Internet Public Opinion Diffusion: A Cross Perspective of Multilayer Network and Multisubject Association

Abstract

How to effectively regulate the spread of Internet public opinion from multisubject is the focus that governments around the world need to face. From the cross perspective of multilayer network and multisubject association, this paper studies the diffusion mechanism of Internet public opinion, constructs the multilayer network of Internet public opinion and its diffusion model, and then systematically analyzes the evolution characteristics of Internet public opinion diffusion. The main conclusions of the study are as follows: it is easier to trigger the spread of Internet public opinion by establishing associations among netizens in a preferred way. When the probability of random connection in different layers increases synchronously, the diffusion intensity of Internet public opinion shows a gradual downward trend. The intensity of government intervention has the greatest impact on the diffusion intensity of Internet public opinion. When the factors of netizens, Internet platform, and government interact, it can effectively reduce the intensity and scope of Internet public opinion diffusion. Once the media participate, it will improve the complexity of Internet public opinion diffusion. The government’s intervention in Internet public opinion is a “double-edged sword,” which needs to set the intervention intensity according to different subjects and their factor characteristics. This study can provide a theoretical reference for formulating strategies to adjust Internet public opinion diffusion from the cross perspective of multilayer network and multisubject association.

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

As the gathering place and intersection of all kinds of information, attitudes, views and behaviors, it is particularly urgent and important to carry out timely and effective management of the Internet [1]. Especially after the outbreak of hot events or problems, it will attract the extensive attention of the majority of netizens [2]. It is very easy to form bad public opinion under the guidance of negative thoughts and spread rapidly through different related media so as to induce public opinion crisis, which is not conducive to social harmony and stability. Therefore, facing the practical problems of the formation and diffusion of Internet public opinion, we should comprehensively consider the behavior of netizens, media views, platform information management, and government regulatory rules so as to realize the effective regulation of the diffusion of Internet public opinion. This is not only the focus that governments all over the world need to face but also the value of this paper.

At present, scholars have studied the influencing factors and evolution characteristics of Internet public opinion diffusion from different participants, including the following four aspects: First, in terms of netizens, existing studies have found that netizens are not only the core participants in Internet public opinion diffusion but also the main perturbs affecting the diffusion intensity [3–5]. Abeles et al. [6] and Dinh et al. [7] proposed that netizens’ attention and awareness of Internet public opinion may act on the whole process of diffusion through their close relationship. Second, in terms of media, there are differences in the existing research on the impact relationship between media and Internet public opinion. Some scholars believe that the reporting intensity and emotional tendency of media will positively affect the diffusion intensity of Internet public opinion [8, 9], but others believe that there is no positive correlation in the strict sense between the two, which may have a more complex impact relationship and needs to be further expanded [10, 11]. Third, in terms of the government, as the main regulator of Internet public opinion, the public opinion regulation measures and systems adopted by the government have always been an important way and means to affect Internet public opinion [12–14]. At the same time, since government regulation often involves multiparty linkage, the impact of the government on Internet public opinion will be more complex. Fourth, in terms of Internet platform, a large number of scholars believe that as an intermediary for information collection, release, and feedback, the information policies adopted by the platform will directly affect the formation and diffusion of Internet public opinion, but it is not clear how to play the specific role mechanism of the platform [15–17]. These studies mainly start from the perspective of a single subject and rarely comprehensively consider the complex impact of the interaction effect of netizens, media, government, and platform. This provides a perspective of multisubject association for this study. This is also the first motivation of this paper.

Complex network theory can simplify the complex system and help to improve the efficiency of analysis [18–22]. Therefore, complex network theory is widely used in many fields [23, 24], such as public opinion diffusion, traffic planning, and risk governance, and so on. At present, there are many researches on Internet public opinion diffusion based on complex network theory, which are mainly divided into two aspects. First is the Internet public opinion diffusion based on single-layer network [25–27]. This kind of research sets the relationship in the Internet public opinion network as the same nature and does not distinguish the possible heterogeneity between the relationships in detail. Although it is easier to simplify and analyze the problem, it is easy to cause network distortion, which may lead to some deviation in the research conclusion. Second is the Internet public opinion diffusion based on multilayer network [19, 28, 29]. Based on the problems existing in the above single-layer network, some scholars try to build a multilayer network of Internet public opinion from the perspective of correlation edge heterogeneity so as to fully reflect the differences of diffusion effects caused by the correlation relationship of different attributes in the process of Internet public opinion diffusion. This provides a multilayer network perspective for this paper, but the difference of this paper is that it constructs a multilayer network through online and offline relationships rather than from different emotional tendencies. At the same time, this paper comprehensively considers the multilayer network perspective and the multisubject correlation perspective, making the perspective more novel and in-depth analysis. This is the second motivation of this paper.

To sum up, this paper refines the Internet public opinion diffusion mechanism from the cross perspective of multilayer network and multisubject association and further simulates and analyzes the influence mechanism of multisubject factors on Internet public opinion diffusion based on the constructed Internet public opinion multilayer network and its diffusion model. The contributions and motivation of this paper are mainly reflected in the following aspects: (1) The model constructed in this paper differs from previous models. In that, first, the Internet public opinion network on which this model is built is endogenous network rather than exogenous network, which makes the correlation characteristics among Internet public opinion subjects closer to reality. Furthermore, the endogenous Internet public opinion network extends from single layer to multilayer and constructs Internet public opinion multilayer network through online and offline correlation attributes. Therefore, the multilayer network of Internet public opinion constructed is more in line with the correlation characteristics of real networks in this paper. Second, the constructed model involves the interaction of multiple subjects rather than the single subject effect. This paper comprehensively considers the complex impact of the multiple interaction effects of netizens, media, government, and Internet platform on the diffusion of Internet public opinion. On this basis, the Internet public opinion diffusion model and its evolution characteristics obtained are more realistic, and the proposed control strategy of Internet public opinion diffusion is more targeted and feasible in this paper. (2) This paper draws some novel and valuable research conclusions: there is a positive “U” correlation between media emotional tendency and Internet public opinion diffusion intensity. When the factors of netizens, Internet platform, and government interact, it can effectively reduce the intensity and scope of Internet public opinion diffusion. Once the media participate, it will improve the complexity of Internet public opinion diffusion. In addition, the government intervention in Internet public opinion is a “double-edged sword,” which needs to set the intervention intensity specifically according to the characteristics of different subjects and their factors.

