Delta }_{s^{n_i-p_i}}$, we obtain

$$\begin{array}{ccc}\hfill {\tilde{\phi }}_i(t;{\tilde{\phi }}_i\left(0\right))& =& {\left[{\left({\left({\delta }_s^r\right)}^{p_i}\right)}^{\top }{\tilde{F}}_i\right]}^t{\tilde{\phi }}_i\left(0\right)\hfill \\ \hfill & =& {\left[{\left({\left({\delta }_s^r\right)}^{p_i}\right)}^{\top }{\tilde{F}}_i\right]}^{s^{n_i-p_i}}{\left[{\left({\left({\delta }_s^r\right)}^{p_i}\right)}^{\top }{\tilde{F}}_i\right]}^{t-s^{n_i-p_i}}{\tilde{\phi }}_i\left(0\right).\hfill \\ \hfill \end{array}$$

Then, according to (10), we derive that

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\left\{{\left[{\left({\left({\delta }_s^r\right)}^{p_i}\right)}^{\top }{\tilde{F}}_i\right]}^t\right\}}_{i,j}=0,i\ne {\psi }_i,j=1,2,\cdots ,s^{n_i-p_i},\hfill \\ ⋮\hfill \\ {\left\{{\left[{\left({\left({\delta }_s^r\right)}^{p_i}\right)}^{\top }{\tilde{F}}_i\right]}^t\right\}}_{i,j}=1,i={\psi }_i,j=1,2,\cdots ,s^{n_i-p_i}.\hfill \end{array}\right.\end{array}$$

Thus, for any initial state ${\tilde{\phi }}_i\left(0\right)\in {\Delta }_{s^{n_i-p_i}}$ and any integer $t\ge s^{n_i-p_i}$, it holds that ${\tilde{\phi }}_i(t;{\tilde{\phi }}_i\left(0\right))={\left({\delta }_s^r\right)}^{n_i-p_i}$. Notice that $Z_i\cup Y_i$ is the set of input and output control players, that is, for any $t\in \mathbb{N}$, we have $v_m\left(t\right)={\delta }_s^r$, $\forall m\in Z_i\cup Y_i$. Then, we obtain ${\phi }_i\left(t;{\tilde{\phi }}_i\left(0\right)\right)={\left({\delta }_s^r\right)}^{p_i}{\tilde{\phi }}_i(t;{\tilde{\phi }}_i\left(0\right))={\left({\delta }_s^r\right)}^{p_i}{\left({\delta }_s^r\right)}^{n_i-p_i}={\left({\delta }_s^r\right)}^{n_i}$, $\forall t\ge s^{n_i-p_i}$. Hence, we conclude that ${\Xi }_i$ achieves the strategy consensus to ${\phi }_{i,e}={\left({\delta }_s^r\right)}^{n_i}\in {\Delta }_{s^{n_i}}$.    □

From Theorem 1, if (7) holds, then the control players set $C_i$ of ${\Xi }_i$ is the set of input and output players, that is, $C_i=Z_i\cup Y_i$. If (7) does not hold, we need to add the control players of ${\Xi }_i$ by fixing the strategy of ${\epsilon }_i^{p_i+1}$ as $\theta ={\delta }_s^r$ and ${\epsilon }_i^{p_i+1}$ is called the control player with the largest degree. Therefore, (6) can be written as

$$\begin{array}{ccc}\hfill {\tilde{\phi }}_i(t+1)& =& {\left({\left({\delta }_s^r\right)}^{p_i}\right)}^{\top }{\tilde{F}}_i{v}_{{\epsilon }_i^{p_i+1}}\left(t\right){⋉}_{l=p_i+2}^{n_i}{v}_{{\epsilon }_i^l}\left(t\right)\hfill \\ \hfill & =& {\left({\left({\delta }_s^r\right)}^{p_i}\right)}^{\top }{\tilde{F}}_i{\delta }_s^r{\widehat{\phi }}_i\left(t\right),\hfill \end{array}$$

