An evolutionary approach to extreme individual impact opinions based on time sunk costs

Abstract

Large-scale studies indicate that the distinct approach to opinion fusion employed by extreme agents exerts a more potent influence on overall opinion evolution when compared to regular agents. The presence of extreme agents within the network tends to undermine the development of opinion neutrality, which is harmful to the guidance of online public opinion. Notably, prior research often overlooks the existence of opinion extreme agents in social networks. However, existing researches seldom consider the time sunk cost in the evolution of opinions. Building upon this foundation, we introduce a temporal dimension to the opinion evolution, integrating the time sunk cost with the opinion evolution process. Furthermore, we devise an agent partitioning method that categorizes agents into four states based on their opinion values: watch state, subjective state, firm state, and extreme state, with extreme state agents generally expressing radical opinions. We constructed an agent network based on the phenomenon of time sunk costs and proposed a model for the evolution of extreme opinions in this network. Our study found that the information sharing among extreme agents significantly influences the extremization of opinions in various networks. After restricting the exchange of opinions on extreme agents, the number of extreme agents in the network decreased by 40% to 50% compared to the initial situation. Additionally, we also discovered that imposing restrictions on extreme agents in the early stages can help increase the possibility of network opinions moving towards neutral positions. When restriction of extreme agents(REA) was performed at the beginning of the experiment compared to REA in the midway of the experiment, the final number of extreme state agents decreased by 15.57%. The results show that extreme agents have a great influence on the spread and evolution of extreme opinions on platforms.

Keywords

Time sunk costs extremists opinion dynamics bounded confidence model social networks opinion evolution

1. Introduction

Public opinion is composed of thousands of individual opinions. The formation and contention of public opinion must rely on the flow, diffusion and gathering of opinion information [1]. From the perspective of information dissemination, during the traditional media era, news media held most channels of information dissemination and had the power of speech. It is easier to create an organized and influential public opinion atmosphere [2]. Therefore, political parties or groups often set agendas, construct frameworks, provide mainstream opinions and form news opinions through the news media. This guides people toward the ideologies or values they advocate and makes public opinion evolve toward the expected direction [3]. However, in the Internet era, this situation fundamentally changed. With the growing popularity of Internet technology applications, the battlefield of public opinion warfare will inevitably extend from traditional media to the Internet domain. The emerging Internet domain brought about convenient information production technology and content, multiple information distribution channels, and a fast flow of various news information and opinions. It led to the overthrow of the information monopoly status of traditional news media production and distribution, resulting in significant changes in media ecology, audience demand, and public opinion dissemination methods [4]. Its impact on traditional journalism, particularly in the areas of public opinion and issue setting, expanded considerably. Influenced by complex factors at home and abroad, the struggle and competition in the ideological field become increasingly intense. Cyberspace become an important position for competition among various forces. The Internet become the main battlefield of public opinion struggle [5].

Past research on public opinion traditionally held that individual opinions interact with each other in ways that either mutually reinforce, absorb, or ignore one another. However, recent experimental results suggest that there can also be mutual exclusion and antagonism in the process of public opinion dissemination and influence. Extremists are the main group responsible for this type of opinion evolution. Extremists are individuals who are obsessed with their opinions and are too stubborn to consider opinions based on internal psychological factors or external information. Individuals tend to embrace beliefs aligned with their desires, often disregarding factual accuracy [6]. Extremists have a strong preference for theoretical systems and abstract thinking. They would even distort the truth to prove their logic and deny the perceived truth. To achieve their goals, the extremists may resort to ignoring common sense and even falsifying facts. Extremists also quickly radicalize opposing opinions. Their unwavering belief in their own ideas leads them to express themselves assertively, often adopting strong language and positions that can elicit feelings of intense irritation in those they encounter [7]. But this overly intense performance often leads to a different effect: the contactee will experience a psychological phenomenon of extreme impatience or rebellion, forming the opposite opinion of the person who is trying to change their mind.

In the domain of social media, achieving opinion neutrality is a shared objective pursued by both governmental bodies and social media corporations. Opinion neutrality fosters social harmony, promotes the development of civilization, and enhances the overall environment of social media, thereby augmenting user engagement and frequency [8]. In addition, the study of opinion evolution has a potential role in advertising promotion [9, 10], political canvassing [11], and opinion analysis [12]. Controlling online public opinion is becoming more and more important. As a government department, it needs to strengthen the control of online public opinion. The first thing it needs to do is to keep up with media information. It should set up a corresponding organization with competent forces, pay attention to all kinds of media in real time, and pay attention to the events and hot spots of public concern in real time [13].

Past researchers considered extremist studies, Zaller proposed the RAS model as the initial demonstration of this phenomenon [14]. The RAS model does not directly study the interaction process among individuals, but focuses on the influence of the external opinion environment on a particular individual [15]. Many researchers have since developed similar models, however, these methods seldom explore extremists individually. In order to facilitate a separate study of extremists, we define four different states of agents with different opinions. In this work, we focus on extreme state agents of extreme opinions.

In the real world, time sunk costs (TSCs) commonly refers to the cost that people have invested and cannot be recovered, the time sunk cost effect refers to the fact that people choose to persist in their previous behavior in order to avoid the loss of time costs previously invested [16]. We find that few people have conducted research in this direction. In order to study the evolution of opinions in a more realistic way, we propose an opinion dynamics model based on time sunk costs (ODTSC). The evolution rule of social opinions mainly depends on the random connection between agents [17, 18]. In ODTSC, agents with connections will increase their sunk cost if they do not change their opinions after their interaction. Secondly, even if they do not participate in the interaction, the sunk cost will increase after each interaction round driven by time. This sunk cost value will reduce the willingness to change opinions in the subsequent interaction.

The continuous opinion evolution model presents a linear representation of opinion values, which is not conducive to our intuitive observation of extreme agents. We need a way to better identify the extremes among agents. To facilitate understanding the process of opinion evolution, we define four different states of opinion: watch state, subjective state, firm state, and extreme state, based on the values of opinions. We simulate different generative networks to explore how extremists influence social networks.