The structure of this paper is organized as follows: the second part analyzes the Internet public opinion diffusion mechanism from the cross perspective of multilayer network and multisubject association. The third part constructs a multilayer network of Internet public opinion. The fourth part constructs the Internet public opinion diffusion model based on multilayer network. The fifth part simulates and analyzes the evolution characteristics of Internet public opinion diffusion. Finally, the research conclusion of this paper is obtained.

2. Internet Public Opinion Diffusion Mechanism

Internet public opinion diffusion refers to the process that the majority of netizens will spread their attitudes, views, and remarks on specific hot spots or focus issues through the Internet. In the process of the formation and diffusion of Internet public opinion, there are correlation relations between different participants and different attributes, which play different roles. In this paper, the mechanism of Internet public opinion diffusion under the interaction of multisubject association and multilayer network is described as shown in Figure 1:

[figure(s) omitted; refer to PDF]

After the occurrence of hot issues on the Internet, the information related to them, on the one hand, makes netizens form online public opinion association through online hot search, stock bar, or platform posting comments, and on the other hand, it makes netizens form offline public opinion association through circle of friends, circle of relatives, mobile communication, social software, etc. According to the complex network theory, these two kinds of associations can form online and offline public opinion association networks reflecting their internal relationship and association nature, respectively. At the same time, due to the inherent correlation between online and offline netizens, it can be further known from the multilayer network theory that the two types of public opinion related networks with different edges can form an Internet public opinion multilayer network including online public opinion related layer network and offline public opinion related layer network. In this multilayer network, because netizens are affected by other netizens in the offline public opinion network layer and online public opinion network layer at the same time, it will cause netizens in the whole network to pay extensive attention to and discuss this hot issue and express their respective attitudes, views, and remarks, resulting in the accelerated evolution of Internet public opinion. Moreover, in view of the irrational behavior of netizens, it will lead to the spread of Internet public opinion and a new round of network hot issues under the herd effect.

However, multiple participants such as netizens, media, government, and Internet platform not only play an important role in the formation and diffusion of Internet public opinion but also further affect the whole process of the formation and diffusion of Internet public opinion through the interaction with the multilayer network of Internet public opinion. On the one hand, the multiple participants will play a disturbing effect directly on the hot issues of widespread concern and discussion of the netizens through the different behavior choices they show and regulate the attitudes, opinions, and statements of netizens in the process. On the other hand, multiple participants will interact and change the formation and diffusion of Internet public opinion through the mutual behavior linkage. To further clarify the influence mechanism of multiple participants on the formation and diffusion of Internet public opinion, according to the existing research, this paper shows the influence mechanism of representative indicators of netizens, media, government, and Internet platform as follows:

(1) Netizen. The influence of netizens on the formation and diffusion of Internet public opinion is mainly reflected in two aspects: attention and awareness. First, the netizen attention $\alpha 0\le \alpha \le 1$ reflects the enthusiasm of netizen to pay attention to hot issues on the Internet and then affects the popularity of Internet public opinion [6, 30]. Therefore, the higher the netizen attention is, the higher the popularity of Internet public opinion is, and the stronger the impact on the formation and diffusion of Internet public opinion is. Second, the netizen recognition $\beta 0\le \beta \le 1$ reflects the ability of netizens to judge hot issues on the Internet and then affects the development direction of Internet public opinion [7]. Therefore, the higher the awareness of netizens is, the more rational they are about the hot issues in the Internet, and then affecting the further evolution of Internet public opinion.

(2) Media. The media reports on Internet public opinion are mainly reflected in two aspects: media coverage intensity and media emotion tendency. First, the media coverage intensity $\delta 0\le \delta \le 1$ reflects the sustainability of media reports, which in turn affects the spread of Internet public opinion [9]. Therefore, the higher the media coverage intensity is, the longer the impact of Internet hot issues is, and then the wider the spread of Internet public opinion is. Second, the media emotion tendency $\lambda 0\le \lambda \le 1$ reflects the different attitudes of the media when reporting hot issues on the Internet and then affects the development trend of Internet public opinion [10, 31]. Among them, the more $\lambda$ tends to 0, the more negative the mood of media reports is, and the more $\lambda$ tends to 1, the more positive the mood of media reports is, $\lambda =0.5$ indicates that the media reports are more objective.

(3) Government. The influence of the government on the formation and diffusion of Internet public opinion is mainly reflected in the intervention intensity and the perfection of government’s early warning mechanism. First, the government intervention intensity $\gamma 0\le \gamma \le 1$ reflects the government influence on Internet public opinion and then acts on the diffusion range of Internet public opinion [14, 32]. Therefore, the higher the government intervention intensity is, the greater the impact on the formation and diffusion of Internet public opinion is. Second, the perfection of government’s early warning mechanism $\kappa 0\le \kappa \le 1$ reflects the perfection of the government's early warning mechanism for Internet public opinion monitoring, which affects the timeliness of Internet public opinion [12]. Therefore, the higher the perfection of government’s early warning mechanism is, the stronger the timeliness of Internet public opinion is.

(4) Internet Platform. The impact of Internet platform on the formation and diffusion of Internet public opinion is mainly reflected in two aspects: the platform openness and the platform audit strength. First, the platform openness $\tau 0\le \tau \le 1$ reflects the free participation of all kinds of people and the free release of all kinds of information, which affects the conditions for the formation and diffusion of Internet public opinion [15]. Therefore, the higher the platform openness is, the stronger the conditions for the formation and diffusion of Internet public opinion are. Second, the platform audit strength $\eta 0\le \eta \le 1$ reflects the audit strength of bad and illegal content, highlights the bottom line rule awareness of the platform, and then affects the compliance of Internet public opinion [17]. Therefore, the stronger the platform audit strength is, the higher the compliance of Internet public opinion is.

