# Modeling of the Public Opinion Polarization Process with the Considerations of Individual Heterogeneity and Dynamic Conformity

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## Abstract

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## 1. Introduction

## 2. Literature Review

## 3. Public Opinion Polarization Model with the Consideration of Individual Heterogeneity and Dynamic Conformity

#### 3.1. The Classic W-D and J-AModels

_{i}(t) represents agent i’s attitude value at time t. In addition, the value x

_{i}(t) ∊ [−1,1] follows continuous distribution. In each iteration, two random pairs of agents are paired up to spread their attitudes. When the attitude distance |x

_{i}− x

_{j}| of the randomly selected agents i and j is less than a given threshold d, the two agents update their own attitudes and interact with each other. But, if the attitude distance |x

_{i}–x

_{j}| is not within the threshold d, the interaction behavior does not occur and the two agents’ attitudes do not change.

_{i}−x

_{j}| ≤ d, there are

_{i}(t + 1) = x

_{i}(t) + μ[x

_{j}(t) − x

_{i}(t)]

x

_{j}(t + 1) = x

_{j}(t) + μ[x

_{i}(t) − x

_{j}(t)]

_{i}− x

_{j}|>d) the interaction does not occur, there are

_{i}(t + 1) = x

_{i}(t)

x

_{j}(t + 1) = x

_{j}(t)

_{1}, that is, when |x

_{i}− x

_{j}|<d

_{1}, then the attitudes of agents i and j change correspondingly, and the updated formulas are:

_{i}(t + 1) = x

_{i}(t) + μ(x

_{j}(t) − x

_{i}(t))

x

_{j}(t +1 ) = x

_{j}(t) + μ(x

_{i}(t) − x

_{j}(t))

_{2}, i.e., when |x

_{i}− x

_{j}| >d

_{2}, the updated formulas of agents i and j are:

_{i}(t + 1) = x

_{i}(t) + μ(x

_{j}(t) − x

_{i}(t))

x

_{j}(t + 1) = x

_{j}(t) + μ(x

_{i}(t) − x

_{j}(t))

_{i}(t + 1) = x

_{i}(t)

x

_{j}(t + 1) = x

_{j}(t)

#### 3.2. A Polarization Model Combining Individual Dynamic Conformity with Heterogeneity

_{i}~N(0,1), and value x

_{i}is in interval [−1,1]. At time t, the attitude value of agent i is expressed as x

_{i}(t), and the x

_{i}(t) ∊ [−1,1]. The social average attitude value T(t) is expressed as follows:

_{j}and agent i’s own authority p

_{i}(usually, the value of p

_{i}is equal to degree centrality of agent i). In a simulation experiment, each agent’s authority is calculated on the basis of the degree centrality, and the magnitude of the influence between i and j is related to the difference of degree centrality between the two. The Z

_{ij}is expressed as agent j’s influence on agent i, shown as follows:

_{j}(t) that agent j transmits to agent i at time t and the influence, Z

_{ij}, between them. The influence degree of agent i by all neighbors is represented by neighbor attitude Z

_{i}(t), which is described as follows:

_{i}(t) received by agent i at each time consists of the received mainstream attitude Π(t) and the neighbor attitude Z

_{i}(t)) transmitted by the neighbor agents of agent i:

_{i}(t) = μΠ(t) + (1 − μ) Z

_{i}(t),

_{i}(t).

