The Analysis of Opinion Evolution and Control Based on the Prisoner’s Dilemma Game in Social Networks

In a social network, a user is greatly inﬂuenced by their neighbors’ opinions, and the user’s opinion updating can be regarded as the prisoner’s dilemma game. In view of such considerations, this paper proposes an opinion evolution and control model based on the prisoner’s dilemma game and gives the corresponding opinion evolution and control algorithm. Under diﬀerent initial positive opinion proportions, diﬀerent opinion control levels, and the same control threshold value and under diﬀerent initial positive opinion proportions, diﬀerent opinion control levels, and diﬀerent opinion control threshold values in a scale-free network, the experiments illustrate the opinion evolution trends and control strategies according to the measures of changing the opinion control levels and opinion control threshold values for network regulators. The experiments show that the lower the initial positive opinion proportion is and the smaller (resp., larger) the control opinion threshold value chosen by the network regulators is, the lower (resp., higher) the opinion control level is; the larger the initial positive opinion proportion is and the larger the control opinion threshold value chosen by the network regulators is, the lower the opinion control level is.


Introduction
In social networks, the process of public opinion propagation is essentially the process of each netizen propagating their opinions in their community. Based on infectious disease models, Daley and Kendall [1] proposed DK model, thought that there is nothing critical in the process of rumors propagation, and analyzed the influence of randomness and certainty on rumor propagation. Maki and omson [2] presented MK model and believed that the first disseminators are controlled to suppress the propagation of rumors. Ising [3] connected the positive and negative of the particle with the positive and negative of the view and put forward Ising model to describe opinion dynamics. Sznajd-Weron and Sznajd [4] further proposed Sznajd model. In their model, they thought that one person will imitate their behavior around people and attract more people to imitate. On the basis of Sznajd model, Deffuant et al. [5,6] established continuous opinion evolution model, i.e., Deffuant model and Hegselmann-Krause (HK) model. Moreno et al. [7,8] made the simulation experiment on MK model and found that the clustering coefficient of network greatly influences the rumor propagation in scale-free networks. ey further proposed a rumor dynamic model and made simulation experiments on homogeneous and heterogeneous networks. Martins [9] considered that all agents have inherent continuous opinions and external discrete behaviors and presented the continuous opinions and discrete actions (CODA) model. DeGroot [10] considered that the individual opinion is updated by the weighted average of all its neighbors' opinions and proposed DeGroot model. Friedkin and Johnsen [11] introduced stubborn individuals, extended the DeGroot model, and proposed the Friedkin-Johnsen (FJ) model. In the FJ model, individual opinion is updated by the weighted average of the convex combination of all its neighbor nodes' opinions and its opinion of innate belief. Gong et al. [12] improved the FJ model, proposed a structural-hole-based approach to control public opinion, and analyzed the influence of ordinary users and structural hole users on opinion evolution. Evidently, these researches mainly focused on the opinion evolution rules and the opinion dynamics environments to depict the public opinion evolution laws in social networks.
To understand the ubiquitous existence of cooperative phenomena in the process of opinion evolution, some scholars introduced the game theory into the process of the opinion evolution and propagation in social networks. Liu et al. [13] thought that opinion propagation process is just the process of different strategy choices and applied the game theory to opinion evolution models. Li et al. [14] studied the rumor propagation based on the evolutionary game and investigated the influences of the penalty coefficient and negative message risk factor on the opinion propagation. Hilbe et al. [15] studied the evolution of extortion in iterated prisoner's dilemma games and found that the extortion is not a stable outcome of evolution but can catalyze the emergence of cooperation. Xu et al. [16] studied the evolution of cooperation structured populations in the context of repeated games by unconditional cooperation, unconditional defection, and extortion strategies and found a nontrivial role of the population structure and the microscopic strategy dynamics in the evolution of cooperation. Nowak and May [17] first adopted the prisoner's dilemma model to study the cooperation evolution of group organizations in rule network and found that the persons with the same strategies are gradually concentrated in a denser group by the self-organization evolution. Tang et al. [18] studied the effects of average degree on cooperation-based prisoner's dilemma game in random networks, small-world networks, and scale-free networks.