3. Multilayer Network Construction of Internet Public Opinion

Under the influence of Internet public opinion communication, the public establish connections through news reports, online hot search, circle of friends, circle of relatives, and so on. At the same time, because Internet public opinion presents two types of online public opinion and offline public opinion in the dissemination process, the public association is manifested as online public opinion association and offline public opinion association. Among them, online public opinion association mainly refers to the public opinion association generated by the public through online hot search, stock bar, or platform posting comments. Offline public opinion association mainly refers to the public opinion association generated by offline circle of friends, circle of relatives, mobile communication, social software, and other means. Therefore, according to different public opinion associations, this part constructs a multilayer network of Internet public opinion, which include two layers: online public opinion association layer and offline public opinion association layer. Moreover, in the multilayer network, each node represents a participant, and the degree of node is affected by both nodes in the same layer and nodes in other layers. This paper describes the formation algorithm of Internet public opinion multilayer network as follows:

(1) In the initial Internet public opinion multilayer network, the online public opinion association layer network $\mathrm{O}\mathrm{N}$ is composed of ${\epsilon }_0^{\mathrm{O}\mathrm{N}}$ participants and ${\omega }_0^{\mathrm{O}\mathrm{N}}$ online public opinion associations, and the offline public opinion association network $\mathrm{O}\mathrm{F}$ is composed of ${\epsilon }_0^{\mathrm{O}\mathrm{F}}$ participants and ${\omega }_0^{\mathrm{O}\mathrm{F}}$ offline public opinion associations.

(2) When the multiple interaction effects between different participants are not considered, the newly added individuals in the online public opinion association layer network $\mathrm{O}\mathrm{N}$ and offline public opinion association layer network $\mathrm{O}\mathrm{F}$ will be randomly connected with the existing individuals with probability ${\mu }_{\mathrm{O}\mathrm{N}}$ and probability ${\mu }_{\mathrm{O}\mathrm{F}}\text{,}$ respectively, and preferentially connected with the existing individuals with probability $1-{\mu }_{\mathrm{O}\mathrm{N}}$ and probability $1-{\mu }_{\mathrm{O}\mathrm{F}}\text{,}$ respectively ($0\le {\mu }_{\mathrm{O}\mathrm{N}},{\mu }_{\mathrm{O}\mathrm{F}}\le 1$). When considering the multiple interaction effects between different participants, the online public opinion association layer network and offline public opinion association layer network are randomly connected, respectively:$$\begin{array}{}\text{(1)}& {\mu }^{\ast }={\mu }_{\mathrm{O}\mathrm{N},}^{\rho }{\mu }^{·}={\mu }_{\mathrm{O}\mathrm{F}}^{1-\rho }\text{.}\end{array}$$

  Among them, $\rho$ is the Internet public opinion diffusion function under the multiple interaction between different participants, and its specific form will be described in detail later.

(3) In the random connection process, in each time period, ${\epsilon }_{\mathrm{O}\mathrm{N}}$ (or ${\epsilon }_{\mathrm{O}\mathrm{F}}$), new individuals join in the online public opinion association layer network $\mathrm{O}\mathrm{N}$ (or offline public opinion association layer network $\mathrm{O}\mathrm{F}$), and each participant has ${\theta }_{\mathrm{O}\mathrm{N}}$ (or ${\theta }_{OF}$) online public opinion association relationships (or offline public opinion association relationships). In the process of preferential connection, when new participants join, the expression of the probability ${\prod }_i^{\mathrm{O}\mathrm{N}}$ of connecting with existing participants $i$ is$$\begin{array}{}\text{(2)}& {\prod }_i^{\mathrm{O}\mathrm{N}}=\frac{k_i^{\mathrm{O}\mathrm{N}}}{{\displaystyle \sum }_j{k}_j^{\mathrm{O}\mathrm{N}}+{\displaystyle \sum }_s{k}_s^{\mathrm{O}\mathrm{F}}}\text{.}\end{array}$$

According to the above algorithm, the change rate of degree $\overline{{\phi }_j^A}=\overline{{\phi }_l^B}=\overline{\phi }$ of participant $m_0^B$ can be expressed as$$\begin{array}{}\text{(3)}& \begin{array}{c}\frac{\partial k_i}{\partial t}={\epsilon }_{\mathrm{O}\mathrm{N}}{\theta }_{\mathrm{O}\mathrm{N}}1-{\mu }^{\ast }+{\epsilon }_{\mathrm{O}\mathrm{F}}{\theta }_{\mathrm{O}\mathrm{F}}1-{\mu }^{·}{\prod }_i^{\mathrm{O}\mathrm{N}}+\frac{{\epsilon }_{\mathrm{O}\mathrm{N}}{\theta }_{\mathrm{O}\mathrm{N}}{\mu }^{\ast }+{\epsilon }_{\mathrm{O}\mathrm{F}}{\theta }_{\mathrm{O}\mathrm{F}}{\mu }^{·}}{{\epsilon }_0^{\mathrm{O}\mathrm{N}}+{\epsilon }_0^{\mathrm{O}\mathrm{F}}+{\epsilon }_{\mathrm{O}\mathrm{N}}+{\epsilon }_{\mathrm{O}\mathrm{F}}t}\\ =\frac{{\epsilon }_{\mathrm{O}\mathrm{N}}{\theta }_{\mathrm{O}\mathrm{N}}1-{\mu }^{\ast }+{\epsilon }_{\mathrm{O}\mathrm{F}}{\theta }_{\mathrm{O}\mathrm{F}}1-{\mu }^{·}{k}_i^{\mathrm{O}\mathrm{N}}}{{\displaystyle \sum }_j{k}_j^{\mathrm{O}\mathrm{N}}+{\displaystyle \sum }_s{k}_s^{\mathrm{O}\mathrm{F}}}+\frac{{\epsilon }_{\mathrm{O}\mathrm{N}}{\theta }_{\mathrm{O}\mathrm{N}}{\mu }^{\ast }+{\epsilon }_{\mathrm{O}\mathrm{F}}{\theta }_{\mathrm{O}\mathrm{F}}{\mu }^{·}}{{\epsilon }_0^{\mathrm{O}\mathrm{N}}+{\epsilon }_0^{\mathrm{O}\mathrm{F}}+{\epsilon }_{\mathrm{O}\mathrm{N}}+{\epsilon }_{\mathrm{O}\mathrm{F}}t}\text{.}\end{array}\end{array}$$

