# How Opinion Leaders Affect Others on Seeking Truth in a Bounded Confidence Model

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## Abstract

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## 1. Introduction

## 2. Models and Methods

_{i}(t). On the other hand, for simulation analysis purpose, the opinion functions are sometimes assumed to change over a set of discrete time points based on certain opinion update mechanism. This paper adopts the extended HK model presented in [13,14], instead of the original HK model, as the basis for opinion update, which is described in Equation (1).

_{L}for leader L, while the weights of normal agents are all assumed being 1, much smaller than R

_{L}, in the following discussion. The influencing power of leaders on other agents is called their appeal [18] and denoted as A

_{L}. The appeal of leaders is reflected by the bounds of confidence of normal agents towards leaders in the following, which is larger than that towards their peers, denoted as T. On the other hand, the leaders are less, or sometimes not, influenced by the other agents representing their stubbornness [33,34], which is reflected by the small bounds of confidence of leaders. Therefore, the stubbornness of opinion leader L is defined as S

_{L}= 1 − T

_{L}, with T

_{L}being the small bound of confidence of leader L. The extremeness of opinion leader comes from the competing nature of two opinion leaders [35,36], and so the extremeness of the leader is reflected by how close the opinion value of the leader is to the extremes of 0 and 1, because there is only one leader in our model who might be competing with the truth. The opinion value of leader L is denoted as x

_{L}in the following. We consider in this paper how the group, especially the normal agents, behave as the three characteristics—reputation; appeal and stubbornness; and the opinion of the leader, representing its extremeness—vary. What is the effect on the agents seeking the truth if the reputation, appeal, or stubbornness of the leader increase or decrease in their values? How will the normal agents act if the leader takes different opinion values?

_{j}> 0 and R > 0 are the weights assigned to agent j and the truth respectively with R, also called reputation of leaders, being much larger than r

_{j}, and the function $g\left(i,t\right)={\sum}_{j\in I\left(i,t\right)}{r}_{j}+f\left({x}_{i}\left(t\right)\right)R$ is the sum of the weights of agents within $I\left(i,t\right)$ and the reputations of truth depending on the value of function $f\left({x}_{i}\left(t\right)\right)$, which is

## 3. Simulation Results and Discussions

#### 3.1. Group without Opinion Leader

#### 3.1.1. Impact of the Reputation of Truth

_{L}= T. The starting value of group bound of confidence T is set to be 0.05 because the agents will be in a great diversity when T is smaller than 0.05, and the impact of truth on the agents will be not so visible.

#### 3.1.2. Impact of the Appeal of Truth

#### 3.2. Group with Opinion Leader

_{L}= 0.25, when exploring the impact of the three characteristics—reputation, stubbornness, and appeal—and vary the leader’s opinion value when investigating the impact of extremeness. It is noted that the value of the leader’s opinion (0.25) is symmetric to the truth value (0.75), about the mid-point 0.5.

#### 3.2.1. Impact of the Reputation of Leader

_{L}= 1 − T, A

_{L}= T, so that the leader only differs from other agents because of reputation, and then we will vary the settings of other characteristics to see their impacts in the following subsections. We then vary the reputation of the leader, in steps of 20, from 1 to 1000 and the group bound of confidence T, in steps of 0.01, from 0.05 to 0.5. Figure 5 shows the simulation results on the average proportion of agents reaching the truth.

_{L}increases from 1 to 50 or 200 depending on the value of T, and then drops down to some stable value when increasing R

_{L}further. The reason for the percentage increasing is similar to the impact of the truth’s reputation, i.e., higher (but not too high) reputation makes the normal agents that fall outside the confidence range of the leader have less chance of being influenced by the leader, and therefore more chance to be influenced by the truth whose value remains unchanged. At the same time, due to the reason that the leader is not stubborn, it will communicate with the other agents and be affected by the truth indirectly, and change its opinion correspondingly, even becoming the follower of the truth in some cases. When R

_{L}goes still higher, the leader has the ability to attract the normal agents within its confidence range to become its followers sharply, and form its own group that is out of the control of the truth when T is smaller than the value about 0.45 as shown in Figure 5b. As a result, the percentage of agents reaching the truth drops down. When R

_{L}is sufficiently high, there will be a trade-off between the impact of the truth and the leader, and the percentage of agents reaching the truth will become stable. The time series plots as shown in Figure 6 help to illustrate the group opinion dynamics.