Specifically, in this paper, we summarizes the main contributions as follows.

We propose a new method that links the evolution of opinions with time sunk costs. It introduces a time mechanism during the evolution process.

Defines four different role states based on the opinion values of agents in the social network(i.e. watch state agent, subjective state agent, firm state agent and extreme state agent), and conducts a focused experimental analysis for the extreme agents of them.

We propose a new model of opinion dynamics based on time sunk costs. Based on this model, we construct an agent network and do a research on extreme state agents in this network.

Through detailed simulation experiments, we prove the harm of extreme state agents to opinion neutralization. The earlier restrictions are imposed on Extreme state agents (ESAs), the more neutral the opinion of the entire social network will be. Delaying REA creates significant costs that are difficult to compensate for.

The rest of this paper is as follows: Section 2 introduces the theory of the bounded confidence model and sunk costs. Section 3 presents a new opinion dynamics model based on the time sunk cost effect. Section 4 conducts simulation experiments by computer, it demonstrates the results and detailed analysis. Section 5 summarizes the conclusions and provides some ideas and suggestions for future research.

2. Related work

2.1 Bounded confidence model

The bounded confidence model is an extension of the DeGroot model, which is considered to be the general classical model [19]. The DeGroot model describes the process of opinion evolution. The common opinion of the people around an agent and the weight of these people for this opinion interaction agent form the agent’s consensus. the DeGroot model [20] has the following opinion evolution process. Let $O_{i}(t)$ represent the opinion of agent $x_{i}$ at time $t$ , the adjusted opinion of $x_{i}$ at time $t+1$ is:

$\displaystyle O_{i}(t+1)=w_{i1}O_{1}(t)+w_{i2}O_{2}(t)+\ldots+w_{im}O_{m}(t),t% =0,1,2\ldots$ (1)

$w_{ij}$ is the weight that $x_{i}$ assigns to $x_{j}$ , which satisfies $\sum_{j=1}^{m}w_{ij}=1,i=1,\ldots,m,j=1,\ldots,m,i\neq j$ . The DeGroot model has been well-studied and has extensions in different directions. If the network representing the relationships is connected, then the convergence of network opinions is equivalent to reaching consensus. The bounded confidence model builds on the ideas of the DeGroot model with some updates. The bounded confidence model considers the influence of psychological factors in interaction rules. An agent’s opinion is only influenced by the opinions of others whose difference in opinion does not exceed a fixed confidence level [21]. If there is a significant difference in opinion between two agents, they will not bother to interact with each other. In extreme cases, individuals may not even attempt to merge their opinions, resulting in the disregard of ideas that are too far removed from their own. The most representative bounded confidence models are the Deffuant-Weisbuch (DW) model [22] and the Hegselmann-Krause (HK) model [23].

In the DW model, agent $x_{i}$ will randomly choose one agent $x_{j}$ with whom they are associated to interact with their opinions in each round. If the difference in opinion between $x_{i}$ and $x_{j}$ is less than the confidence threshold, they will perform an opinion update based on each other’s opinions, as defined by the following rules:

$\displaystyle O_{i}(t+1)=O_{i}(t)+\mu\cdot(O_{j}(t)-O_{i}(t))$ (2)

The two agents accept each other’s opinions when the difference between their opinions is less than or equal to this value, i.e., $|O_{i}(t)-O_{j}(t)|\leqslant r$ , where $r$ is the confidence level in the bounded confidence model. The parameter $\mu$ is a predetermined learning coefficient. The confidence level reflects the degree of acceptance of the other opinions by the agents. Generally, the value of confidence is set to (0, 0.5]. In the traditional DW model [22], all agents have the same bounded confidence threshold r and the same opinion learning rate $\mu$ , and such a social network environment is called a homogeneous network. Sobkowicz [24] introduced a heterogeneous bounded confidence threshold in the network, which resulted in significant changes in opinion evolution. The DeGroot model involves incorporating group opinions, while the DW model relies on individual agent-to-agent opinion interactions.

The biggest difference between the HK model and the DW model is the number of other agents in the model when agents conduct opinion interactions. The original DW model typically considers only one person’s idea and opinion when performing opinion fusion. In contrast, the HK model unifies all contacts whose opinion difference is less than the bounded confidence level at that time. Then, it shapes their own ideas based on the opinion difference and their original opinion. This process is called the communication interaction means and mechanism. The following equation gives the HK model:

$\displaystyle O_{i}(t+1)=\frac{1}{|N_{i}(t)|}\sum_{j\in N_{i}(t)}O_{j}(t)$ (3)

where $N_{i}(t)$ is the set of the number of neighbors of agent $x_{i}$ at discrete time $t$ . The neighbors are a set of agents with opinions within a confidence radius, i.e., $|O_{j}(t)-O_{i}(t)|\leqslant r$ , which includes $x_{i}$ itself. $r$ is the confidence radius if all agents have the same confidence radius, then such an HK model is called a homogeneous bounded confidence HK model. Previous research has found that individual opinions and the network structures in which they are embedded can interact and co-evolve to obtain consistency or fragmentation of group opinion [25]. Conversely, we call it the heterogeneous bounded confidence HK model [26].

But as the study progressed, researchers found that people do not always agree with and accept others’ opinions or ignore those that are very different from their own, and they also oppose and reject those extreme opinions. However, the previous DW model only considered agents’ acceptance of each other, but not their opposition to each other. When agents are confronted with other agents who are completely opposite and stubborn to them, they will show behaviors that do not accept and deepen their own opinions instead. In order to better study agents’ aversion to the extreme opinions of other extreme agents, we performed a opinion state division for different opinion values.

2.2 Sunk costs

Sunk costs, also known as retrospective costs, are costs that have already been incurred and cannot be recovered. Arkes and Blumer [27] first proposed sunk costs on the basis of social psychology research. The sunk cost effect is manifested in a greater tendency to continue an endeavor once an investment in money, effort, or time has been made. Persistence owing to prior investment can occur even when the future consequences of an alternative course of action are more favorable [28]. In a field study, researchers found that customers who initially paid more for a season subscription to a theater series attended more plays in the following 6 months. This could be attributed to their higher sunk cost in the season tickets. Similarly, agents who spend more time and effort initially forming a particular opinion tend to pay more attention to information that confirms that opinion in the future. As a result, they are more likely to continue to believe it. Under a rational setting, they should not be considered when evaluating alternative courses of action. However, as a matter of fact, their existence is a driver of organizational inertia and inhibits change [29]. Thereby, sunk costs are generally considered a barrier to innovation [30].