When $t⟶\infty$ and ${\epsilon }_0^{\mathrm{O}\mathrm{N}}+{\epsilon }_0^{\mathrm{O}\mathrm{F}}+{\epsilon }_{\mathrm{O}\mathrm{N}}+{\epsilon }_{\mathrm{O}\mathrm{F}}t\approx {\epsilon }_{\mathrm{O}\mathrm{N}}+{\epsilon }_{\mathrm{O}\mathrm{F}}t$, Equation (3) can be expressed as$$\begin{array}{}\text{(4)}& \frac{\partial k_i}{\partial t}=\frac{{\epsilon }_{\mathrm{O}\mathrm{N}}{\theta }_{\mathrm{O}\mathrm{N}}1-{\mu }^{\ast }+{\epsilon }_{\mathrm{O}\mathrm{F}}{\theta }_{\mathrm{O}\mathrm{F}}1-{\mu }^{·}{k}_i^{\mathrm{O}\mathrm{N}}}{{\displaystyle \sum }_j{k}_j^{\mathrm{O}\mathrm{N}}+{\displaystyle \sum }_s{k}_s^{\mathrm{O}\mathrm{F}}}+\frac{{\epsilon }_{\mathrm{O}\mathrm{N}}{\theta }_{\mathrm{O}\mathrm{N}}{\mu }^{\ast }+{\epsilon }_{\mathrm{O}\mathrm{F}}{\theta }_{\mathrm{O}\mathrm{F}}{\mu }^{·}}{{\epsilon }_{\mathrm{O}\mathrm{N}}+{\epsilon }_{\mathrm{O}\mathrm{F}}t}\text{.}\end{array}$$

In addition, at $t$, due to ${\displaystyle \sum }_j{k}_j^{\mathrm{O}\mathrm{N}}=2{\epsilon }_{\mathrm{O}\mathrm{N}}{\theta }_{\mathrm{O}\mathrm{N}}t+{\epsilon }_0^{\mathrm{O}\mathrm{N}}$ and ${\displaystyle \sum }_s{k}_s^{\mathrm{O}\mathrm{F}}=2{\epsilon }_{\mathrm{O}\mathrm{F}}{\theta }_{\mathrm{O}\mathrm{F}}t+{\epsilon }_0^{\mathrm{O}\mathrm{F}}$ and when $t⟶\infty$, Equation (4) can be further expressed as$$\begin{array}{c}\text{(5)}& \frac{\partial k_i}{\partial t}=\frac{{\epsilon }_{ON}{\theta }_{ON}1-{\mu }^{\ast }+{\epsilon }_{\mathrm{O}\mathrm{F}}{\theta }_{\mathrm{O}\mathrm{F}}1-{\mu }^{·}{k}_i^{\mathrm{O}\mathrm{N}}}{2{\epsilon }_{\mathrm{O}\mathrm{N}}{\theta }_{\mathrm{O}\mathrm{N}}t+{\epsilon }_0^{\mathrm{O}\mathrm{N}}+{\epsilon }_{\mathrm{O}\mathrm{F}}{\theta }_{\mathrm{O}\mathrm{F}}t+{\epsilon }_0^{OF}}+\frac{{\epsilon }_{\mathrm{O}\mathrm{N}}{\theta }_{\mathrm{O}\mathrm{N}}{\mu }^{\ast }+{\epsilon }_{\mathrm{O}\mathrm{F}}{\theta }_{\mathrm{O}\mathrm{F}}{\mu }^{·}}{{\epsilon }_{\mathrm{O}\mathrm{N}}+{\epsilon }_{\mathrm{O}\mathrm{F}}t}=\frac{{\epsilon }_{\mathrm{O}\mathrm{N}}{\theta }_{\mathrm{O}\mathrm{N}}1-{\mu }^{\ast }+{\epsilon }_{\mathrm{O}\mathrm{F}}{\theta }_{\mathrm{O}\mathrm{F}}1-{\mu }^{·}{k}_i^{ON}}{2t{\epsilon }_{\mathrm{O}\mathrm{N}}{\theta }_{\mathrm{O}\mathrm{N}}+{\epsilon }_{\mathrm{O}\mathrm{F}}{\theta }_{\mathrm{O}\mathrm{F}}}+\frac{{\epsilon }_{\mathrm{O}\mathrm{N}}{\theta }_{\mathrm{O}\mathrm{N}}{\mu }^{\ast }+{\epsilon }_{\mathrm{O}\mathrm{F}}{\theta }_{\mathrm{O}\mathrm{F}}{\mu }^{·}}{{\epsilon }_{\mathrm{O}\mathrm{N}}+{\epsilon }_{\mathrm{O}\mathrm{F}}t}\text{.}\end{array}$$