To the best of our knowledge, few scholars studied that the opinion evolution and control based on the prisoner's dilemma game.
is paper aims to establish the opinion evolution and control model and corresponding opinion evolution and control algorithm and finds the relationships among the initial positive opinion proportions, opinion control levels, and opinion control threshold values, providing a wide variety of control strategies for network regulators in social networks. e remainder of this paper is organized as follows. Section 2 establishes the opinion evolution and control model and the corresponding algorithm based on prisoner's dilemma game in detail. Section 3 carefully makes some experiments under the different initial positive opinion proportions, different opinion control levels, and same or different opinion control threshold values in scale-free networks. Finally, Section 4 summarizes this paper and gives some future research directions.

Opinion Evolution and Control Based on the Prisoner's Dilemma Game
In a social network, users mainly propagate two different opinions: positive opinion and negative opinion. Considering that a user is greatly influenced by its neighbors' opinions and inspired by the prisoner's dilemma game [15], we take two arbitrary adjacent users as two players, and consider two strategies: positive opinion propagation and negative opinion propagation for them. When two users choose the positive opinion propagation (resp., negative opinion propagation), each player obtains the payoff R (resp., P). When two players choose different strategies, the player who chooses the cooperation strategy gains the payoff S, and the other player obtains the payoff T. In general, the relationship among the four payoffs is T > R > P > S and 2R > T + S. In particular, the donation game is a game where the player who chooses cooperation strategy pays a cost c to provide a benefit b for the other player with 0 < c < b, resulting in the parameters T � b, R � bc, P � 0, and S � c. For simplicity, we set b − c � 1 in the following. For two adjacent users u and v, let Pf uv denote by the payoff of the user u from the user v, and let the values of negative, neutral, and positive opinions of u be denoted by o u � − 1, 0, 1, respectively. Because the users holding neutral opinions are not susceptible to or influenced by others, we only consider the users holding positive and negative opinions and their interaction. en, we can obtain the payoff of user u from user v: where o u , o v ∈ − 1, 1 { }. Denote the set of all neighbors of user u by N(u). en the total payoff of user u, that is, from its all neighbors, denoted by PF u , is the following: Let St u and deg u stand for the strategy, i.e., positive opinion propagation or negative opinion propagation, and the number of the neighbors of user u, respectively. During the game, user u continues to update its strategy based on its neighbors' payoffs and its payoff. en we assume that u randomly chooses a strategy from one of their neighbors, say v with the strategy St v , as their updating strategy, and the updating probability can be defined as In a social network excluding the nodes of neutral opinions, let pp (resp., pn) denote the proportion of the users holding positive opinions (resp., negative opinions), simplified as positive (negative) opinion proportion in the following, at some time. As users continue to change their opinion propagation strategies, the values pp and pn vary 2 Complexity continuously. If the values pp > 0.5, pp � 0.5, and pp < 0.5 in the network, we call the network as positive, neutral, and negative opinion networks, respectively. For the neutral opinion network and negative opinion network, we need to employ some control strategies to change the networks into positive opinion networks. When the proportion of the users holding positive opinions pp (resp., the proportion of the users holding negative opinions 1 − pp) reaches a control threshold value, denoted by L ∈ [0, 1], the network regulators should adopt the control strategy, such as persuading users, providing some positive information to change some users' opinions, to reduce the proportion of negative opinions in the social network. For user u holding a negative opinion, we assume that the penalty variable C(k) that represents opinion control level k for the payoff of u is where c is a parameter, deg stands for the average degree of the network, and k � 0, 1, 2, 3, 4, 5. e higher the opinion control level, the greater the costs that the network regulators need. Obviously, when k � 0, C(k) � 0 implies that the user is not controlled by network regulators in the network. en, we further get a new payoff with opinion control level k for user u, denoted by PF u (k) as follows: where k � 0, 1, 2, 3, 4, 5. Obviously, when k � 0, we have PF u (0) � PF u . Similar to the former user's strategy updating principle, a user u randomly chooses the strategy of one of their neighbors, say v with the strategy St v , as their updating strategy, and the updating probability is Now, we propose an opinion evolution and control algorithm based on prisoner's dilemma game (PDG-OEC algorithm) as follows (Algorithm 1).