_{L}= 50 in the following subsections when exploring the impact of the other characteristics.

#### 3.2.2. Impact of the Stubbornness of Leader

_{L}= 50, and vary its stubbornness S

_{L}, in steps of 0.01, from 0.8 to 1 and the group bound of confidence T, in steps of 0.01, from 0.05 to 0.5, with its appeal A

_{L}= T. Figure 7 shows the simulation results on the average proportion of agents who reached the truth.

_{L}is around 0.9. The results for high values of stubbornness are consistent with the findings in [38], where a conflicting source to the truth is assumed to be completely stubborn and it is shown that no agents can converge on the truth for sufficiently high values of confidence. Figure 7b shows a critical point at about 0.2, before which all stubbornness values (from 0.8 to 1) make the same proportion of agents reach the truth, while thereafter the leader with stubbornness value larger than 0.9 pulls all the other agents away from the truth and that with stubbornness smaller than 0.8 helps the others to reach the truth. However, when T is between 0.2 and 0.4, the group dynamics are very sensitive, especially when S

_{L}= 0.85 where the percentage of agents reaching the truth fluctuates between 0 and 70% as shown in Figure 7b. The time series plots on the opinion update process as shown in Figure 8 helps to better understand the group opinion dynamics. Note that the group behavior is sensitive when T is around 0.25, the agents usually divide into more than three groups as shown in Figure 8f, but sometimes form two groups similar to that in Figure 8e.

_{L}= 1, as exploring how the leader influencing the agents seeking the truth in terms of its appeal and opinion value.

#### 3.2.3. Impact of the Appeal of Leader

_{L}= 50 and its stubbornness S

_{L}= 1, and vary the group bound of confidence T, in steps of 0.01, from 0.05 to 0.5 and the appeal of the leader on the normal agents A

_{L}, in steps of 0.01, from 0.25 to 0.5. Figure 9 shows the simulation results on the average proportion of agents who reached the truth.

_{L}also larger than 0.3. We have selected some typical time series plots as shown in Figure 10 to further illustrate the above results. It can then be concluded that increasing the appeal of the leader whose opinion is opposite to the truth has a straightforward impact, i.e., normally prevents the normal agents from finding the truth. On the other hand, increasing T also makes the agents who start out close to the truth move away from the truth if there is an opinion leader opposite to the truth. This can be explained as the agents are more likely to be influenced by both the truth and the leader when T is larger, and will normally converge to an intermediate opinion. Therefore, it means that the opinion of the leader is quite important in affecting the normal agents’ ability to reach the truth. We then vary the opinion of the leader and see how the group behave in the following section, while letting the appeal take the middle value, i.e., A

_{L}= 0.4 for comparison purposes.

#### 3.2.4. Impact of the Opinion Value of Leader

_{L}= 50, its stubbornness S

_{L}= 1 and its appeal A

_{L}= 0.4, and vary the group bound of confidence T, in steps of 0.01, from 0.05 to 0.5 and the leader’s opinion x

_{L}, in steps of 0.05, from 0 to 1. Figure 11 shows the simulation results on the average proportion of agents who reach the truth.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Average proportion of agents reaching the truth, taken over 200 runs, with respect to (

**a**) reputation of truth R with different bounds of confidence, and (

**b**) group bound of confidence T with different reputations of the truth, where A

_{L}= T.

**Figure 2.**Different simulation results (

**a**,

**b**) on opinion updating process with the same parameter setting R = 50, A = T = 0.25.

**Figure 3.**Average proportion of agents reaching the truth, taken over 200 runs, with respect to (

**a**) appeal of the truth A with different bounds of confidence, and (

**b**) group bound of confidence T with some different appeals of the truth, where R = 50.

**Figure 4.**Simulation results on opinion updating process with R = 50, and (

**a**) T = 0.1, A = 0.25, (

**b**) T = 0.1, A = 0.35, (

**c**) T = 0.2, A = 0.25, (

**d**) T = 0.2, A = 0.35, (

**e**) T = 0.3, A = 0.25, (

**f**) T = 0.3, A = 0.35.

**Figure 5.**Average proportion of agents reaching the truth with respect to (

**a**) reputation R

_{L}with different bounds of confidence, and (

**b**) group bound of confidence T with different reputations of the leader, where S

_{L}= 1 − T, A

_{L}= T. Averages taken over 200 runs.