where ${\widehat{\phi }}_i\left(t\right)={⋉}_{l=p_i+2}^{n_i}{v}_{{\epsilon }_i^l}\left(t\right)$. Then, we have $\left({\delta }_s^r\right){\widehat{\phi }}_i(t+1)={\left({\left({\delta }_s^r\right)}^{p_i}\right)}^{\top }{\tilde{F}}_i{\delta }_s^r{\widehat{\phi }}_i\left(t\right)$. According to ${\left({\delta }_s^r\right)}^{\top }\left({\delta }_s^r\right)=1$, we obtain

$$\begin{array}{ccc}\hfill & & {\widehat{\phi }}_i(t+1)={\left({\delta }_s^r\right)}^{\top }{\left({\left({\delta }_s^r\right)}^{p_i}\right)}^{\top }{\tilde{F}}_i{\delta }_s^r{\widehat{\phi }}_i\left(t\right)\hfill \\ \hfill & =& {\left({\left({\delta }_s^r\right)}^{\top }\right)}^{p_i+1}{\tilde{F}}_i{\delta }_s^r{\widehat{\phi }}_i\left(t\right)\hfill \\ \hfill & =& {\widehat{F}}_i{\widehat{\phi }}_i\left(t\right),\hfill \end{array}$$

(11)

where ${\widehat{F}}_i={\left({\left({\delta }_s^r\right)}^{\top }\right)}^{p_i+1}{\tilde{F}}_i{\delta }_s^r\in {\mathcal{L}}_{s^{n_i-p_i-1}\times s^{n_i-p_i-1}}$.

In the following, we continue to verify whether

$$\begin{array}{c}\hfill Ro{w}_{K_i}{\left({\widehat{F}}_i\right)}^{s^{n_i-p_i-1}}={\mathbf{1}}_{s^{n_i-p_i-1}}^{\top }\end{array}$$

(12)

holds, where $K_i=1+(r-1){\displaystyle \frac{s^{n_i-p_i-1}-1}{s-1}}$. Denote $C_i^{*}$ as the set of input and output players of ${\Xi }_i$, that is, $C_i^{*}=Z_i\cup Y_i$, and define ${\widehat{C}}_i$ as the set of control players with the largest degree. If (12) holds, then we derive that $C_i=C_i^{*}\cup {\widehat{C}}_i=Z_i\cup Y_i\cup \left\{{\epsilon }_i^{p_i+1}\right\}$. Otherwise, we control several players with larger degrees at the same time, in which we control the first ${\eta }_i$ players belonging to the set $V_i\setminus Y_i$, where $1<{\eta }_i\le n_i-p_i$, fixing their strategies to $\theta ={\delta }_s^r$. Similarly, we can obtain

$$\begin{array}{ccc}\hfill & & {\widehat{\phi }}_i(t+1)={\widehat{F}}_i{\widehat{\phi }}_i\left(t\right)\hfill \\ \hfill & =& {\widehat{F}}_i{\left({\delta }_s^r\right)}^{{\eta }_i-1}{⋉}_{l=p_i+{\eta }_i+1}^{n_i}{v}_{{\epsilon }_i^l}\left(t\right)\hfill \\ \hfill & =& {\widehat{F}}_i{\left({\delta }_s^r\right)}^{{\eta }_i-1}{\overline{\phi }}_i\left(t\right),\hfill \end{array}$$

where ${\overline{\phi }}_i\left(t\right)$$={⋉}_{l=p_i+{\eta }_i+1}^{n_i}$$v_{{\epsilon }_i^l}\left(t\right)$. Then, we have ${\left({\delta }_s^r\right)}^{{\eta }_i-1}{\overline{\phi }}_i(t+1)={\widehat{F}}_i{\left({\delta }_s^r\right)}^{{\eta }_i-1}{\overline{\phi }}_i\left(t\right)$. According to ${{\left({\delta }_s^r\right)}^{{\eta }_i-1}}^{\top }{\left({\delta }_s^r\right)}^{{\eta }_i-1}=1$, we obtain that

$$\begin{array}{ccc}\hfill & & {\overline{\phi }}_i(t+1)={\left({\left({\delta }_s^r\right)}^{{\eta }_i-1}\right)}^{\top }{\widehat{F}}_i{\left({\delta }_s^r\right)}^{{\eta }_i-1}{\overline{\phi }}_i\left(t\right)\hfill \\ \hfill & =& {\overline{F}}_i{\overline{\phi }}_i\left(t\right),\hfill \end{array}$$

(13)

where ${\overline{F}}_i={\left({\left({\delta }_s^r\right)}^{{\eta }_i}\right)}^{\top }{\widehat{F}}_i{\left({\delta }_s^r\right)}^{{\eta }_i}\in {\mathcal{L}}_{s^{n_i-p_i-{\eta }_i}\times s^{n_i-p_i-{\eta }_i}}$.