There are many types of sunk costs, but in most cases, in the evolution of opinions, there will only be the impact of time sunk costs. Time is involved in every activity we engage in, and cannot be recovered. Navarro and Fantino [31] provided compelling evidence for the sunk cost effect, particularly in their behavioral experiments where participants invested real time in an activity. The participants completed a jigsaw puzzle with either a short or long time invested, with task completion being either obligatory or voluntary, to assess the role of personal responsibility in the sunk cost effect. Overall, the sunk cost effect was observed, and its occurrence was dependent on a high level of personal responsibility. In the evolution of opinions, this time sunk effect depends on the individual’s attention to the event and the number of interactions.

Figure 1.

The framework of opinion dynamics model based on time sunk costs.

3. ODTSC

3.1 Agent state

Agents in a social network continuously change their opinions through interactions with other agents. Furthermore, agents in different states behave distinctively when interacting with different agents. The opinion value is the centralized reflection of agents’ opinions in the social network [32]. We define four different types of opinion states as follows:

Watch state agents (WSAs) are the most placid group of people in the social network. They are just beginning to understand the event or have not yet formed their own ideas, but have a general understanding of the event. The opinion value of WSAs is $-0.3\leqslant O_{i}\leqslant 0.3$ , so they are willing to listen to others’ suggestions, adopt others’ opinions, and accept different opinions.

Subjective state agents (SSAs) are individuals in a social network who have formed their own opinions after obtaining a certain degree of understanding of an event. These individuals will indicate their initial positions during the communication process. They are good guides for the WSAs and are able to tolerate others’ ideas and opinions in the face of other different opinions. They have a opinion value of $-0.7\leqslant O_{i}<-0.3$ or $0.3<O_{i}\leqslant 0.7$ , and they are the group with the largest number of state agents in the initial network.

Firm state agents (FSAs) are hard to be influenced by others, but they are not incapable of changing their opinions. They are the agents closest to the extreme states and the group of agents most likely to be extremists, with firm attitudes and opinion values of $-0.9\leqslant O_{i}<-0.7$ or $0.7<O_{i}\leqslant 0.9$ .

ESAs are the agents with opinion values $O_{i}<-0.9$ or $O_{i}>0.9$ , also called extremists, whose ideas have become too entrenched to be changed [33, 34]. Their influence on the evolution of social network opinions is enormous. Therefore, this article focuses on studying these extreme agents.

[H] : agent partitioning algorithmInput: $O_{i}(0)$ : the initial opinion value of each agentOutput: $S_{i}(0)$ : the state of each agents; number of agents in each stateinitialization

$v_{i}$ in V $-0.3\leqslant O_{i}\leqslant 0.3$ set $S_{i}$ watch state agent;number of watch state agents $++$ ;continue; $-0.7\leqslant O_{i}<-0.3$ or $0.3<O_{i}\leqslant 0.7$ set $S_{i}$ subjective state agent number of subjective state agents $++$ continue $-0.9\leqslant O_{i}<-0.7$ or $0.7<O_{i}\leqslant 0.9$ set $S_{i}$ firm state agent number of firm state agents $++$ continue $O_{i}<-0.9$ or $O_{i}>0.9$ set $S_{i}$ extreme state agent number of extreme state agents $++$ continue;

Algorithm 3.1 demonstrates the method of partitioning agent states. Generally, to reach opinion unity, agents exchange opinions with other agents in social networks for the purpose of mutual understanding to reach opinion unity. The bounded confidence model takes into account a specific limitation in the formation of individual opinions, only the opinions of agents within the confidence threshold are considered. However, the exchange with ESAs breaks this rule. When there is opinion interaction between agents beyond the confidence level, the extremists influence the ideas of another agent in reverse. In our analysis, we observed the interactions between agents in each state. When the WSA interacts with the SSA, it evolves towards an intermediate opinion between them. This process has a positive effect on opinion neutralization. Similarly, when the WAS interacts with the FSA, it has a facilitative effect. On the other hand, the interaction between the WSA and the ESA evolves in the opposite direction outside the interval between their opinions. This has a reverse negative effect on opinion neutrality. When interacting with a FSA, the SSA has a positive effect on evolving towards a neutral opinion. Conversely, when interacting with an ESA, the SSA has a negative effect on evolving towards a neutral opinion. The FSA has some negative effect after interacting with the ESA. In this model, extreme agents have a negative effect on opinion neutralization. We incorporated the influence factor of time sunk costs, which had not been considered by previous academic workers. In a simulated network with a social structure, we discuss the interaction of opinions between agents with extreme agents. There are two possibilities for agents to merge their opinions with other agents: neutral evolution of opinions and extreme evolution of opinions. As shown in Fig. 2, the fusion of opinions between A state agents and B state agents is developing towards opinion neutrality. Similarly, there are also A state agents and C state agents, A state agents and A state agents. However, the fusion of opinions between D state agents and other state agents is developing in the direction of extreme opinions. Specially, there is no interaction between two extreme states agents.

Figure 2.

Interaction between different state agents.agentA: WSA. agentB: SSA. agentc: FSA. agentD: ESA.

3.2 Opinions evolution under time sunk cost

Agents in different states exhibit different behaviors when interacting with other agents. During opinion interaction, when the two agents involved are distant, the sunk cost effect can influence their decision-making as they integrate information. If they choose to accept the opposing agent’s opinion, they would need to abandon their previous preparations for their own opinion, and the cost of this preparation increases with time and the number of interactions. These agents are hesitant to believe each other’s opinions during their interactions, leading to a reduction in the reference to others’ ideas. This ultimately results in a decrease in the learning efficiency of both parties.

