# Modeling Multi-Dimensional Public Opinion Process Based on Complex Network Dynamics Model in the Context of Derived Topics

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## Abstract

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## 1. Introduction

## 2. Literature Review

## 3. Model Construction

_{i}is at a high interval, it will generate a derived subtopic.

_{0}at time T

_{i}, the individual will participate in the discussion of the sub-topic. Infection rate and immunity rate change based on Brownian motion. The subtopic information evolves according to the SIR model of the new propagation parameters.

_{0}of a derived subtopic reaches a certain threshold is it considered that the subtopic forms a derived public opinion, and the newly emerging derived public opinion is intertwined with the initial public opinion to form a multi-dimensional public opinion.

#### 3.1. Initial Public Opinion Propagation Model

#### 3.2. The Formation Process of Derived Subtopics

#### 3.2.1. The Degree of Variation P

#### 3.2.2. The Formation of Subtopics

_{U}as the threshold. When 0 < P < P

_{U}, the degree of variation is lower; when P

_{U}< P < 1, the degree of variation is higher. The specific process is shown in Figure 3.

_{U}< P < 1 at time T

_{i}and the degree of variation P

_{U}exceeds the variation threshold, a derived subtopic is generated. Here, the variation threshold P

_{U}is 0.8.

#### 3.3. The Propagation Process of Derived Subtopics

#### 3.3.1. The Propagation Parameter of Derived Subtopics

**Definition**

**1**

**[32].**

_{t}is continuous, and B

_{0}= 0.

- (1)
- For all real numbers s, t satisfies 0 ≤ s ≤ t, B
_{t}−B_{s}and F_{s}are independent; - (2)
- When 0 ≤ s ≤ t, B
_{t}− B_{s}obeys the normal distribution N(0, t − s), the normal distribution satisfies the mean value of 0 and the variance is t − s.

_{i}(t) is the independent Brownian motion, α

_{0}and ${\beta}_{0}$ is the infection rate and immunity rate of the initial public opinion, respectively, and σ

_{i}> 0 (0 < σ

_{i}< 1) is a constant, which represents the amount of information of the current subtopic, that is, the intensity of the disturbance. The greater the intensity, the greater the change in infection rate.

#### 3.3.2. The Propagation Model of Derived Subtopics

_{inf}are used to describe the degree of individual attention to derived subtopics. Setting k

_{inf}as the number of neighboring nodes infected around a node with degree k, then the attention degree function of a node with degree kin the network [34] is expressed by the following Formula (6):

_{i}, the individuals with high attention to that subtopic will continue to participate in the propagation of the subtopic, while individuals with lower attention will lose interest in the subtopic and will no longer participate in the discussion of the subtopic, resulting in reduced topic spread. It is defined that when g

_{i}> g

_{0}at the time T

_{i}, individuals in the network will participate in the discussion of the subtopic and carry out the propagation of the subtopic. These individuals propagate according to the SIR model with the new propagation parameters and are intertwined with the initial public opinion to form a multi-layer SIR propagation network. The schematic diagram of their propagation transformation is shown in Figure 4.

#### 3.4. The Formation of Multi-Dimensional Derived Public Opinion

_{i}, and α and β refer to the infection rate and immunity rate, respectively. In the infectious disease model, an important critical point of R

_{0}is R

_{0}= 1. The larger the value of R

_{0}is, the more difficult it is to control the epidemic. When R

_{0}< 1, the infectious disease will gradually disappear. When R

_{0}= 1, the infectious disease will become endemic. When R

_{0}> 1, infectious diseases will spread exponentially. According to the propagation of derived subtopics, the basic reproduction number R

_{0}of each derived subtopic is dynamically changing. It is defined here that when the basic reproduction number R

_{0}of the subtopic is larger than 1 and lasts for a period of time, the subtopic forms a network public opinion, i.e., a dimension is added to the initial public opinion.