When ${\eta }_i=2,3,\cdots ,n_i-p_i$, we need to consider whether

$$\begin{array}{c}\hfill Ro{w}_{{\sigma }_i}{\left({\overline{F}}_i\right)}^{s^{n_i-p_i-{\eta }_i}}={\mathbf{1}}_{s^{n_i-p_i-1}}^{\top }\end{array}$$

(14)

holds, respectively, where ${\sigma }_i$ = $1+(r-1)$${\displaystyle \frac{s^{n_i-p_i-{\eta }_i}-1}{s-1}}$. Denote the minimum value of ${\eta }_i$ that makes (14) hold as ${\alpha }_i$, that is, $\left|{\widehat{C}}_i\right|={\alpha }_i$. Thus, the control players set of ${\Xi }_i$ is

$$C_i=C_i^{*}\cup {\widehat{C}}_i=Z_i\cup Y_i\cup \{{\epsilon }_i^{p_i+1},{\epsilon }_i^{p_i+2},\cdots ,{\epsilon }_i^{p_i+{\alpha }_i}\},$$

such that ${\Xi }_i$ achieves the strategy consensus to ${\phi }_{i,e}$.

Based on the above analysis, we present Algorithm 1 that briefly summarizes how to design the set of control players for ${\Xi }_i$.

Algorithm 1 Calculate the set of control players for ${\Xi }_i$

Input: $C_i^{*}$, ${\widehat{C}}_i=\{{\epsilon }_i^{p_i+1},\cdots ,{\epsilon }_i^{l_i}\}$, ${\left({\delta }_s^r\right)}^{n_i}$

Output: $C_i$

$C_i\Leftarrow \varnothing$

if  $Ro{w}_{1+(r-1){\displaystyle \frac{s^{n_i-p_i}-1}{s-1}}}{\left[{\left({\left({\delta }_s^r\right)}^{p_i}\right)}^{\top }{\tilde{F}}_i\right]}^{s^{n_i-p_i}}={\mathbf{1}}_{s^{n_i-p_i}}^{\top }$ then

$C_i=C_i^{*}$

else

for $j=1:\left|{\widehat{C}}_i\right|$ do

$\overline{F}={\left[{\left({\delta }_s^r\right)}^{p_i+j}\right]}^{\top }{\tilde{F}}_i{\left({\delta }_s^r\right)}^j$

if $Ro{w}_{1+(r-1){\displaystyle \frac{s^{n_i-p_i-j}-1}{s-1}}}{\left({\overline{F}}_i\right)}^{s^{n_i-p_i-j}}={\mathbf{1}}_{s^{n_i-p_i-j}}^{\top }$ then

$C_i=C_i^{*}\cup \{{\epsilon }_i^{p_i+1},\cdots ,{\epsilon }_i^{p_i+j}\}$

end if

end for

end if

In what follows, based on the control design procedure for achieving strategy consensus of subnetworked evolutionary games, we give a sufficient condition for achieving the strategy consensus of large-scale NEGs.

Denote $C={\bigcup }_{i=1}^k{C}_i$, where $C_i$ is the set of control players of ${\Xi }_i$ derived from the above algorithm. Then, a sufficient condition is given below for the strategy consensus of large-scale NEGs.

Theorem 2.

An NEG achieves strategy consensus to ${\theta }^n={\delta }_s^{{\xi }_n}$ if the strategies of all players in the set C are fixed to $\theta \in {\Delta }_s$.

Proof. 