Figure 3.

An example of opinion evolution structure under time sunk cost.

3.2.1 Sunk coefficient

An example of opinion evolution structure under time sunk cost is shown in the Fig. 3. As shown in Fig. 3, Agent A obtains information and then continues to gather relevant information on the Internet to form an initial opinion. Agent A interacts with other agents in the community who share similar opinions, such as Agent B, to update their opinions. This process continues over time and incurs a time cost. When Agent A interacts with Agent C, who holds distant opinions, Agent A’s opinions may be swayed. If Agent A decides to accept the new opinion, he must gives up the time he have invested in this matter, that is, the time sunk cost. We use $R_{i}(t)$ to quantify the time sunk cost of agents, $R_{i}(t)$ is formed based on the interaction frequency and time steps of agents. Then, based on the proportion of time sunk impact factor and the total number of time rounds, we use sunk coefficient $S_{i}(t)$ to represent the reduced willingness to change opinions. We express the equation of $S_{i}(t)$ as follow:

$\displaystyle S_{i}(t)=1-\frac{1}{2}\frac{\ln R_{i}(t)}{\ln\#R}$ (4)

where $\#R$ denotes the total number of opinion exchanges. In this study, sunk coefficient is dependent on the time and agents’ activity. Time sunk cost describes that people will consider the past time input when making decisions, the use of $S_{i}(t)$ makes up for the gap in the past research on the opinion evolution. The use of $S_{i}(t)$ enables the ODTSC to be closer to reality.

3.2.2 Opinion agency network

The network graph in a social network represents a network of relations. Let the undirected graph $G(V,E,O)$ be a social network, where each point represents an agent, $V=\{v_{1},v_{2},\ldots,v_{n}\}$ is the set of $n$ agents, $E=\{e_{1},e_{2},\ldots,e_{m}\}$ is the set of $m$ edges, and the opinion of agent $i$ at time point $t$ is denoted as $O_{i}(t)$ , then the set of opinions of all agents at time $t$ is denoted as $O=\{O_{1}(t),O_{2}(t),\ldots,O_{n}(t),O_{j}(t)\in[-1,1],j\in V\}$ , with $t=0$ being the initial opinion of that agent. The closer the opinion value is to 0, the more neutral the opinion. Conversely, if it is closer to $-$ 1 or 1, the agent’s opinion is more intense and polarized. Opinions are divided by 0 and categorized into negative opinions from [ $-$ 1, 0) and positive opinions from (0, 1]. If two opinions are both positive or negative, we call them similar opinions. Conversely, if two opinions are a positive opinion and a reverse opinion, we call them distant opinions. Edges represent the connections between agents, if two agents are related, it means that the corresponding points are connected to each other. The set of edges between all points $V$ in the graph is denoted by $E$ . The effect of extremists on the evolution of opinions in social networks is studied in different networks.

The initialization process includes the generation of agents’ initial opinions and the generation of network relations, which both form the initial network. The initial opinion values are generated randomly. The initial time sunk impact factor $R_{i}(t)=1,i=1,2,\ldots,2000$ is set to 1. As shown in Fig. 1, the first step is to generate a community by using the above network. In the community, two connected agents are randomly selected to express their opinions to each other. If these two agents interact with each other, they update their opinions. If their opinions are out of each other’s acceptance range, they return to the community to continue selecting agents. During the process of updating agents’ opinions, the time sunk cost effect affects them. If the two agents have similar opinions, their interaction is not affected. However, if they have distant opinions, their interaction becomes more difficult.

3.2.3 Opinion evolution process in opinion agency network

This model considers both learning rate and individual time sunk cost. The first step is to select agents for opinion interaction. Then, calculate the time sunk coefficients and perform opinion fusion. Finally, the next round is initiated by increasing the time cost impact factor. Next is the specific fusion process of various state agents. The first case of opinion fusion process involves agents that belong to the same watch state. WSAs are easily influenced, and they exchange opinions with each other. Assuming that opinion values of two agents at moment t are $O_{i}(t)$ and $O_{j}(t)$ , and they are distant from each other. Their opinion evolution is formed by multiplying the average of their opinions by a sunk coefficient to form their opinions for the next time point. Specially, the agent with higher opinion value has higher Learning rate. The expressions of their opinion fusion process are as follows:

$\displaystyle\left\{\begin{array}[]{l}O_{i}(t+1)=\left[1-\frac{1}{2}\left(% \frac{\ln R_{i}(t)}{\ln\#R}\right)\right](1+\rho)\frac{O_{i}(t)+O_{j}(t)}{2}\\ \\ O_{j}(t+1)=\left[1-\frac{1}{2}\left(\frac{\ln R_{j}(t)}{\ln\#R}\right)\right]% \frac{O_{i}(t)+O_{j}(t)}{2}\\ \end{array}\right.$ (5)

where $|O_{i}(t)|>|O_{j}(t)|$ , $R_{i}(t)$ denotes the number of similar opinion evolution of agent $i$ at time $t$ . If the opinion does not change direction after each interaction between agents, it remains either positive or negative before and after the evolution of the opinion, i.e., $O_{i}(t)*O_{i}(t+1)>0$ , it means that the time sunk cost is continuing to increase, then $R_{i}(t+1)=R_{i}(t)+1$ , and conversely if the opinion changes in direction, it means before and after the evolution of opinions, one is positive and the other is negative, i.e. $O_{i}(t)*O_{i}(t+1)<0$ , then $R_{i}(t+1)$ resets. $\#R$ denotes the total number of opinion exchanges. $\rho=0.4$ represents the trust level of the higher opinion value between the two WSAs towards themselves. And if the two agents are similar in attitude, they do not need to abandon previous efforts when interacting. The opinion fusion expression is as follows.