Therefore, the solution of Equation (5) can be obtained as$$\begin{array}{}\text{(6)}& k_i^{\mathrm{O}\mathrm{N}}t=C\cdot t^{\frac{{\epsilon }_{\mathrm{O}\mathrm{N}}{\theta }_{\mathrm{O}\mathrm{N}}1-{\mu }^{\ast }+{\epsilon }_{\mathrm{O}\mathrm{F}}{\theta }_{\mathrm{O}\mathrm{F}}1-{\mu }^{·}}{2{\epsilon }_{\mathrm{O}\mathrm{N}}{\theta }_{\mathrm{O}\mathrm{N}}+{\epsilon }_{\mathrm{O}\mathrm{F}}{\theta }_{\mathrm{O}\mathrm{F}}}}-\frac{2{\epsilon }_{\mathrm{O}\mathrm{N}}{\theta }_{\mathrm{O}\mathrm{N}}+{\epsilon }_{\mathrm{O}\mathrm{F}}{\theta }_{\mathrm{O}\mathrm{F}}{\epsilon }_{\mathrm{O}\mathrm{N}}{\theta }_{\mathrm{O}\mathrm{N}}{\mu }^{\ast }+{\epsilon }_{\mathrm{O}\mathrm{F}}{\theta }_{\mathrm{O}\mathrm{F}}{\mu }^{·}}{{\epsilon }_{\mathrm{O}\mathrm{N}}{\theta }_{\mathrm{O}\mathrm{N}}1-{\mu }^{\ast }+{\epsilon }_{\mathrm{O}\mathrm{F}}{\theta }_{\mathrm{O}\mathrm{F}}1-{\mu }^{·}{\epsilon }_{\mathrm{O}\mathrm{N}}+{\epsilon }_{\mathrm{O}\mathrm{F}}}\text{.}\end{array}$$

According to the given initial condition $k_i^{ON}t={\epsilon }_{ON}{\theta }_{ON}+{\epsilon }_{OF}{\theta }_{OF}$, the particular solution of Equation (6) is$$\begin{array}{c}\text{(7)}& k_i^{\mathrm{O}\mathrm{N}}t={\epsilon }_{\mathrm{O}\mathrm{N}}{\theta }_{\mathrm{O}\mathrm{N}}+{\epsilon }_{\mathrm{O}\mathrm{F}}{\theta }_{\mathrm{O}\mathrm{F}}+\frac{2{\epsilon }_{\mathrm{O}\mathrm{N}}{\theta }_{\mathrm{O}\mathrm{N}}+{\epsilon }_{\mathrm{O}\mathrm{F}}{\theta }_{\mathrm{O}\mathrm{F}}{\epsilon }_{\mathrm{O}\mathrm{N}}{\theta }_{\mathrm{O}\mathrm{N}}{\mu }^{\ast }+{\epsilon }_{\mathrm{O}\mathrm{F}}{\theta }_{\mathrm{O}\mathrm{F}}{\mu }^{·}}{{\epsilon }_{\mathrm{O}\mathrm{N}}{\theta }_{\mathrm{O}\mathrm{N}}1-{\mu }^{\ast }+{\epsilon }_{\mathrm{O}\mathrm{F}}{\theta }_{\mathrm{O}\mathrm{F}}1-{\mu }^{·}{\epsilon }_{\mathrm{O}\mathrm{N}}{\theta }_{\mathrm{O}\mathrm{N}}+{\epsilon }_{\mathrm{O}\mathrm{F}}{\theta }_{\mathrm{O}\mathrm{F}}}{\frac{t}{t_i}}^{{\epsilon }_{\mathrm{O}\mathrm{N}}{\theta }_{\mathrm{O}\mathrm{N}}1-{\mu }^{\ast }+{\epsilon }_{\mathrm{O}\mathrm{F}}{\theta }_{\mathrm{O}\mathrm{F}}1-{\mu }^{·}/2{\epsilon }_{\mathrm{O}\mathrm{N}}{\theta }_{\mathrm{O}\mathrm{N}}+{\epsilon }_{\mathrm{O}\mathrm{F}}{\theta }_{\mathrm{O}\mathrm{F}}}\\ -\frac{2{\epsilon }_{\mathrm{O}\mathrm{N}}{\theta }_{\mathrm{O}\mathrm{N}}+{\epsilon }_{\mathrm{O}\mathrm{F}}{\theta }_{\mathrm{O}\mathrm{F}}{\epsilon }_{\mathrm{O}\mathrm{N}}{\theta }_{\mathrm{O}\mathrm{N}}{\mu }^{\ast }+{\epsilon }_{\mathrm{O}\mathrm{F}}{\theta }_{\mathrm{O}\mathrm{F}}{\mu }^{·}}{{\epsilon }_{\mathrm{O}\mathrm{N}}{\theta }_{\mathrm{O}\mathrm{N}}1-{\mu }^{\ast }+{\epsilon }_{\mathrm{O}\mathrm{F}}{\theta }_{\mathrm{O}\mathrm{F}}1-{\mu }^{·}{\epsilon }_{\mathrm{O}\mathrm{N}}+{\epsilon }_{\mathrm{O}\mathrm{F}}}\text{.}\end{array}$$

In this network, the probability that the degree $k_i^{ON}t$ of participant $i$ is less than $k$ is$$\begin{array}{}\text{(8)}& \begin{array}{c}{P}^{\mathrm{O}\mathrm{N}}{k}_i^{\mathrm{O}\mathrm{N}}t<k=1\\ -P^{\mathrm{O}\mathrm{N}}{t}_i\le t{\frac{k{\epsilon }_{\mathrm{O}\mathrm{N}}+{\epsilon }_{\mathrm{O}\mathrm{F}}{\epsilon }_{\mathrm{O}\mathrm{N}}{\theta }_{\mathrm{O}\mathrm{N}}1-{\mu }^{\ast }+{\epsilon }_{\mathrm{O}\mathrm{F}}{\theta }_{\mathrm{O}\mathrm{F}}1-{\mu }^{·}+2{\epsilon }_{\mathrm{O}\mathrm{N}}{\theta }_{\mathrm{O}\mathrm{N}}+{\epsilon }_{\mathrm{O}\mathrm{F}}{\theta }_{\mathrm{O}\mathrm{F}}{\epsilon }_{\mathrm{O}\mathrm{N}}{\theta }_{\mathrm{O}\mathrm{N}}{\mu }^{\ast }+{\epsilon }_{\mathrm{O}\mathrm{F}}{\theta }_{\mathrm{O}\mathrm{F}}{\mu }^{·}}{2{\epsilon }_{\mathrm{O}\mathrm{N}}+{\epsilon }_{\mathrm{O}\mathrm{F}}{{\epsilon }_{\mathrm{O}\mathrm{N}}{\theta }_{\mathrm{O}\mathrm{N}}+{\epsilon }_{\mathrm{O}\mathrm{F}}{\theta }_{\mathrm{O}\mathrm{F}}}^2+{\epsilon }_{\mathrm{O}\mathrm{N}}{\theta }_{\mathrm{O}\mathrm{N}}{\mu }^{\ast }+{\epsilon }_{\mathrm{O}\mathrm{F}}{\theta }_{\mathrm{O}\mathrm{F}}{\mu }^{·}}}^{2{\epsilon }_{\mathrm{O}\mathrm{N}}{\theta }_{\mathrm{O}\mathrm{N}}+{\epsilon }_{\mathrm{O}\mathrm{F}}{\theta }_{\mathrm{O}\mathrm{F}}/{\epsilon }_{\mathrm{O}\mathrm{N}}{\theta }_{\mathrm{O}\mathrm{N}}1-{\mu }^{\ast }+{\epsilon }_{\mathrm{O}\mathrm{F}}{\theta }_{\mathrm{O}\mathrm{F}}1-{\mu }^{·}-}\text{.}\end{array}\end{array}$$