Consider any initial state ${\phi }_{i,\lambda }={\delta }_{s^{n_i}}^{{\lambda }_i}\in {\Delta }_{s^{n_i}}$ of ${\Xi }_i$. If the strategies of the players in the set $C_i$ are fixed to $\theta \in {\Delta }_s$, then, based on Theorem 1 and Definition 5, it follows that ${\Xi }_i$ will achieve strategy consensus to ${\phi }_{i,e}={\theta }^{n_i}={\delta }_{s^{n_i}}^{{\xi }_i}$ at time ${\tau }_i$. Let $\tau =max\{{\tau }_1,\cdots ,{\tau }_k\}$. Define $T_{i,a_i}=\{{\delta }_{s^{n_i}}^{l_{a_i,0}}\to {\delta }_{s^{n_i}}^{l_{a_i,1}}\to \cdots \to {\delta }_{s^{n_i}}^{l_{a_i,\tau }}\}$ as the $\tau$-step strategy trajectory of ${\Xi }_i$ and ${\tilde{T}}_{i,a_i}=\{{\delta }_{s^{n_i}}^{l_{a_i,0}},{\delta }_{s^{n_i}}^{l_{a_i,1}},{\delta }_{s^{n_i}}^{l_{a_i,2}},\cdots ,{\delta }_{s^{n_i}}^{l_{a_i,\tau -1}}\}$ as the ordered set of states corresponding to $T_{i,a_i}$, where ${\delta }_{s^{n_i}}^{l_{a_i,0}}={\delta }_{s^{n_i}}^{{\lambda }_i}$, ${\delta }_{s^{n_i}}^{l_{a_i,\tau }}={\delta }_{s^{n_i}}^{{\xi }_i}$. Since the strategies of all players in the set C are fixed to $\theta \in {\Delta }_s$, for each $T_{i,a_i}$, the corresponding input sequence ${\omega }_{i,a_i}=\{\underset{\tau }{\underbrace{{\delta }_{s^{q_i}}^{{\overline{w}}_i},{\delta }_{s^{q_i}}^{{\overline{w}}_i},\cdots ,{\delta }_{s^{q_i}}^{{\overline{w}}_i}}}\}$ is fixed. Then, we obtain that

$$\begin{array}{ccc}\hfill {\sigma }_{V_i,Y_i\cap Z_i}\left({\tilde{T}}_{i,a_i}\right)& =& {\sigma }_{Z_j,Z_j\cap Y_i}\left({\omega }_{j,a_j}\right)\hfill \\ \hfill & =& \{{\theta }^{Y_i\cap Z_i},{\theta }^{Y_i\cap Z_i},\cdots ,{\theta }^{Y_i\cap Z_i}\}\hfill \end{array}$$

is satisfied for any $i\ne j$, $i,j=1,2,\cdots ,k$, that is, there exists a corresponding $\tau$-matching input sequence $\{{\omega }_{i,a_i}:i=1,2,\cdots ,k\}$ such that for any $l\in V_i$ and any initial state $v_{\lambda }$, we have $\{v_l\left(0\right),\cdots ,v_l\left(\tau \right)\}={\sigma }_{V_i,\left\{l\right\}}\left({\tilde{T}}_{i,a_i}\right)$. Hence, we derive the $\tau$-step strategy trajectory $T=\left\{v\right(0)\to v(1)\to \cdots \to v(\tau \left)\right\}$, where $v\left(0\right)=v_{\lambda }$, $v_{\tau }=v_e$. In other words, $v_{\lambda }$ can reach $\theta$ at the $\tau$th step. Then, for any $i=1,2,\cdots ,k$, we have ${\phi }_i\left(t\right)={\phi }_i\left(\tau \right)={\phi }_{i,e}$, $\forall t\ge \tau$. Thus, $v\left(t\right)={⋉}_{i=1}^k$${\phi }_i\left(t\right)$$=v\left(\tau \right)=v_e$ holds for any integer $t\ge \tau$. Therefore, the NEG reaches the strategy consensus to $v_e={\theta }^n={\delta }_{s^n}^{\xi }$. □

Remark 5.

By the above algorithm, we can calculate the set of control players $C_i$ for each ${\Xi }_i$. Therefore, we obtain that the set of control players for the large-scale network is $C={\bigcup }_{i=1}^k{C}_i$.

Remark 6.

Different partitions of a network will result in different subnetworks, and the final set of control players C may also be different.

Finally, we use two examples to illustrate Theorem 2.