_{i}(t) is close to the total received attitude X

_{i}(t) (in the assimilation effect zone), it will be encouraged to reinforce its attitude and adjust the attitude closer to X

_{i}(t). On the contrary, when the difference is very large (in the repulsion effect zone), the reverse psychology will strengthen their attitude value and adjust the attitude far from X

_{i}(t). At the same time, agent I only interacts directly with its neighboring agents; that is, the opinion is only approved or disapproved by the neighboring agents. When the agent i and the neighbor’s attitude Z

_{i}(t) have the same positive or negative tendency, they will be affirmed, thus enhancing the certainty degree about their own attitudes. At time t, agent i selects the assimilation rule, the repulsion rule, or the neutral rule according to the difference calculated by total received attitude X

_{i}(t) and his own attitude to adjust his attitude at the next time, shown as follows:

_{i}(t) − x

_{i}(t)| ≤ d

_{1}, there are

_{i}(t + 1) = f

_{i}(t) * x

_{i}(t) + ζ

_{i}(t) * (X

_{i}(t) − x

_{i}(t)),

_{i}(t) = (e

^{k}

^{1i(t)/ki(t)}− 1) + Y

_{i},

_{i}(t) = 1 − f

_{i}(t),

_{i}(t) indicates the degree that agent i confirms his attitude at time t. If agent i confirms his attitude when interacting with neighbor agents, he will strengthen his attitude; otherwise, he will doubt the correctness of attitude because of being attacked all the time. This article assumes that all agents in the network have a one-to-many interaction at each moment (agent i interacts with all neighbor agents at the same time). After the interaction occurs, agent i adjusts his attitude according to dominant social attitude Π(t)and neighbor attitude Z

_{i}(t) passed to him by all the neighbor nodes it receives. k

_{i}(t) denotes the number of times agent i receives neighbor attitude Z

_{i}(t). Because each agent interacts one-to-many with its neighbors at each moment, the k

_{i}(t) of all agents is the same, and its value is equal to the number of iterations t.k

_{1i}(t) indicates that during the initial time to time t, agent i is positively counted when performing attitude interaction (assuming that agent i’s attitude x

_{i}(t) has the same positive and negative tendency as the received neighbor attitude Z

_{i}(t) when interacting with neighbor agents); k

_{1i}(t)/k

_{i}(t) indicates the positive probability, which changes as the interaction goes on; and Y

_{i}indicates the inherent confidence of agent i, which is an inherent property of agent i and does not change with the interaction. If agent i is very confident about his attitude, and the f

_{i}(t) is at a relatively large value, then their conformity ζ

_{i}(t) will be reduced, so ζ

_{i}(t) is inversely related to f

_{i}(t), as shown in the Formula (14). At time = 0, since the agent does not participate in the interaction and not be influenced by other agents’ attitude, so the conformity is only related to himself, and the conformity of agent i is ζ

_{i}(0) = 1 − Y

_{i}.

_{i}(t) − x

_{i}(t)|≥d

_{2}, there is

_{i}(t + 1) = f

_{i}(t) * x

_{i}(t) − ζ

_{i}(t) * (X

_{i}(t) − x

_{i}(t)),

_{i}(t + 1) = x

_{i}(t),

## 4. Numerical Simulation Experiments

#### 4.1. The Influence of Individual Dynamic Conformity

_{i}(t) represented by formulas (14) and (15) at time t = 0, the conformity ζ

_{i}(0) = 1 − Y

_{i}. So in the formula of static conformity, agent i’s conformity of all moments, is 1 − Y

_{i}, as shown in formulas (18) and (19). For visualization, Y

_{i}takes 0.6, and the result is shown in Figure 2, Figure 3, Figure 4, Figure 5, Figure 6 and Figure 7. The network used in the simulation is fully connected network, and the other parameters are set as follows: n = 500,μ = 0.1, d

_{1}=0.10, andd

_{2}=0.70.

_{i}(t + 1) = Y

_{i}* x

_{i}(t) + (1 − Y

_{i}) * (X

_{i}(t) − x

_{i}(t)),

_{i}(t + 1) = Y

_{i}* x

_{i}(t) − (1− Y

_{i}) * (X

_{i}(t) − x

_{i}(t)).