_{0}> 1, it is considered that the subtopic forms a derived public opinion. When multiple derived subtopics form a derived public opinion, this and the initial public opinion form a multi-dimensional public opinion.

## 4. Simulation Experiments

#### 4.1. The Impact of Information Alienation on the Formation of Multi-Dimensional Public Opinion

_{U}= 0.8, g

_{0}= 0.6. In order to facilitate the observation of the formation of multi-dimensional public opinion, the subtopics are set to be derived from the variation of the initial public opinion information. At this time, each derived subtopic presents a positive correlation with the initial public opinion. Taking the comprehensive visualization into consideration, other parameters are set as: δ = 9, θ~N(0.9, 0.3), σ~N(0.9, 0.3), and the evolution time is set to T = 40.

#### 4.1.1. The Impact of Information Alienation on the Amount of Derived Subtopics and Propagation Process

#### 4.1.2. The Impact of Information Alienation on the Degree of Multi-Dimensional Public Opinion

#### 4.2. The Impact of Environmental Forces on the Formation of Multi-Dimensional Public Opinion

_{U}= 0.7, g

_{0}= 0.6, ρ = 0.85, T = 40.

#### 4.2.1. The Impact of Environmental Forces on the Number of Derived Subtopics and the Propagation Process

#### 4.2.2. The Impact of Environmental Forces on the Dimensions of Multi-Dimensional Public Opinion

_{0}and the multidimensional public opinion dimensions of the subtopics under different environmental forces are simulated here and the results are shown in Figure 16 and Figure 17.

_{0}reaches 6. However, due to the increasing environmental force, the peak of the second derived topic is 3. This shows that the environmental force has a relatively small impact on the first derived subtopic, and it has a greater impact on the second and third derived subtopics, and the basic regeneration number of the second subtopic increases with the increase in environmental force. However, when the environmental force increases to a certain level, the increase in the basic regeneration rate begins to slow down. From Figure 17, it can be seen that when the environmental force is 3, although three derived subtopics are generated, the final public opinion dimension is still 1, and no multi-dimensional public opinion is formed. At the same time, the increase of environmental forces has a positive correlation with the increase of the final public opinion dimension.

#### 4.2.3. Analysis of the Combination of Factors Influencing the Number of Derived Subtopics

#### 4.3. The Impact of Topic Correlation on the Formation Process of Multi-Dimensional Public Opinion

_{i}, individuals in the network will pay attention to certain subtopics and participate in the discussion of that subtopic according to their own characteristics and environment. The degree of individual attention will be affected by the degree of correlation between the current subtopic and the initial public opinion and the number of surrounding infected neighbors. Therefore, by analyzing the degree of correlation between the subtopic and the initial public opinion, we can further explain which subtopics can form derived public opinions. The simulation results are as follows.

_{0}of the first subtopic is also the highest, and the final public opinion dimension is 2. When θ~N(0.5, 0.3), the first and third derived subtopics have a relatively high relevance. From Figure 20b and Figure 21b, we can see that the peak value of the basic reproduction number R

_{0}of the first, second, and third subtopics is higher, and the final public opinion dimension is 4. When θ~N(0.9, 0.3), the correlation degree of each derived subtopic with the initial public opinion is relatively large, and the correlation degree of the second, fourth, sixth, and seventh subtopics even reaches 1. From Figure 20c and Figure 21c, the peak value of the basic reproduction number R

_{0}of the first, second, third, and fourth subtopics is higher, but the peak value reached by the curve has decreased, and the final public opinion dimension is 5. When θ~U[0, 1], the second, sixth, and seventh derived subtopics have a higher degree of correlation. From Figure 20d and Figure 21d, it can be seen that the peak value of the basic reproduction number R

_{0}of the second subtopic is higher. The final public opinion dimension is 4. It can be seen from the above phenomenon that the peak of the basic reproduction number of topics with a high degree of correlation is correspondingly higher, which means that it is easier to form derived public opinion. When all the subtopics have a relatively high degree of correlation with the initial public opinion, the subtopic with the highest degree of correlation is relatively high. Topics may not be able to form derived public opinion, but the subtopics generated in the early stage are more likely to become hot topics and form derived public opinions. The reason may be that when the correlation of subtopics is relatively high, an individual will have a sense of freshness in the early subtopics, and as time goes by, they will gradually lose interest in the initial public opinion, even if there is a degree of correlation with initial public opinion in the later period. The subtopics with higher correlation are not very popular, leading to the failure to form a new public opinion.