**Figure 6.**Simulation results on opinion updating process with S

_{L}= 1 − T, A

_{L}= T, and (

**a**) T = 0.25, R

_{L}= 1, (

**b**) T = 0.25, R

_{L}= 50, (

**c**) T = 0.25, R

_{L}= 100, (

**d**) T = 0.25, R

_{L}= 200.

**Figure 7.**Average proportion of agents reaching the truth with respect to (

**a**) stubbornness S

_{L}with different bounds of confidence, and (

**b**) group bound of confidence T with the leader’s stubbornness taking different values, where R

_{L}= 50, A

_{L}= T. Averages taken over 200 runs.

**Figure 8.**Simulation results on opinion updating process with R

_{L}= 50, A

_{L}= T, and (

**a**) T = 0.3, S

_{L}= 0.85, (

**b**) T = 0.3, S

_{L}= 0.9, (

**c**) T = 0.3, S

_{L}= 0.95, (

**d**) T = 0.3, S

_{L}= 1, (

**e**) T = 0.2, S

_{L}= 1, (

**f**) T = 0.25, S

_{L}= 1, (

**g**) T = 0.35, S

_{L}= 1, (

**h**) T = 0.4, S

_{L}= 1.

**Figure 9.**Average proportion of agents reaching the truth with respect to (

**a**) appeal A

_{L}with different bounds of confidence, and (

**b**) group bound of confidence T with different appeals of the leader, where R

_{L}= 50, S

_{L}= 1. Averages taken over 200 runs.

**Figure 10.**Simulation results on opinion updating process with R

_{L}= 50, S

_{L}= 1, and (

**a**) T = 0.2, A

_{L}= 0.25, (

**b**) T = 0.25, A

_{L}= 0.25, (

**c**) T = 0.3, A

_{L}= 0.25, (

**d**) T = 0.4, A

_{L}= 0.25, (

**e**) T = 0.25, A

_{L}= 0.3, (

**f**) T = 0.3, A

_{L}= 0.3, (

**g**) T = 0.3, A

_{L}= 0.4, (

**h**) T = 0.4, A

_{L}= 0.4.

**Figure 11.**Average proportion of agents reaching the truth taken over 200 runs with respect to (

**a**) leader’s opinion x

_{L}with different bounds of confidence, and (

**b**) group bound of confidence T with the leader’s opinion taking different values, where R

_{L}= 50, S

_{L}= 1, A

_{L}= 0.4.

**Table 1.**Average percentages of agents reaching the truth with respect to T, where R = 1000. Average taken of 200 runs.

Group Bound of Confidence T | Average Percentage (%) |
---|---|

0.1 | 21.2 |

0.15 | 37.9 |

0.2 | 46.3 |

0.25 | 51.4 |

0.3 | 56.3 |

0.35 | 61.8 |

0.4 | 79.6 |

**Table 2.**Average percentages of agents reaching the truth under 200 runs with respect to T, where R = 50 and R

_{L}= 1000. Averages taken over 200 runs.

Group Bound of Confidence T | Average Percentage (%) |
---|---|

0.1 | 64.3 |

0.15 | 59.3 |

0.2 | 54.4 |

0.25 | 49.8 |

0.3 | 44.8 |

0.35 | 39.7 |

0.4 | 30.5 |

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**MDPI and ACS Style**

Chen, S.; Glass, D.H.; McCartney, M.
How Opinion Leaders Affect Others on Seeking Truth in a Bounded Confidence Model. *Symmetry* **2020**, *12*, 1362.
https://doi.org/10.3390/sym12081362

**AMA Style**

Chen S, Glass DH, McCartney M.
How Opinion Leaders Affect Others on Seeking Truth in a Bounded Confidence Model. *Symmetry*. 2020; 12(8):1362.
https://doi.org/10.3390/sym12081362

**Chicago/Turabian Style**

Chen, Shuwei, David H. Glass, and Mark McCartney.
2020. "How Opinion Leaders Affect Others on Seeking Truth in a Bounded Confidence Model" *Symmetry* 12, no. 8: 1362.
https://doi.org/10.3390/sym12081362