_{i}, one of the important parameters in the dynamic conformity function ζ

_{i}(t), has a crucial influence on the polarization, representing the affirmation of the attitude of individual i at the initial time, thus it affects the conformity degree. To study the influence of Y

_{i}on polarization, four different values of 0.2, 0.4, 0.6, and 0.8 were set as Y

_{i}, and the proportion of individuals polarized under different values of Y

_{i}was observed, as shown in Figure 8. The network used in the simulation was a fully connected network, and the other parameters were set as follows: n = 500, μ = 0.3, d

_{1}= 0.2, and d

_{2}= 0.70.

_{i}after all agents in the network have fully interacted. We set the social average attitude value T(0) = 0 in this simulation. When the value of Y

_{i}is small, due to the “spiral of silence” effect [36], the individual is not confident enough to hold his own attitude, so it is easy for a stronger view to overcome the other side. However, as Y

_{i}increases, the strength of the stronger view weakens, and gradually, both the polarization levels are evenly matched.

_{i}, there is the influence parameter of social conformity μ, defined formula (12), which indicates the probability of the tendency to be consistent with the dominant social attitude Π(t). In order to analyze the influence of parameter μ on polarization phenomenon, four different values, 0.2, 0.4, 0.6, and 0.8, were set, and the results are shown in Figure 9 and Figure 10. Figure 9a–d shows the evolution of public opinion under four different values μ. Figure 10a–d shows the polarization probability of the proportion of all individuals in the network forming the opinion polarization under four different values of μ. In the fully connected network used above, all individuals in the network were directly connected; that is, all individuals were neighbors of each other, and the difference between dominant social attitude Π(t) and connected neighbor agent’s attitude Z

_{i}(t)could not be accurately analyzed. Therefore, this section uses the Barabasi-Albert (BA) scale-free network in the simulation (the other parameters are set as follows: the number of nodes is n = 500, the clustering coefficient is 0.12839, and the average degree is 35.872).

#### 4.2. The Influence of Individual Heterogeneity

#### 4.2.1. The Influence of Initial Cognitive Heterogeneity

_{i}~U(0,1), μ = 0.2,d

_{1}= 0.2, and d

_{2}= 0.70.

_{i}, and observed the evolution of the viewpoints of 10 groups of individuals representing different initial attitudes in the experiment. The results are shown in Figure 14:

_{i}< −0.8, 0.8 < x

_{i}< −0.6, −0.6 < x

_{i}< −0.4, −0.4 < x

_{i}< −0.2, −0.2 < x

_{i}< 0, 0 < x

_{i}< 0.2, 0.2 < x

_{i}< 0.4, 0.4 < x

_{i}< 0.6, 0.6 < x

_{i}< 0.8, and 0.8 < x

_{i}< 1. Figure 14a–d is a set of four data indicating the initial time attitude value x

_{i}< −0.2. In the evolution of public opinion, they are accompanied by polarization phenomena, all of which are transformed into negative polarization, and according to Figure 14a–d sequence, the occurrence time of polarization phenomenon decreases in turn, indicating that the individuals whose attitude value is close to the polarization value at the initial time are more likely to be polarized and the individuals whose polarization degree is weaker will be induced toward a pole in the public opinion evolution. Figure 14g–j is a set of four data indicating the initial time attitude value x

_{i}> 0.2, similar to the negative polarity, and the time for the positive polarization of Figure 14g–j is also reduced in turn. Figure 14e,f shows that the individuals do not have strong polarization at the initial time. The individual attitude values in Figure 14e are gradually divided into two parts as the public opinion evolves; that is, one part finally forms negative polarization, the other continues to be neutral. Figure 14f means one part form positive polarization and others maintain neutrality, and the existence of these neutral individuals is essential for intervening in the polarization process.