In the multilayer network of Internet public opinion constructed in this paper, at $t$, ${\epsilon }_{ON}$ and ${\epsilon }_{OF}$ new individuals join in the online public opinion association layer network $ON$ and offline public opinion association layer network $OF\text{,}$ respectively. Therefore, the probability density of participant $k$ selected at $k_i^{A}t$ is$$\begin{array}{}\text{(9)}& P_i=\frac{1}{{\epsilon }_0^{\mathrm{O}\mathrm{N}}+{\epsilon }_0^{\mathrm{O}\mathrm{F}}+{\epsilon }_{\mathrm{O}\mathrm{N}}+{\epsilon }_{\mathrm{O}\mathrm{F}}t}\text{.}\end{array}$$

By substituting Equations (9) into (8), we get$$\begin{array}{}\text{(10)}& \begin{array}{c}{P}^{\mathrm{O}\mathrm{N}}{k}_i^{\mathrm{O}\mathrm{N}}t<k=1-\frac{t}{{\epsilon }_0^{ON}+{\epsilon }_0^{OF}+{\epsilon }_{ON}+{\epsilon }_{OF}t}\\ \times {\frac{k{\epsilon }_{\mathrm{O}\mathrm{N}}+{\epsilon }_{\mathrm{O}\mathrm{F}}{\epsilon }_{\mathrm{O}\mathrm{N}}{\theta }_{\mathrm{O}\mathrm{N}}1-{\mu }^{\ast }+{\epsilon }_{\mathrm{O}\mathrm{F}}{\theta }_{\mathrm{O}\mathrm{F}}1-{\mu }^{·}+2{\epsilon }_{\mathrm{O}\mathrm{N}}{\theta }_{\mathrm{O}\mathrm{N}}+{\alpha }_{\mathrm{O}\mathrm{F}}{\theta }_{\mathrm{O}\mathrm{F}}{\epsilon }_{\mathrm{O}\mathrm{N}}{\theta }_{\mathrm{O}\mathrm{N}}{\mu }^{\ast }+{\epsilon }_{\mathrm{O}\mathrm{F}}{\theta }_{\mathrm{O}\mathrm{F}}{\mu }^{·}}{2{\epsilon }_{\mathrm{O}\mathrm{N}}+{\epsilon }_{\mathrm{O}\mathrm{F}}{{\epsilon }_{\mathrm{O}\mathrm{N}}{\theta }_{\mathrm{O}\mathrm{N}}+{\epsilon }_{\mathrm{O}\mathrm{F}}{\theta }_{\mathrm{O}\mathrm{F}}}^2+{\epsilon }_{\mathrm{O}\mathrm{N}}{\theta }_{\mathrm{O}\mathrm{N}}{\mu }^{\ast }+{\epsilon }_{\mathrm{O}\mathrm{F}}{\theta }_{\mathrm{O}\mathrm{F}}{\mu }^{·}}}^{{\epsilon }_{\mathrm{O}\mathrm{N}}{\theta }_{\mathrm{O}\mathrm{N}}{\mu }^{\ast }-3+{\epsilon }_{\mathrm{O}\mathrm{F}}{\theta }_{\mathrm{O}\mathrm{F}}{\mu }^{·}-3/{\epsilon }_{\mathrm{O}\mathrm{N}}{\theta }_{\mathrm{O}\mathrm{N}}1-{\mu }^{\ast }+{\epsilon }_{\mathrm{O}\mathrm{F}}{\theta }_{\mathrm{O}\mathrm{F}}1-{\mu }^{·}}\text{.}\end{array}\end{array}$$