$\displaystyle\left\{\begin{array}[]{l}O_{i}(t+1)=(1+\rho)\frac{O_{i}(t)+O_{j}(% t)}{2}\\ O_{j}(t+1)=\frac{O_{i}(t)+O_{j}(t)}{2}\end{array}\right.$ (6)

If there is no change in the direction of their opinions, there is no time sunk cost to consider. In the second case, WSA interacts with SSA. At this point, the WSA is passive. He is not as well informed about the situation as the SSA, and his own trust is not as high as the SSA. The WSA chooses to trust the opinion of the SSA, in this case, SSA holds their own opinions while WSA listens to SSA. WSA absorbs the opinions to a certain extent based on the difference in opinion between the two and combined his original opinion to form a new one:

$\displaystyle\left\{\begin{array}[]{l}O_{i}(t+1)=O_{i}(t)+\mu(O_{j}(t)-O_{i}(t% ))\\ O_{j}(t+1)=O_{j}(t)\end{array}\right.$ (7)

This is the evolutionary process when two agents have a similar attitude. Here, agent $v_{j}$ represents the subjective state, agent $v_{i}$ represents the watch state, and $\mu$ represents the learning rate of the watch agent, which is usually set to 0.4 [22]. The WSA will adjust its opinion based on the difference in opinion with the SSA, along with the learning rate. Meanwhile, the SSA will not modify its opinion when dealing with the WSA. The subjective agent’s opinion does not change when facing the watch agent. If the two agents are distant attitudes, it is necessary to consider the time sunk cost:

$\displaystyle\left\{\begin{array}[]{l}O_{i}(t+1)=O_{i}(t)+\mu\left[1-\frac{1}{% 2}\left(\frac{\ln R_{i}(t)}{\ln\#R}\right)\right](O_{j}(t)-O_{i}(t))\\ O_{j}(t+1)=O_{j}(t)\end{array}\right.$ (8)

The firm state is more contagious than the subjective state and has a more assertive attitude. When the WSA interacts with the FSA, it has the same evolutionary rules as when the SSA interacts. Then there is the case of the WSA and the ESA. The interaction with the ESA does not neutralize the opinion. If $|O_{j}(t)|>0.9$ , the WSA $v_{i}$ and the ESA $v_{j}$ do not allow the opinion of the WSA to evolve in the direction of the ESA’s opinion due to conflicting opinions. Instead, it will allow the opinions to move towards more extreme. In particular, agents pay more attention to the number of ESAs surrounding them when they interact with ESAs. The surrounding ESAs transmit negative emotions through connections. Each agent further resists the neutralization of opinions based on the proportion of surrounding ESAs.

$\displaystyle\left\{\begin{array}[]{l}O_{i}(t+1)=O_{i}(t)-\mu\frac{N_{i}+\sum_% {k=1}^{N_{i}}\delta_{k}(t)}{N_{i}}(O_{j}(t)-O_{i}(t))\\ O_{j}(t+1)=O_{j}(t)\end{array}\right.$ (9)

This is the process of opinion evolution when similar attitudes are present, where $\delta=\left\{{\begin{array}[]{ll}{1,}&{{O_{k}}(t)>|\eta|}\\ {0,}&\text{otherwise}\\ \end{array}}\right.$ , $\eta=0.9$ is the threshold for determining an agent’s extreme state. When an agent $v_{j}$ has a watch value $O_{j}(t)>0.9$ or $O_{j}(t)<-0.9$ at moment $t$ , this agent is to be in an extreme state. $v_{i}$ is the WSA, $v_{j}$ is the ESA, and $N_{i}$ represents the number of neighbors of agent $i$ . The more extreme agents around, the more the non-extreme agents will be affected. Extremists remain immovable and steadfast in their own opinions, even when interacting with other individuals. In the case of distant opinions, the time sunk cost effect is also added:

$\displaystyle\left\{\begin{array}[]{l}O_{i}(t+1)=O_{i}(t)-\mu\left[1-\frac{1}{% 2}\left(\frac{\ln R_{i}(t)}{\ln\#R}\right)\right]\frac{N_{i}+\sum_{k=1}^{N_{i}% }\delta_{k}(t)}{N_{i}}(O_{j}(t)-O_{i}(t))\\ O_{j}(t+1)=O_{j}(t)\end{array}\right.$ (10)

The above formula describes the interaction process between the WSA and all state agents. Additionally, it introduces Algorithm 3.2.3, a proposed model algorithm that integrates individual and others’ opinions and incorporates time sunk costs.

When both agents are subjective, they interact based on their own opinions. Both agents then revise their opinions based on each other’s opinions. If the opinions are similar, we use the DW model [22]:

$\displaystyle O_{i}(t+1)=O_{i}(t)+\theta(O_{j}(t)-O_{i}(t))$ (11)

FSAs also aim to exchange opinions with SSAs who share similar attitudes. Their update methods are consistent with those between subjective agents, where both sides update their opinions based on each other’s opinions. However, based on the fact that the agents already have certain knowledge and ideas, the learning rate will not be as high as that of the WSAs. When the two agents are distant attitude, both agents need to consider the time sunk cost:

$\displaystyle O_{i}(t+1)=O_{i}(t)+\theta\left[1-\frac{1}{2}\left(\frac{\ln R_{% i}(t)}{\ln\#R}\right)\right](O_{j}(t)-O_{i}(t))$ (12)

[H] : WSA opinion calculation algorithmInput: $O_{i}(0)$ : original $O_{i}(0),O_{j}(0),R_{i}(0)$ , $\mu$ : WSA learning rate, $\#R$ : total number of agent opinion exchanges, $N_{i}$ , $\rho$ Output: calculated opinion $O_{i}(t+1),O_{j}(t+1)$

$|O_{i}(t)|\leqslant 0.3\cap|O_{j}(t)|\leqslant 0.3$ $O_{i}(t)*O_{j}(t)>0$ calculate $O_{i}(t+1)$ and $O_{j}(t+1)$ by Eq. (6) calculate $O_{i}(t+1)$ and $O_{j}(t+1)$ by Eq. (5) $|O_{i}(t)|\leqslant 0.3\cap 0.3<|O_{j}(t)|\leqslant 0.9$ $O_{i}(t)*O_{j}(t)>0$ calculate $O_{i}(t+1)$ and $O_{j}(t+1)$ by Eq. (7) calculate $O_{i}(t+1)$ and $O_{j}(t+1)$ by Eq. (8) $|O_{i}(t)|\leqslant 0.3\cap|O_{j}(t)|>0.9$ $O_{i}(t)*O_{j}(t)>0$ calculate $O_{i}(t+1)$ and $O_{j}(t+1)$ by Eq. (9) calculate $O_{i}(t+1)$ and $O_{j}(t+1)$ by Eq. (10) $R_{i}(t+1)$ = $R_{i}(t)$ + 1