#### 4.4. The Impact of the Amount of Information Contained in Subtopics on the Formation Process of Multi-Dimensional Public Opinion

_{0}is low, and the final public opinion dimension is only 2. When σ~N(0.5, 0.3), the second and third derived subtopics contain more information. From Figure 23b and Figure 24b, it can be seen that the peak value of the basic reproduction number R

_{0}of the second and third subtopics is also the highest, and the final public opinion dimension is 4. When σ~N(0.9, 0.3), each derived subtopic contains a large amount of information. Among them, the amount of information contained in the first, third, fourth, and ninth subtopics even reaches 1. From Figure 23c and Figure 24c, we can see that the peak value of the basic reproduction number R

_{0}of the first, third, fourth, and fourth subtopics is also higher, and the final public opinion dimension is 6. When σ~U[0, 1], the third and fourth subtopics contain more information. From Figure 23d and Figure 24d, it can be seen that the peak of the basic reproduction number R

_{0}of the third and fourth subtopics is also higher, and the final public opinion dimension is 3. From the above phenomenon, it can be seen that the peak value of the basic reproduction number R

_{0}of the subtopic with a large amount of information is correspondingly higher, and it is easier to form a new one-dimensional derived topic.

#### 4.5. Combination Analysis of Various Factors Affecting the Multi-Dimensional Public Opinion

#### 4.6. The Influence of Network Topology on the Formation Process of Public Opinion Dimensions

_{U}= 0.7, g

_{0}= 0.6, N = 1000. The simulation results are shown in Figure 29 and Figure 30.

## 5. Empirical Analysis

## 6. Conclusions

- (1)
- When the degree of information alienation reaches a certain threshold, derived subtopics will be generated. In addition, when the degree of information alienation is high, the earliest derived subtopics may not necessarily form derived public opinion but later derived subtopics may also be generated. Derived public opinion is formed, and the degree of information alienation has a greater impact on the number of derived subtopics but has small impact on the dimensions of the final state of public opinion.
- (2)
- Environmental forces and the amount of information contained in subtopics are the key factors that affect the formation of multi-dimensional public opinion. Among them, environmental forces have a greater impact on the early subtopics, and the amount of information contained in subtopics is key to forming derived public opinion.
- (3)
- Subtopics that are highly related to the initial public opinion topic are more likely to form derived public opinions. When all subtopics are highly correlated with the initial public opinion, the subtopic with the highest degree of correlation may not necessarily form a derived public opinion, but subtopics generated in the early stage are more likely to form a derived public opinion.
- (4)
- The network topology does not have much impact on the number of subtopics, but it has a greater impact on the number of individuals participating in the discussion of the subtopics, and the dimensions of multidimensional public opinion formed by the network topology with a high aggregation coefficient and short average path length are greater.

- (1)
- This paper does not consider the influence of external information intervention in the study of the formation process of multi-dimensional public opinion and subsequent intervention mechanisms which can be introduced to study the influence of external information on initial public opinion and derived public opinion.
- (2)
- The paper considers the situation of static nodes without considering the increase or withdrawal of Internet users’ nodes [38]. In reality, individuals participating in discussion of derived subtopics often increase and decrease. Therefore, the multi-dimensional public opinion evolution mechanism under the dynamic network can be considered in the follow-up research.
- (3)
- In this paper, the dynamic equations for complex networks only considers the node average degree of uniform networks and cannot reflect the connections between each node and its neighbor. Therefore, more complex dynamic equations should be considered to simulate the infection process of each node in future.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 8.**The number of agents participating in subtopic communication under different information alienation. (