Therefore, it can be obtained that the degree distribution function of participant $i$ in the online public opinion association layer network $ON$ in the whole Internet public opinion multilayer network at $t$ is$$\begin{array}{c}\text{(11)}& P^{\mathrm{O}\mathrm{N}}k=\frac{\partial P^{\mathrm{O}\mathrm{N}}{k}_{i}t<k}{\partial k}=\frac{t}{{\epsilon }_0^{\mathrm{O}\mathrm{N}}+{\epsilon }_0^{\mathrm{O}\mathrm{F}}+{\epsilon }_{\mathrm{O}\mathrm{N}}+{\epsilon }_{\mathrm{O}\mathrm{F}}t}\times \\ {\frac{k{\epsilon }_{\mathrm{O}\mathrm{N}}+{\epsilon }_{\mathrm{O}\mathrm{F}}{\epsilon }_{\mathrm{O}\mathrm{N}}{\theta }_{\mathrm{O}\mathrm{N}}1-{\mu }^{\ast }+{\epsilon }_{\mathrm{O}\mathrm{F}}{\theta }_{\mathrm{O}\mathrm{F}}1-{\mu }^{·}+2{\epsilon }_{\mathrm{O}\mathrm{N}}{\theta }_{\mathrm{O}\mathrm{N}}+{\alpha }_{\mathrm{O}\mathrm{F}}{\theta }_{\mathrm{O}\mathrm{F}}{\epsilon }_{\mathrm{O}\mathrm{N}}{\theta }_{\mathrm{O}\mathrm{N}}{\mu }^{\ast }+{\epsilon }_{\mathrm{O}\mathrm{F}}{\theta }_{\mathrm{O}\mathrm{F}}{\mu }^{·}}{2{\epsilon }_{\mathrm{O}\mathrm{N}}+{\epsilon }_{\mathrm{O}\mathrm{F}}{{\epsilon }_{\mathrm{O}\mathrm{N}}{\theta }_{\mathrm{O}\mathrm{N}}+{\epsilon }_{\mathrm{O}\mathrm{F}}{\theta }_{\mathrm{O}\mathrm{F}}}^2+{\epsilon }_{\mathrm{O}\mathrm{N}}{\theta }_{\mathrm{O}\mathrm{N}}{\mu }^{\ast }+{\epsilon }_{\mathrm{O}\mathrm{F}}{\theta }_{\mathrm{O}\mathrm{F}}{\mu }^{·}}}^{{\epsilon }_{\mathrm{O}\mathrm{N}}{\theta }_{\mathrm{O}\mathrm{N}}{\mu }^{\ast }-3+{\epsilon }_{\mathrm{O}\mathrm{F}}{\theta }_{\mathrm{O}\mathrm{F}}{\mu }^{·}-3/{\epsilon }_{\mathrm{O}\mathrm{N}}{\theta }_{\mathrm{O}\mathrm{N}}1-{\mu }^{\ast }+{\epsilon }_{\mathrm{O}\mathrm{F}}{\theta }_{\mathrm{O}\mathrm{F}}1-{\mu }^{·}}.\end{array}$$

Moreover, when $t⟶\infty$, Equation (11) can be further expressed as$$\begin{array}{c}\text{(12)}& P^{\mathrm{O}\mathrm{N}}k=\frac{\partial P^{\mathrm{O}\mathrm{N}}{k}_{i}t<k}{\partial k}=\frac{1}{{\epsilon }_{\mathrm{O}\mathrm{N}}+{\epsilon }_{\mathrm{O}\mathrm{F}}}\times \\ {\frac{k{\epsilon }_{\mathrm{O}\mathrm{N}}+{\epsilon }_{\mathrm{O}\mathrm{F}}{\epsilon }_{\mathrm{O}\mathrm{N}}{\theta }_{\mathrm{O}\mathrm{N}}1-{\mu }^{\ast }+{\epsilon }_{\mathrm{O}\mathrm{F}}{\theta }_{\mathrm{O}\mathrm{F}}1-{\mu }^{·}+2{\epsilon }_{\mathrm{O}\mathrm{N}}{\theta }_{\mathrm{O}\mathrm{N}}+{\epsilon }_{\mathrm{O}\mathrm{F}}{\theta }_{\mathrm{O}\mathrm{F}}{\epsilon }_{\mathrm{O}\mathrm{N}}{\theta }_{\mathrm{O}\mathrm{N}}{\mu }^{\ast }+{\epsilon }_{\mathrm{O}\mathrm{F}}{\theta }_{\mathrm{O}\mathrm{F}}{\mu }^{·}}{2{\epsilon }_{\mathrm{O}\mathrm{N}}+{\epsilon }_{OF}{{\epsilon }_{\mathrm{O}\mathrm{N}}{\theta }_{\mathrm{O}\mathrm{N}}+{\epsilon }_{\mathrm{O}\mathrm{F}}{\theta }_{\mathrm{O}\mathrm{F}}}^2+{\epsilon }_{\mathrm{O}\mathrm{N}}{\theta }_{\mathrm{O}\mathrm{N}}{\mu }^{\ast }+{\epsilon }_{\mathrm{O}\mathrm{F}}{\theta }_{\mathrm{O}\mathrm{F}}{\mu }^{·}}}^{{\epsilon }_{\mathrm{O}\mathrm{N}}{\theta }_{\mathrm{O}\mathrm{N}}{\mu }^{\ast }-3+{\epsilon }_{\mathrm{O}\mathrm{F}}{\theta }_{\mathrm{O}\mathrm{F}}{\mu }^{·}-3/{\epsilon }_{\mathrm{O}\mathrm{N}}{\theta }_{\mathrm{O}\mathrm{N}}1-{\mu }^{\ast }+{\epsilon }_{\mathrm{O}\mathrm{F}}{\theta }_{\mathrm{O}\mathrm{F}}1-{\mu }^{·}.}\end{array}$$

For the same reason, the degree distribution function of participant $i$ in the offline public opinion association layer network $\mathrm{O}\mathrm{F}$ is similar to Equation (12) at $t$. Therefore, in this paper, the degree distribution function $Pk$ of participant $i$ in Internet public opinion multilayer network is expressed by Equation (12).

4. Internet Public Opinion Diffusion Model Based on Multilayer Network

The Internet public opinion diffusion model constructed in this paper is mainly based on the epidemic model in the complex network theory. The epidemic model can accurately depict the mutual transformation mechanism among subjects with different natures. At the same time, the process of Internet public opinion diffusion also involves the mutual transformation among subjects with different natures. This shows that the epidemic model is applicable to the Internet public opinion diffusion. In addition, when drawing on the epidemic model, according to the reality of Internet public opinion diffusion, this paper expands the classic SIR model to the SIRS model, which makes the conversion mechanism involved in Internet public opinion diffusion more complex and reflects the nature of different subjects.