Example 1.

Consider an NEG whose network graph is shown in Figure 1. Select control players to guarantee that the considered NEG achieves strategy consensus to $v_e={\left({\delta }_2^2\right)}^9={\delta }_{512}^{512}$, where the strategy set $S=\{1,2\}$, the payoff matrix $\pi =\left[\begin{array}{cc}1& -1\\ 2& 0\end{array}\right]$, and the SUR is an unconditional imitation with fixed priority (Cheng et al., 2015).

As shown in Figure 1, we divide the whole network into two subnetworks ${\Xi }_1$ and ${\Xi }_2$. For subnetwork ${\Xi }_1$, we have

$$Z_1=\{{\check{\epsilon }}_{1,1},{\check{\epsilon }}_{1,2}\}=\{5,6\},$$

$$Y_1=\{{\widehat{\epsilon }}_{1,1},{\widehat{\epsilon }}_{1,2}\}=\{3,4\},$$

$$V_1=\{{\epsilon }_1^1,{\epsilon }_1^2,{\epsilon }_1^3,{\epsilon }_1^4\}=\{3,4,1,2\}.$$

Fixing the strategy of the players in the set $Z_1\cup Y_1=\{3,4,5,6\}$ to be ${\delta }_2^2$, the SPDs of ${\Xi }_1$ are obtained as

$$\begin{array}{ccc}\hfill {\phi }_1(t+1)& =& F_1{\gamma }_1\left(t\right){\phi }_1\left(t\right)\hfill \\ \hfill & =& F_1{\gamma }_1\left(t\right)P_1\left(t\right){⋉}_{i=3}^4{v}_{{\epsilon }_1^i}\left(t\right)\hfill \\ \hfill & =& F_1{\left({\delta }_2^2\right)}^4{\tilde{\phi }}_1\left(t\right)\hfill \\ \hfill & =& Bl{k}_{16}\left(F_1\right){\tilde{\phi }}_1\left(t\right),\hfill \end{array}$$

where ${\gamma }_1\left(t\right)=v_5\left(t\right)v_6\left(t\right)$, $P_1\left(t\right)=v_3\left(t\right)v_4\left(t\right)$ and ${\tilde{\phi }}_1\left(t\right)=v_{{\epsilon }_1^3}\left(t\right)v_{{\epsilon }_1^4}\left(t\right)$. Thus, we have

$${\tilde{\phi }}_1(t+1)={\left({\delta }_4^4\right)}^{\top }Bl{k}_{16}\left(F_1\right){\tilde{\phi }}_1\left(t\right)={\tilde{F}}_1{\tilde{\phi }}_1\left(t\right),$$

where ${\tilde{F}}_1={\left({\delta }_4^4\right)}^{\top }Bl{k}_{16}\left(F_1\right)$.

According to the unconditional imitation with fixed priority, the average payoffs of players 1, 2, 3 and the strategy update dynamics of player 1 and player 2 in subnetwork ${\Xi }_1$ are obtained, as shown in

Table 1. From average payoffs to dynamics in ${\Xi }_1$.

$\mathrm{Average}\mathrm{Payoff}\setminus \mathrm{Profile}$	$11$	$12$	$21$	$22$
$c_1$	$-1$	$-1$	0	0
$c_2$	$-1$	0	$-1$	0
$c_3$	${\displaystyle \frac{4}{3}}$	${\displaystyle \frac{2}{3}}$	${\displaystyle \frac{2}{3}}$	0
$\mathrm{Dynamic}\setminus \mathrm{Profile}$	$11$	$12$	$21$	$22$
$f_1$	2	2	2	2
$f_2$	2	2	2	2

.

By a simple calculation, we have ${\tilde{F}}_1={\delta }_4\left[4444\right]$. Since

$$Ro{w}_4{\left({\tilde{F}}_1\right)}^4=Ro{w}_4{\left({\tilde{F}}_1\right)}^{\top }={\mathbf{1}}_4^{\top },$$

we derive that ${\Xi }_1$ achieves strategy consensus to ${\phi }_{1,e}={\delta }_{16}^{16}$ at $t=1$, and the set of control players of ${\Xi }_1$ is $C_1=Z_1\cup Y_1=\{3,4,5,6\}$.