**a**) $\rho $ = 0.75. (

**b**) $\rho $ = 0.85. (

**c**) $\rho $ = 0.95.

**Figure 9.**Evolution of initial public opinion and subtopics under different information alienation settings. (

**a**) $\rho $ = 0.75. (

**b**) $\rho $ = 0.85. (

**c**) $\rho $ = 0.95.

**Figure 10.**Changes of subtopics’ R

_{0}values under different information alienation settings. (

**a**) $\rho $ = 0.75. (

**b**) $\rho $ = 0.85. (

**c**) $\rho $ = 0.95.

**Figure 11.**Changes of the number of topics under different information alienation settings. (

**a**) $\rho $ = 0.75. (

**b**) $\rho $ = 0.85. (

**c**) $\rho $ = 0.95.

**Figure 14.**The number of agents participating in subtopic communication under different environmental forces. (

**a**) δ = 3. (

**b**) δ = 5. (

**c**) δ = 7. (

**d**) δ = 9.

**Figure 15.**Evolution of initial public opinion and subtopics under different environmental forces. (

**a**) δ = 3. (

**b**) δ = 5. (

**c**) δ = 7. (

**d**) δ = 9.

**Figure 16.**The change of R

_{0}under different environmental forces. (

**a**) δ = 3. (

**b**) δ = 5. (

**c**) δ = 7. (

**d**) δ = 9.

**Figure 17.**The change of the number of subtopics under different environmental forces. (

**a**) δ = 3. (

**b**) δ = 5. (

**c**) δ = 7. (

**d**) δ = 9.

**Figure 19.**The correlation between derived subtopic number and initial public opinion. (

**a**) θ~N(0.3, 0.3). (

**b**) θ~N(0.5, 0.3). (

**c**) θ~N(0.9, 0.3). (

**d**) θ~U[0, 1].

**Figure 20.**The change of R

_{0}under different distributions of correlation. (

**a**) θ~N(0.3, 0.3). (

**b**) θ~N(0.5 0.3). (

**c**) θ~N(0.9, 0.3). (

**d**) θ~U[0, 1].

**Figure 21.**The change of the number of topics under different distributions of correlation. (

**a**) θ~N(0.3, 0.3). (

**b**) θ~N(0.5, 0.3). (

**c**) θ~N(0.9, 0.3). (

**d**) θ~U[0, 1].

**Figure 22.**The amount of information of each subtopic under different distributions. (

**a**) σ

_{i}~N(0.1. 0.3). (

**b**) σ

_{i}~N(0.5, 0.3). (

**c**) σ

_{i}~N(0.9, 0.3). (

**d**) σ

_{i}~U[0, 1].

**Figure 23.**The change of R

_{0}under different distributions of the amount of information. (

**a**) σ

_{i}~N(0.1, 0.3). (

**b**) σ

_{i}~N(0.5, 0.3). (

**c**) σ

_{i}~N(0.9, 0.3). (

**d**) σ

_{i}~U[0, 1].

**Figure 24.**The change of the number of topics under different distributions of the amount of information. (

**a**) σ

_{i}~N(0.1, 0.3). (

**b**) σ

_{i}~N(0.5, 0.3). (

**c**) σ

_{i}~N(0.9, 0.3). (

**d**) σ

_{i}~U[0, 1].