A FSA already has a clear attitude and will only interact with another FSA with a similar opinion. There will not be any fusion of opinions with a FSA who has a distant opinion. When a FSA $v_{i}$ and another FSA $v_{j}$ try to make contact, the opinion interaction will take place when and only when the two have similar opinion attitudes, i.e., $O_{i}(t)*O_{j}(t)>0$ . Other cases will be such that the two will not consider each other’s attitudes because the opinion gap is too large. The ESA has a stronger idea and does not change his or her mind, but will influence the SSA in reverse:

$\displaystyle\left\{\begin{array}[]{l}O_{i}(t+1)=O_{i}(t)-\theta\left[1-\frac{% 1}{2}\left(\frac{\ln R_{i}(t)}{\ln\#R}\right)\right]\frac{N_{i}+\sum_{k=1}^{N_% {i}}\delta_{k}(t)}{N_{i}}(O_{j}(t)-O_{i}(t))\\ O_{j}(t+1)=O_{j}(t)\end{array}\right.$ (13)

where $O_{i}$ is the opinion of the SSA, $O_{j}$ is the opinion of the ESA, and $\theta$ represents the learning rate of the subjective agent, which is set to 0.2 here. As with the WSA and the ESA, the more ESAs around, i.e., the larger $\sum_{k=1}^{N_{i}}\delta_{k}(t)$ , the greater the negative influence on the SSA. In the case of similar attitudes, there is no need to consider the time sunk cost:

$\displaystyle\left\{\begin{array}[]{l}O_{i}(t+1)=O_{i}(t)-\theta\frac{N_{i}+% \sum_{k=1}^{N_{i}}\delta_{k}(t)}{N_{i}}(O_{j}(t)-O_{i}(t))\\ O_{j}(t+1)=O_{j}(t)\end{array}\right.$ (14)

Table 1

The initial and final opinion of agent in subjective state through different evolution with agents in different states

Agent	$O_{i}(0)$	$v_{1}\&v_{2}$	$v_{1}\&v_{3}$	$v_{1}\&v_{4}$	$v_{1}\&v_{5}$
$v_{1}$	0.541	0.541	0.480	0.600	0.444
$v_{2}$	$-$ 0.081	0.148	–	–	–
$v_{3}$	$-$ 0.345	–	$-$ 0.199	–	–
$v_{4}$	0.837	–	–	0.789	–
$v_{5}$	0.976	–	–	–	0.976

As shown in Table 1, the first column of the table lists the agent nodes in various states throughout the social network. The second column shows the initial opinions of the agents in various states. We randomly choose initial opinions from the range of [ $-$ 1, 1] and select one neighbor in different states. Among them, agents $v_{1},v_{2}$ are WSAs, agent $v_{3}$ is a SSA, agent $v_{4}$ is a FSA, and agent $v_{5}$ is an ESA. The third column shows the two-sided opinions of agents $v_{1}$ and $v_{2}$ after their interaction. The fourth column is the two-sided opinions of agents $v_{1}$ and $v_{3}$ after their interaction. The fifth column is the two-sided opinions after the interaction between agents $v_{1}$ and $v_{4}$ . The sixth column is the two-sided opinions of agents $v_{1}$ and $v_{5}$ after their interaction. Agents in the firm state are not negatively influenced by extreme agent who have similar attitudes. However, agents in the extreme state with distant attitudes can negatively influence the firm agent to such an extent that they make both sides of the distant attitude extreme agents. There will be no opinion interaction between the extreme agents. The following is the process of interaction between the FSA and the ESA of the distant attitude.

$\displaystyle\left\{\begin{array}[]{l}O_{i}(t+1)=C\\ O_{j}(t+1)=O_{j}(t)\end{array}\right.$ (15)

where ${{C}}=\left\{{\begin{array}[]{ll}{1,}&{{O_{i}}(t)>0}\\ {-1,}&\text{otherwise}\end{array}}\right.$ . When the FSA is the positive attitude and the ESA is the negative attitude, the FSA’s opinion will move toward the positive extreme. When the FSA has a negative attitude and the ESA has a positive attitude, the opinion of the FSA moves toward the negative extreme. During the interaction, the FSA $O_{i}$ also polarizes their opinion toward the extreme agent. In contrast, ESA $O_{j}$ will maintain their extreme opinions at moment $t+1$ .

4. Simulation and results

In this section, to demonstrate the effect of extremists on opinion polarization, we conducted simulation experiments to represent the values of agents’ opinions of people at different times and analyzed the results of the experiments. In Section 4.2, we examine the impact of introducing REA in different social networks. In Section 4.3, we alter the timing of introducing REA. Within Section 4.3, we investigate the costs associated with compensating for the delayed implementation of REA. Our source code are publicly. available.1

4.1 Simulation design

In the interaction of group opinions, different social networks tend to have different evolutionary processes. To explore the influence of extremists in different communities, we utilize three initial social networks: Erdös-Rényi random graph [35], scale-free network [36] and small world network [37] all consisting of 2,000 agents and 200,000 connections. These networks are used to demonstrate the harmfulness of extremists in the evolution of network opinions towards neutralization.

First, experiment 1 demonstrates the evolution of the overall social network perspective at the macro level to demonstrate the harmfulness of extreme agents in various social networks. Specifically, we set the same initial node $v=2000$ and the same connected edge $e=200000$ in three networks. The specific experiment is to restrict the connections between extreme agents (REA) and others, reduce the output of extreme agents’ opinions, then observe the evolution of the overall opinions.