_{i}< −0.8, 0.8 < x

_{i}< −0.6, −0.6 < x

_{i}< −0.4, −0.4 < x

_{i}< −0.2, −0.2<x

_{i}<0, 0<x

_{i}<0.2, 0.2 < x

_{i}< 0.4, 0.4 < x

_{i}< 0.6, 0.6 < x

_{i}< 0.8, and 0.8 < x

_{i}< 1. Comparing Figure 15 and Figure 16 with Figure 13 and Figure 14, it can be seen that there is a clear difference between the evolutionary processes of public opinion in the network when the attitude tends to be clear and unclear at the initial time.

_{i}< 0.2, which, by comparison, is similar to the individual’s attitude value at the initial time is −0.2 < x

_{i}< 0, shown in Figure 16e in evolution. The individuals with attitudes close to neutral values at the initial time show a completely neutral attitude after many interactions (the attitude value is 0), indicating that if we need to prevent polarization by artificial intervention, then the existence of these neutral individuals is crucial.

#### 4.2.2. The Different Conformity Influences of Heterogeneous Individuals

_{i}in the dynamic conformity function ζ

_{i}(t), and that each individual participating in the interaction is heterogeneous and has different inherent confidence. In this section, we simulate a network with 500 nodes, and set the social average attitude value T(0) = 0 at the initial time. It was assumed that in the simulation experiment, that the value of each individual’s intrinsic self-confidence was Y

_{i}~U(0,1). The polarization probability of the network public opinion evolution is shown in Figure 17. The network used in the simulation was a fully connected network, and the other parameters were set as follows: n = 500, Y

_{i}= 0.3, μ = 0.6, d

_{1}= 0.20, and d

_{2}= 0.70.

_{i}< 0.25, 0.25 ≤ Y

_{i}< 0.5, 0.5 ≤ Y

_{i}< 0.75, and 0.75 ≤ Y

_{i}< 1 accounts for 25% of the total amount. Figure 18a–d indicates the polarization probability of these four groups of individuals in the evolution.

_{i}s were 0 ≤ Y

_{i}< 0.25, 0.25 ≤ Y

_{i}< 0.5, 0.5 ≤ Y

_{i}< 0.75, and 0.75 ≤ Y

_{i}< 1 had different times of reaching polarization, and the proportion of the final polarizations were different in this simulation. Additionally, according to the different values of Y

_{i}, there is a certain regularity. According to Figure 17, the proportion of negative polarization is greater than that of positive polarization at initial stage (time < 15), and is reversed by positive polarization at time = 15, and thereafter, the proportion of the positive polarization of individuals becomes slightly greater than that of negative polarization. In Figure 18, the case where the positive polarization is greater than the negative polarization is only shown in Figure 18a,b indicating that individuals with different inherent self-confidence in network evolution have different positive and negative tendencies, and individuals with high inherent self-confidence are more likely to have a negative attitude. In addition, according to Figure 18a–c, the polarization rate formed in the network’s evolution increases with the increase of inherent self-confidence, indicating that when the individual has a small inherent self-confidence, his conformity is larger, and the polarization rate finally formed is larger. However, the polarization probability decreases when the inherent confidence increases to a certain threshold, as shown in Figure 18c,d.

#### 4.3. The Influence of Network Structure

_{i}= 0.6, μ = 0.05, d

_{1}= 0.20, and d

_{2}= 0.70.

_{i}= 0.6, μ = 0.1, d

_{1}= 0.20, and d

_{2}= 0.80.

## 5. Real Case Study and Analysis

_{i}= 0.4. Because the “Mimeng Event” itself was negative, the social average attitude value T(0) = −0.2 was set at the initial time; the initial polarization rates were set to be ρ(0) = 1.2, d

_{1}= 0.18, and d

_{2}= 0.70. In this simulation, 500 nodes were selected. In the experiment, the network aggregation was set to be 0.33392, which is closer to the Sina Weibo’s characteristic, with an average degree of 118. The selection of relevant parameters was based on the actual situation, giving them a certain practical reference value. The results are shown in Figure 26, Figure 27 and Figure 28.