For the network regulators, they can control opinion evolution trends in social networks by flexibly changing opinion control levels and opinion control threshold values based on the PDG-OEC algorithm.

Experiment
As the connections of nodes (nodes' degrees) in social networks obey power law distribution, we will adopt the scale-free network [19] to make simulation experiments on the proposed opinion evolution and control model and the PDG-OEC algorithm. To illustrate the effect of network topology on our model, we set the parameters of scale-free networks with 1000 nodes as follows. e number of the initial nodes is m 0 � 20; the number of newly added nodes in each time is m � 1, 3, 5, 7, 9, respectively. On the five scalefree networks, when the proportion of the initial positive opinion (initial proportion) is pp � 0.5, we obtain five opinion evolution trends as shown in Figure 1. By comparison, it is found that the positive opinion proportions are slowly descending to a stable state (even tend to zero) in all networks. e reason is that, under the opinion evolution model based on prisoner's dilemma game, if two adjacent users choose different strategies, then the user who chooses negative opinion propagation gains more payoff compared to another user, resulting in users tending to choose negative opinion propagation in the networks. In fact, compared to true information, rumors (false information) are more easier to propagate from one user to others and to be accepted by users to some extent. When m � 5, the positive opinion proportion in the corresponding network declines the slowest compared with three other networks. For our experiment purpose, the scale-free network with the parameters m 0 � 20 and m � 5 will be chosen. For the newly added nodes, their initial strategies are randomly chosen from positive opinion propagation and negative opinion propagation.
Next, we analyze the opinion evolution trends in the scale-free network under the different initial proportions, opinion control levels, and opinion control threshold value L � 0.5 (see Figure 2). In Figure 2(a), when the initial proportion pp � 0.1, if C � 2, it has no effect on the trend of network opinions; if C � 4, 6, 8, the positive opinion proportions are increasing to a small extent and are controlled within the range of 0.1 to 0.2, respectively; if C � 10, the control effect is significant, and the positive opinion proportion continues to rise more than 0.5. It can be seen that, under the initial proportion pp � 0.1, the network regulators should adopt the opinion control level 5 to achieve positive opinion network. In Figure 2(b), when the initial proportion pp � 0.2, if C � 2, the positive opinion proportion is controlled about 0.15, a stable state; if C � 4, the positive opinion proportion increases continually in excess of 0.5; if C � 6, 8, 10, the positive opinion proportions go up to 1, keeping a stable state, respectively, and the larger C is, the faster it goes up. It can be seen that, under the initial proportion pp � 0.2, the network regulators should adopt the opinion control level 2 to achieve positive opinion network. In Figure 2 For simplicity, in the following experiments, we only consider opinion control levels 1, 3, and 5 (i.e., C � 2, 6, 10), called low, medium, and high control levels, respectively.
For the initial proportion pp � 0.1, from Figure 3 is implies that, under the initial proportion pp � 0.1, the network regulators using the low control level cannot change 1234567891 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 2 0 2 1 2 2 2 3 2 4 2 5 2 6 2 7 2 8 2 9     proportion rises to the stable state at the first speed. is presents that the network regulators should opt L � 0.5 to start to control the network state, a low cost, for C � 10. Combining the above analysis, it is concluded that if the initial proportion pp � 0.1, the network regulators should select medium control level and L � 0.5, which needs a low cost to change the negative opinion network into the positive opinion network.