**Figure 28.**Combination analysis of environmental forces, correlation and the amount of information contained in subtopics. (

**a**) The relationship among dimension, environmental forces and correlation. (

**b**) The relationship among dimension, environmental forces and amount of information.

**Figure 29.**The number of agents participating in the discussion of derived subtopics under different network topologies. (

**a**) WS small world network. (

**b**) BA network. (

**c**) Fully connected network.

**Figure 30.**The change of number of topics under different network topologies. (

**a**) WS small world network. (

**b**) BA network. (

**c**) Fully connected network.

**Figure 32.**Simulation of derived subtopics for this public opinion. (

**a**)The numberof agents discussing derived subtopics. (

**b**) Change inthe number of topics.

**Figure 33.**Simulation of derived subtopics for this public opinion used the model proposed in this paper. (

**a**) The number of agents discussing derived subtopics. (

**b**) Change in the number of topics.

Parameters | Description |
---|---|

α | Infection rate |

β | Immunity rate |

ρ | Information alienation rate |

δ | Environmental forces |

θ_{i} | The topic correlation between the ith derived subtopic and the initial public opinion |

σ_{i} | The amount of information contained in the ith derived subtopic |

P_{U} | The parameter of highly variable degree threshold |

g_{0} | The attention threshold |

Variable | Description |
---|---|

S(t) | The numberof susceptible individuals in the network at time t |

I(t) | The number of infective individuals in the network at time t |

R(t) | The number of recovered individuals in the network at time t |

P | The degree of variation of initial public opinion |

g_{i} | Individual i’s attention to subtopics |

R_{0} | Basic reproduction number |

Network Name | Average Path Length | Aggregation Coefficient | Average |
---|---|---|---|

WS small world network | 5.3719 | 0.0088 | 4 |

BA network | 4.0282 | 0.034 | 3.964 |

Fully connected network | 1 | 1 | 999 |

Release Time | Topic | Reading Volume | Discussion Volume | Topic Number |
---|---|---|---|---|

11.09 | #One new local confirmed case of COVID-19 in Shanghai # | 240 m | 6607 | 0 |

11.10 | #Shanghai Epidemic Prevention and Control Work Conference# | 300 m | 18,000 | 1 |

11.21 | #One Community in Pudong New Area Was Upgraded to Medium Risk Degree# | 23.868 m | 594 | 2 |

11.21 | #One Community in Pudong New Area Will Be Upgraded to Medium Risk Degree Tomorrow# | 130 m | 3452 | 3 |

11.21 | #4015 people in Shanghai Pudong Hospital have been quarantined# | 450 m | 17,000 | 4 |

11.21 | #83 people that once contacted with infected person were tracked# | 100 m | 6383 | 5 |

11.21 | #1 new COVID-19 case confirmed among15,416 people in Shanghai# | 32.914 m | 1319 | 6 |

11.23 | #2 new local confirmed cases of COVID-19 in Shanghai# | 710 m | 28,000 | 7 |

11.23 | #One COVID-19 patient once exposed toan aviation container# | 310 m | 8109 | 8 |

11.29 | #Shanghai Songjiang # | 12.469 m | 3106 | 9 |

11.29 | #No COVID-19 in Shanghai Songjiang # | 6.118 m | 431 | 10 |

1129 | #Reasults of 6 local confirmed case of COVID-19 in Shanghai# | 190 m | 3670 | 11 |

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**MDPI and ACS Style**

Chen, T.; Yin, X.; Yang, J.; Cong, G.; Li, G.
Modeling Multi-Dimensional Public Opinion Process Based on Complex Network Dynamics Model in the Context of Derived Topics. *Axioms* **2021**, *10*, 270.
https://doi.org/10.3390/axioms10040270

**AMA Style**

Chen T, Yin X, Yang J, Cong G, Li G.
Modeling Multi-Dimensional Public Opinion Process Based on Complex Network Dynamics Model in the Context of Derived Topics. *Axioms*. 2021; 10(4):270.
https://doi.org/10.3390/axioms10040270

**Chicago/Turabian Style**

Chen, Tinggui, Xiaohua Yin, Jianjun Yang, Guodong Cong, and Guoping Li.
2021. "Modeling Multi-Dimensional Public Opinion Process Based on Complex Network Dynamics Model in the Context of Derived Topics" *Axioms* 10, no. 4: 270.
https://doi.org/10.3390/axioms10040270