## 6. Conclusions

- (1)
- When one extreme attitude dominates in the network, the individual with the other extreme attitude will gradually change his attitude and then become neutral through enough interactions.
- (2)
- The degree of individual attitude change is limited in the evolution of the network, and it is difficult for individuals who have one directional attitude at the initial time to change into another opposite attitude through interactions.
- (3)
- Different individuals have different conformability and individuals with low conformability are likely to form polarization phenomena within a certain threshold.
- (4)
- Through comparisons with the J-A model and the static conformity model, the model proposed in this article was demonstrated to be more valuable in theory and application.

- (1)
- Combined with the real case, it can be seen that the spread of hot events in the network is a dynamically changing process, and the number of netizens participating in the discussion increases gradually along with the spread of hot events, but decreases gradually with a reduction of the popularity of hot events. Therefore, it is necessary to study apolarization phenomenon in dynamic networks by considering the increase and decrease of network nodes (netizens). In addition, another important research focus is to understand the feedback loop amongthe two.
- (2)
- Due to the virtual nature of the network, it is difficult for netizens to distinguish the inductive information. In addition, with continuous disclosure of the truth, the reversal of public opinion occurs. Therefore, it is necessary to study the influence of public opinion’s reversal upon polarization.
- (3)
- This article concludes that the probability of public opinion’s polarization is related to individual conformity, the social influence parameter, and the intrinsic self-confidence parameter. However, for the parameters mentioned above, this article only discussed them specifically. In fact, the polarization phenomenon of public opinion is composed of many factors and interactions, so the compositional effects of these factors should be discussed in future.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 14.**A three-dimensional graph of the attitude evolution of individuals with different initial values.

**Figure 16.**A three-dimensional graph of the attitude evolution of individuals with different initial values.

**Figure 21.**Polarization probabilities at different times in small world networks with different characteristics and a fully connected network.

**Figure 22.**Polarization probability under four different clustering coefficients of Barabasi-Albert (BA) networks.

**Figure 32.**The simulation graph of the static conformity model for public opinion’s evolution in the “Mimeng Event”.

Network Type | Number of Edges | Average Path Length | Clustering Coefficient | Average Degree | Reconnection Probability |
---|---|---|---|---|---|

Small World network | 2500 | 3.4731 | 0.3541 | 10 | 0.2 |

Small World network | 2500 | 3.13 | 0.16133 | 10 | 0.4 |

Small World network | 2500 | 2.9853 | 0.059089 | 10 | 0.6 |

Small World network | 2500 | 2.9488 | 0.025154 | 10 | 0.8 |

Fully connected network | 124,750 | 1 | 1 | 499 |

Serial Number | Number of Edges | Clustering Coefficient | Average Degree |
---|---|---|---|

1 | 4711 | 0.092597 | 18.842 |

2 | 9001 | 0.12714 | 36.002 |

3 | 20,041 | 0.21929 | 80.162 |

4 | 33,723 | 0.3151 | 134.89 |

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## Share and Cite

**MDPI and ACS Style**

Chen, T.; Li, Q.; Yang, J.; Cong, G.; Li, G.
Modeling of the Public Opinion Polarization Process with the Considerations of Individual Heterogeneity and Dynamic Conformity. *Mathematics* **2019**, *7*, 917.
https://doi.org/10.3390/math7100917

**AMA Style**

Chen T, Li Q, Yang J, Cong G, Li G.
Modeling of the Public Opinion Polarization Process with the Considerations of Individual Heterogeneity and Dynamic Conformity. *Mathematics*. 2019; 7(10):917.
https://doi.org/10.3390/math7100917

**Chicago/Turabian Style**

Chen, Tinggui, Qianqian Li, Jianjun Yang, Guodong Cong, and Gongfa Li.
2019. "Modeling of the Public Opinion Polarization Process with the Considerations of Individual Heterogeneity and Dynamic Conformity" *Mathematics* 7, no. 10: 917.
https://doi.org/10.3390/math7100917