Second, the start time of REA is an important time point. If it does not produce significant effects after using REA, the control effect of his application to public opinion will be inefficient. Therefore, we designed experiment 2: a restriction on extreme agents at different time points in the evolution of public opinion to determine the optimal REA time point by observing the convergence of opinions in the social network. We repeated experiment 2 500 times to reduce random errors and presented the numerical averages of all experimental results in a visualized form.

To reduce the evolution of opinion polarization and alert social media to concerns regarding the control of public opinion, it is important to control the intervention time of REA. To achieve this, it is necessary to study the impact of the intervention time of REA on the polarization of overall social network opinions. If delaying the intervention time of REA causes irreversible deterioration of public opinion on social media and public platforms, it leads to the need for platforms to use coercive means to control public opinion. This would inevitably make the platforms’ credibility decline, and such a consequence would be unbearable. Experiment 3 consisted two sets of comparison experiments. The first set intervened in the event at the pre-fermentation stage of public opinion. the second set postponed this REA to intervene and gradually increased the intervention to achieve a similar effect of opinion control as the first set of experiments.

Figure 4.

Evolution process of agents’ opinions in various networks.

4.2 Influence of extreme agent

In Fig. 4a and b are the opinions evolution processes of the agents under simulation for the networks generated by ER random graph and ER random graph using REA. In the simulation experiments, the initial distributions in both networks are uniformly distributed so that various opinion values are present in the networks. In Fig. 4b, we changed the treatment of extreme agents to have a 50% chance of terminating opinion interactions when extreme agents are in contact with others.

Table 2
Number of people in each network and in various states under the influence of REA

State	Initial	ER	BA	SW	ER&REA	BA&REA	SW&REA
Watch	600	302	239	302	461	437	506
Subjective	800	718	629	711	861	898	887
Firm	400	348	344	370	316	306	294
Extreme	200	632	788	617	362	359	313

Among them, Fig. 4a and b illustrate that when left untreated in the ER network, some agents’ opinions evolve rapidly toward extremes and some agents’ opinions evolve toward neutrality, but the effect is not significant. When restrictions on extreme agents are imposed, there is a significant decrease in the development of agents’ opinions toward extremes in the network and a substantial increase in the number of agents with opinions $O\in[-0.5,0.5]$ . ER networks with restrictions on extreme agents are all more neutral in overall opinion than when there are no restrictions on extremes, and extreme opinions are less likely to appear. Figure 4c–f show the evolution of agents’ opinions under simulation for the scale-free and small-world models and the networks generated after their use of REA. Limiting the interaction of extreme agents’ opinions leads to a more neutral overall network opinion, and decreases the probability of extreme opinions appearing, although not by the same proportion and degree. Table 2 shows the results of the evolution of the number of various state agents in the three networks and their addition of REA. The first column of the table shows the four state agents in the social network, and the second column shows the number of agents in the initial network. The third to fifth columns show the number of each state agent in the ER network, BA network, and SW network after a time node of $t=20$ , respectively. The sixth to eighth columns then show the final results of opinion evolution in these networks after the addition of REA at the time node of $t=0$ . It can be seen that after restricting extreme state agents, the number of extreme agents in the network has decreased by 40% to 50% compared to the initial situation. In summary, extreme agents have a small negative influence on opinion neutrality across the different networks. However, individuals begin to influence the social network environment at different times, and the size of their influence varies. Additionally, the optimal time for intervention by the platform and the government in shaping public opinion also differs. Therefore, it is necessary to analyze the influence of extreme agents’ interventions at different points in time on the overall environment.

4.3 Influence of extreme agent in different time

Figure 5.

Evolution of network opinions after REA at different time points.

To explore the negative impact of extreme agents on overall opinion evolution over time in a social network, we set different time limits for extreme agents and conducted 100 simulations to eliminate the influence of randomness. We then obtained the average number of overall opinion changes in society after introducing REA at different time points, as shown in Fig. 5. According to Fig. 5a, the number of agents with a final opinion value in the range of (0.8, 1] or [ $-$ 1, $-$ 0.8] did not change significantly when REA intervened at time $t=1$ . There is a significant decrease in the number of agents with opinion values $\in(0.6,0.8]\bigcup(-0.8,-0.6]$ and opinion values $\in(-0.2,0.2]$ , and most of the agents with opinion values belong to $\in(0.2,0.6]\bigcup(-0.6,-0.2]$ . When REA intervenes at time point $t=5$ , the number of agents with opinion values $\in(0.8,1]\bigcup[-1,-0.8]$ eventually rises slightly, and the number of agents with opinion values $\in(0.2,0.6]\bigcup(-0.6,-0.2]$ remains in the majority by 20 time points later. At REA’s intervention time point $t=10$ , the number of agents with opinion value $\in(0.8,1]\bigcup[-1,-0.8]$ rises rapidly until time $t=10$ . However, the number of agents with opinion values $\in(0.8,1]\bigcup[-1,-0.8]$ in this range still rises slowly, even after limiting the extreme agents. The number of agents with opinion values $\in(0.2,0.6]\bigcup(-0.6,-0.2]$ decreases. When delaying the REA intervention until $t=15$ , the interval with the highest number of agents is the interval with opinion values $\in(0.8,1]\bigcup[-1,-0.8]$ . If the REA intervention is delayed until $t=20$ , the number of agents in the network with opinion values biased towards the extremes is even higher.

Delaying intervention time for extreme agents increases the number of agents with polarized opinions in the final network. This result suggests that polarization of opinions in the network increases with time, leading to more negative effects. Therefore, limiting extreme individuals at an earlier time is more effective in controlling public opinion.

4.4 Impact of delayed REA intervention

Figure 5 illustrates the optimal time period for the REA intervention to slowly increase the number of ESAs across the social network. In Table 3 are the number of ESAs with positive opinions and the number of ESAs with negative opinions after the intervention at different time points. To reduce the randomness, we reduce such effects by averaging 100 simulation experiments. The first column represents different time points, while the second column indicates the number of ESAs who hold a forward opinion. The third column represents the number of ESAs who hold a backward opinion. The social network has 200 ESAs at the initial state. The number of ESAs grows naturally to 286 before $t=5$ . With the REA intervention at $t=5$ , the number of ESAs grows to 328 at $t=10$ after 5 moments. The growth number is 42 ESAs. After indulging in the entire social network’s opinion at moment $t=5$ without any means of treatment, then receiving REA intervention at moment $t=15$ , the number of agents in an extreme state increased by 65 at time point $t=20$ relative to moment $t=15$ .