Similarly, for subnetwork ${\Xi }_2$, we have

$$Z_2=\{{\check{\epsilon }}_{2,1},{\check{\epsilon }}_{2,2}\}=\{3,4\},$$

$$Y_2=\{{\widehat{\epsilon }}_{2,1},{\widehat{\epsilon }}_{2,2}\}=\{5,6\},$$

$$V_2=\{{\epsilon }_2^1,{\epsilon }_2^2,{\epsilon }_2^3,{\epsilon }_2^4,{\epsilon }_2^5\}=\{6,5,8,7,9\}.$$

Fixing the strategy of the players in the set $Z_2\cup Y_2=\{3,4,5,6\}$ to be ${\delta }_2^2$, the SPDs of ${\Xi }_2$ are obtained as

$${\tilde{\phi }}_2(t+1)={\left({\delta }_4^4\right)}^{\top }Bl{k}_{16}\left(F_2\right){\tilde{\phi }}_2\left(t\right)={\tilde{F}}_2{\tilde{\phi }}_2\left(t\right),$$

where ${\tilde{\phi }}_2\left(t\right)={⋉}_{i=3}^5{v}_{{\epsilon }_2^i}\left(t\right)$ and ${\tilde{F}}_2={\left({\delta }_4^4\right)}^{\top }Bl{k}_{16}\left(F_2\right)$. Thus, we have ${\tilde{F}}_2={\delta }_8\left[78388888\right]$. Since $Ro{w}_8{\left({\tilde{F}}_2\right)}^8\ne {\mathbf{1}}_8^{\top }$ and $deg\left({\epsilon }_2^3\right)>deg\left({\epsilon }_2^4\right)=deg\left({\epsilon }_2^5\right)$, we need to control ${\epsilon }_2^3$ such that $v_8\left(t\right)=\theta ={\delta }_2^2$ holds for any $t\in \mathbb{N}$. Then, we obtain

$${\overline{\phi }}_2(t+1)={\left({\delta }_2^2\right)}^{\top }Bl{k}_2\left({\tilde{F}}_2\right){\overline{\phi }}_2\left(t\right)={\overline{F}}_2{\overline{\phi }}_2\left(t\right),$$

where ${\overline{\phi }}_2\left(t\right)={⋉}_{i=4}^5{v}_{{\epsilon }_2^i}\left(t\right)$. By a simple calculation, we have ${\overline{F}}_2={\delta }_4\left[4444\right]$.

It is verified that $Ro{w}_4{\left({\overline{F}}_2\right)}^4=Ro{w}_4{\left({\overline{F}}_2\right)}^1={\mathbf{1}}_2^{\top }$ holds, and thus ${\Xi }_2$ achieves strategy consensus to ${\delta }_{32}^{32}$ at $t=1$. Moreover, the set of control players of ${\Xi }_2$ is $C_2=Z_2\cup Y_2\cup \left\{{\epsilon }_2^3\right\}=\{3,4,5,6,8\}$.

Therefore, the set of control players for the whole network is $C=C_1\cup C_2=\{3,4,5,6,8\}$. Controlling the players in C with a fixed strategy $\theta ={\delta }_2^2$ allows the above NEG to achieve strategy consensus to ${\delta }_{512}^{512}$ at $\tau =1$ (see Figure 2).

Remark 7.

When we define the strategy “1” as “non-confession” and the strategy “2” as “confession”, the game of Example 1 can be regarded as a special case of the prisoner’s dilemma.

Example 2.

Consider an NEG whose network graph is shown in Figure 3. The policy set, payment matrix and SUR are the same as those in Example 1. Select control players to guarantee that the considered NEG achieves strategy consensus to $v_e={\left({\delta }_2^2\right)}^{50}$.