For the initial proportion pp � 0.2, from Figure 4(a), it is shown that if the network regulators adopt the low control level (C � 2), while L � 0.5 and 0.2, the positive opinion proportions decline about 0.15, a stable state; meanwhile, 1 234567891 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 2 0 2 1 2 2 2 3 2 4 2 5 2 6 2 7 2 8 2 9  is implies that, under the initial proportion pp � 0.3, the network regulators using the low control level cannot change the network opinion trends into a positive opinion network whenever L � 0.5, 0.4, 0.3, 0.2, and 0.1. From Figure 5(b), if the network regulators choose the medium control level (C � 6), while L � 0.5, 0.4, and 0.3, the positive opinion proportions continuously increase over 0.5 and then reach the stable value 1 at almost the same time step; compared with the former three cases, while L � 0.2, the positive opinion proportion slowly goes up to 0.9, a stable state; when L � 0.1, the positive opinion proportion increases to 0.51 at the lowest speed, a stable state. It is concluded that, in this case, the network regulators should choose L � 0.5 to control the network opinion trends into a positive opinion network, a low cost. From Figure 5(c), if the network regulators choose the high control level (C � 10), while L � 0.5 and 0.4, the positive opinion proportions continuously increase over 0.5 and then reach the stable value 1 at almost the same time step; compared with the former two situations, while L � 0.3, 0.2, the positive opinion proportions go up to 0.94 at almost the same growth rate, a stable state; when L � 0.1, the positive opinion proportion slowly increases over 0.5 and does not reach a stable state. It concludes that, in this case, the network regulators should choose L � 0.5 to control the network opinion trends into a positive opinion network, a low cost. Combining the above analysis, it is concluded that if the initial proportion pp � 0.3, the network regulators should select medium control level and L � 0.5, which  Practically speaking, while L � 0.5 and 0.4, the positive opinion proportions rise to the stable value 1 at almost the same speed; when L � 0.3, the positive opinion proportion increases to the stable value 0.98; when L � 0.2, the positive opinion proportion goes up to more than 0.9 but does not reach a stable state; when L � 0.1, the positive opinion proportion presents a linear increase and eventually exceeds 0.5.
is shows that, under this case, the network regulators should choose L � 0.5 to change the network opinion trends into a positive opinion network, a low cost. Combining with the analysis, if the initial proportion pp � 0.4, to reduce network control cost, the network regulators should choose medium control level and L � 0.5 to change the current opinion network into the positive opinion network.
According to the above experiments, it is concluded that, for the network, the lower the initial positive proportion is, the lower (higher) the opinion control level is, while the control opinion threshold value is chosen smaller (larger); the higher the initial positive proportion is, the lower the opinion control level is, while the control opinion threshold value is chosen larger.

Conclusion
As a user's neighbors impact the opinion of the user strongly and the process of users' opinion evolution can be considered the process of the prisoner's dilemma game, this paper proposes an opinion evolution and control model based on the prisoner's dilemma game and gives the corresponding opinion evolution and control algorithm, i.e., the PDG-OEC algorithm. For our purpose, based on the PDG-OEC algorithm, we first analyze the parameter selection of our experimental scale-free network.
en we make two types of simulation experiments in the scale-free networks. Under the different initial proportions, opinion control levels, and the same control threshold value, and under the different positive opinion proportions, opinion control levels, and opinion control threshold values in the scale-free network, the experiments show that if the initial positive proportion is lower, then the opinion control level needs to be lower (higher), while the control opinion threshold value is adopted smaller (larger); if the initial positive proportion is higher, then the opinion control level could be chosen lower, while the control opinion threshold value is chosen larger. e future work is finding more compatible game models to depict the opinion evolution and control in social networks and making some opinion evolution and control game models in the real social networks.

Data Availability
Data sharing is not applicable to this article as no datasets were generated.