Table 3
Changes in extreme agents at five time points after different REA interventions

Time points	Number of positive opinions of ESA	Number of negative opinions of ESA
Initial	100	100
$t=15$ after REA intervention at $t=5$	161	162
$t=20$ after REA intervention at $t=5$	177	179
$t=5$	143	143
Five time points after REA intervention at $t=5$	164	164
$t=15$ after REA intervention at $t=5$	187	188
$t=20$ after REA intervention at $t=5$	207	208
No previous operation for $t=15$	250	248
Five time points after REA $=$ 50% intervention at $t=15$	283	280
Five time points after REA $=$ 60% intervention at $t=15$	273	273
Five time points after REA $=$ 70% intervention at $t=15$	270	266
Five time points after REA $=$ 75% intervention at $t=15$	265	263

Figure 6.

Network opinion distribution.

As shown in Fig. 6a, the dots in the figure represent the opinion of each agent. The closer the color of the dot is to red, the more extreme their opinion is, and conversely the closer the color is to white, the more neutral their opinion is, i.e., the closer it is to 0. The size of the dot in the figure represents the size of their initial opinion value, and the more extreme the initial opinion value is, the larger the shape of the dot will be. Figure 6a–c show the initial social network opinion values, the opinion values at $t=5$ intervention and the opinion values of the social network at $t=20$ after the final evolution. Figure 6d shows the final evolved social network opinion values at $t=15$ REA intervention. In contrast to Fig. 6c, more nodes in the social network go to the extremes and we should avoid this situation.

Figure 6e–h shows the results of the evolution of the final opinions of the social network intervening in different REA efficiencies at $t=15$ , with REA efficiencies of 60%, 70%, 80%, and 90%, respectively. As the proportion of REA restrictions increases, the growth of ESAs in the society at $t=20$ slows down significantly. In the social network where REA $=$ 50% restriction was performed at $t=5$ , 47 ESAs increased in this time period from $t=15$ to $t=20$ . In the network that performs REA $=$ 50% restriction at $t=15$ , 65 ESAs are added in this time period from $t=15$ to $t=20$ . And after performing REA $=$ 60% restriction at $t=15$ , 48 ESAs were added in this time period from $t=15$ to $t=20$ . This increase in the number of ESAs is essentially the same as performing REA $=$ 50% at $t=5$ . In the social network where the REA $=$ 50% restriction was performed at $t=0$ , 33 ESAs were added in this time period from $t=15$ to $t=20$ . After performing REA $=$ 70% restriction at $t=15$ , 38 ESAs were added in this time period from $t=15$ to $t=20$ . similarly, after performing the REA $=$ 70% restriction at $t=15$ , 30 ESAs were added in this time period from $t=15$ to $t=20$ . Performing the REA $=$ 70%–75% restriction at $t=15$ has the same effect as performing the REA $=$ 50% restriction at $t=0$ , which increases the number of ESAs in this time period from $t=15$ to $t=20$ .

In short, delaying REA’s intervention requires more efforts to achieve the public opinion control effect achieved by intervening in the pre-fermentation stage of the event. This is a huge damage to the credibility of the public platform and an irreparable loss to the enterprise. Therefore, public opinion should be controlled by intervening as early as possible. The later the intervention, the more difficult it is to control the direction of public opinion. Social platforms should effectively control the extremists in the early stage when public opinion first starts to achieve a more toward neutralized opinion of public opinion.

5. Conclusions and future works

This paper defines four different role states based on the opinion values of agents in the social network: watch state, subjective state, firm state, and extreme state. This method provides a more compositional understanding of opinion evolution in a social network among individuals. We propose a model of opinion evolution based on the sunk cost effect. Through computer simulation, we analyze the negative impact of ESAs on the neutralization of opinion evolution in the whole social network. We also examine the impact of restricting ESAs at different points in opinion evolution. The results of the study provide certain reference value for the control strategy of platform opinion in social networks.

In addition, our simulation analysis led to these conclusions: (1) ESAs have certain negative effects on the evolutionary process of opinions toward neutrality in various social networks. With certain restrictions on ESAs, the overall opinions in social networks evolve toward a more neutral aspect. In the process of human-to-human information interaction, extreme agents may behave aggressively and use sharp language that exceeds the effective range of information acceptance of the interaction target. This behavior can lead to dissatisfaction and have negative effects on the interaction target. (2) It was found that restrictions on ESAs would have a positive effect on influencing the overall opinion evolution toward neutral evolution in social networks. Comparative experiments show that placing restrictions on ESAs earlier results in more neutral opinions across the social network, which is helpful for controlling public opinion. (3) Delaying REA creates significant costs that are difficult to recover.

The research on extreme opinions can help the public to enhance their understanding of extreme opinions, promote rational and inclusive discussions and exchanges. It can also help platforms and policy makers to understand the impact of these opinions, so as to formulate more fair and rational policies. However, there are still some limitations. Firstly, the real situation of each country and platform is different, we need more detailed real data for the experiment. Secondly, more in-depth research is needed on how to control the impact of extreme opinions. As future work, it would be a very good research point about how to go about better controlling the negative influence of extremists based on time sunk cost of opinion evolution model.

Footnotes

https://github.com/Diregie/Time-sunk-cost.

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An evolutionary approach to extreme individual impact opinions based on time sunk costs

Abstract

Keywords

1. Introduction

2. Related work

2.1 Bounded confidence model

3.1 Agent state

3.2.3 Opinion evolution process in opinion agency network

4.1 Simulation design

Table 2 Number of people in each network and in various states under the influence of REA

Table 3 Changes in extreme agents at five time points after different REA interventions

Footnotes

References

Table 2
Number of people in each network and in various states under the influence of REA

Table 3
Changes in extreme agents at five time points after different REA interventions