As shown in Figure 3, we divide the network into five parts. For subnetwork ${\Xi }_1$, we have

$$\begin{array}{ccc}\hfill Z_1& =& \{{\check{\epsilon }}_{1,1},{\check{\epsilon }}_{1,2},{\check{\epsilon }}_{1,3},{\check{\epsilon }}_{1,4},{\check{\epsilon }}_{1,5},{\check{\epsilon }}_{1,6},{\check{\epsilon }}_{1,7},{\check{\epsilon }}_{1,8}\}=\{10,11,12,13,19,20,21,22\},\hfill \\ \hfill Y_1& =& \{{\widehat{\epsilon }}_{1,1},{\widehat{\epsilon }}_{1,2},{\widehat{\epsilon }}_{1,3},{\widehat{\epsilon }}_{1,4},{\widehat{\epsilon }}_{1,5},{\widehat{\epsilon }}_{1,6}\}=\{1,2,3,4,5,6\},\hfill \\ \hfill V_1& =& \{{\epsilon }_1^1,{\epsilon }_1^2,{\epsilon }_1^3,{\epsilon }_1^4,{\epsilon }_1^5,{\epsilon }_1^6,{\epsilon }_1^7,{\epsilon }_1^8,{\epsilon }_1^9\}=\{1,2,3,4,5,6,7,8,9\}.\hfill \end{array}$$

Fixing the strategy of the players in the set $Z_1\cup Y_1=\{1,2,3,4,5,6,10,11,12,13,19,20,21,22\}$ to be ${\delta }_2^2$, the SPD of ${\Xi }_1$ is obtained as

$$\begin{array}{ccc}\hfill {\phi }_1(t+1)& =& F_1{\gamma }_1\left(t\right){\phi }_1\left(t\right)\hfill \\ \hfill & =& F_1{\gamma }_1\left(t\right)P_1\left(t\right){⋉}_{i=7}^9{v}_{{\epsilon }_1^i}\left(t\right)\hfill \\ \hfill & =& F_1{\left({\delta }_2^2\right)}^{14}{\tilde{\phi }}_1\left(t\right),\hfill \end{array}$$

where ${\gamma }_1\left(t\right)=v_{10}\left(t\right)v_{11}\left(t\right)v_{12}\left(t\right)v_{13}\left(t\right)v_{19}\left(t\right)v_{20}\left(t\right)v_{21}\left(t\right)v_{22}\left(t\right)$, $P_1\left(t\right)={⋉}_{i=1}^6{v}_i\left(t\right)$ and ${\tilde{\phi }}_1\left(t\right)={⋉}_{i=7}^9{v}_{{\epsilon }_1^i}\left(t\right)$. Thus, we have

$${\tilde{\phi }}_1(t+1)={\left({\left({\delta }_2^2\right)}^6\right)}^{\top }{F}_1{\left({\delta }_2^2\right)}^{14}{\tilde{\phi }}_1\left(t\right)={\tilde{F}}_1{\tilde{\phi }}_1\left(t\right),$$

where ${\tilde{F}}_1={\left({\left({\delta }_2^2\right)}^6\right)}^{\top }{F}_1{\left({\delta }_2^2\right)}^{14}$. By a simple calculation, we have ${\tilde{F}}_1={\delta }_8\left[88888888\right]$. Since

$$Ro{w}_8{\left({\tilde{F}}_1\right)}^8={\mathbf{1}}_8^{\top },$$

according to Theorem 1, we derive that the set of control players of ${\Xi }_1$ is

$$C_1=Z_1\cup Y_1=\{1,2,3,4,5,6,10,11,12,13,19,20,21,22\}.$$

For $C_i$, $i=2,\cdots ,5$, the calculation process is similar to that of $C_1$. To avoid redundancy, the detailed calculations are not presented here. Finally, the set of control players for the whole network is

$$C=\{1,2,3,4,5,6,10,11,12,13,19,20,21,22,35,36,39,43,44,45,46\}.$$

5. Conclusions

In this paper, we analyzed the strategy consensus of large-scale NEGs via network aggregation and pinning control. Based on the semi-tensor product of matrices, we gave the algebraic form of the subnetworked evolutionary games. Then, we established a criterion for achieving strategy consensus in subnetworks and presented a control design procedure based on that criterion. Finally, by using network aggregation, we provided a sufficient condition for NEGs to achieve strategy consensus. Through the network aggregation method, the control set of a larger network can be identified by studying the properties of subnetworks, which can effectively reduce the computational complexity. In future studies, we can focus on finding the optimal aggregation method such that the number of control players in the large network is minimized.