4.1. Internet Public Opinion Diffusion Rules

With the emergence of hot issues in the Internet, netizens present three communication states in the process of public opinion formation and diffusion: one is susceptible netizen (SN), that is, netizens who pay attention to hot issues but have not formed motivation and action to spread Internet public opinion, or actively spread Internet public opinion under the influence of associated infected netizens; infectious netizen (IN) is highly concerned about hot issues and has the motivation and action to actively spread Internet public opinion; recovery netizen (RN) has a clear understanding of hot issues and has a high ability to judge the attitude, words, and deeds of associated infected netizens without spreading Internet public opinion. In this paper, we use $\mathrm{S}\mathrm{N}$, $\mathrm{I}\mathrm{N}$, and $\mathrm{R}\mathrm{N}$ to represent the number of these three types of netizens on the Internet. In this paper, the Internet public opinion diffusion rules caused by the transformation of the communication state of these three types of netizens are expressed as follows:

(1) After the emergence of hot issues in the Internet, it is assumed that susceptible netizen may be transformed into infectious netizen with $\rho$ ($0\le \rho \le 1$) probability. At the same time, susceptible netizen may be transformed into recovery netizen with $\chi$ ($0\le \chi \le 1$) probability.

(2) With the continuous evolution of hot issues and their public opinion, infectious netizen may change into recovery netizen with $\phi$ ($0\le \phi \le 1$) probability. At the same time, some recovery netizen may be affected by environmental changes and changes into susceptible netizen with $\varphi$ ($0\le \varphi \le 1$) probability.

(3) It is assumed that in each time period, new netizens enter the Internet public opinion multilayer network with $\xi$ ($0\le \xi \le 1$) probability, and some netizens exit with $\alpha \sigma$ ($0\le \sigma \le 1$) probability.

4.2. Internet Public Opinion Diffusion Model

This paper assumes that $N$ is the total number of netizens in the Internet public opinion multilayer network. $sn$, $in$, and $rn$ account for the proportion of the susceptible netizen, the infectious netizen, and the recovery netizen, respectively, namely, $sn=\mathrm{S}\mathrm{N}/N$, $iN=\mathrm{I}\mathrm{N}/N$, $\mathrm{r}\mathrm{n}=\mathrm{R}\mathrm{N}/N$, and $\mathrm{s}\mathrm{n}+\mathrm{i}\mathrm{n}+\mathrm{r}\mathrm{n}=1$ ($0\le \mathrm{s}\mathrm{n},\mathrm{i}\mathrm{n},\mathrm{r}\mathrm{n}\le 1$). Moreover, it is assumed that at time $t$, the density of infectious netizen with degree $k$ is $i{n}_{k}t$, and the probability of connection between susceptible netizen and infectious netizen is $\mathrm{\Theta }t$. Based on the above analysis and the setting method of risk contagion function by Wang et al. [33], this paper defines the Internet public opinion diffusion function $G^{\mathrm{P}\mathrm{O}}\alpha ,\beta ,\delta ,\lambda ,\gamma ,\kappa ,\tau ,\eta$ as$$\begin{array}{}\text{(13)}& G^{PO}\alpha ,\beta ,\delta ,\lambda ,\gamma ,\kappa ,\tau ,\eta ={\alpha }^{\beta }{e}^{-1/6\delta }{{\lambda }^2-\lambda +1}^2{\gamma }^{-1}1-e^{-1/{\kappa }^2}{\tau }^{1/4e}{e}^{-3\eta }\text{.}\end{array}$$

Therefore, according to Equation (13), the Internet public opinion diffusion rate $\rho$ under the multisubject association is expressed as$$\begin{array}{}\text{(14)}& \rho =G^{PO}\alpha ,\beta ,\delta ,\lambda ,\gamma ,\kappa ,\tau ,\eta ={\alpha }^{\beta }{e}^{-1/6\delta }{{\lambda }^2-\lambda +1}^2{\gamma }^{-1}1-e^{-1/{\kappa }^2}{\tau }^{-1/4}{e}^{-3\eta }\text{.}\end{array}$$

According to Internet public opinion diffusion rules and mean field theory [33, 34], the differential equations of Internet public opinion diffusion model under the multisubject association are expressed as$$\begin{array}{}\text{(15)}& \begin{array}{c}\frac{ds{n}_{k}t}{dt}=\xi -k\rho s{n}_{k}t\Theta t-\chi s{n}_{k}t+\varphi r{n}_{k}t,\\ \frac{di{n}_{k}t}{dt}=k\rho s{n}_{k}t\Theta t-\phi i{n}_{k}t\text{,}\begin{array}{cc}& \end{array}\\ \frac{dr{n}_{k}t}{dt}=\phi i{n}_{k}t+\chi s{n}_{k}t-\sigma r{n}_{k}t-\varphi r{n}_{k}t.\end{array}\end{array}$$

According to Equation (15), for the steady-state condition $di{n}_{k}t/dt=0$, the steady-state value becomes $i{n}_{k}t$ as follows$$\begin{array}{}\text{(16)}& i{n}_{k}t=\frac{k\rho s{n}_{k}t\Theta t}{\phi }=\frac{k\xi \rho \sigma +\varphi \Theta t}{\sigma \phi \chi +k\sigma \phi \rho \Theta t}\text{.}\end{array}$$

The average density of the infectious netizen becomes $in={\sum }_{k}Pki{n}_{k}t$. Based on equation (16), $\mathrm{\Theta }t$ becomes$$\begin{array}{}\text{(17)}& \Theta t={\displaystyle \sum }_k\frac{kPki{n}_{k}t}{{\displaystyle \sum }_{s}sPs}=\frac{1}{k}{\displaystyle \sum }_{k}kPki{n}_{k}t\text{,}\end{array}$$where $k$ represents the average degree of Internet public opinion diffusion under the action of multiple participants.

Given that $<k>={\sum }_{k}kPk$ and $<k^2>={\sum }_k{k}^{2}Pk$, Equations (16) and (17) can be combined as follows:$$\begin{array}{}\text{(18)}& \Theta t=\frac{1}{k}{\displaystyle \sum }_{k}kPk\frac{k\xi \rho \sigma +\varphi \Theta t}{\sigma \phi \chi +k\sigma \phi \rho \Theta t}\text{.}\end{array}$$

Given that $\mathrm{\Theta }=\mathrm{\Theta }t$, equations (18) has a trivial solution: $\mathrm{\Theta }=0$. If equation (18) has a nontrivial solution, $\mathrm{\Theta }\ne 0$, then the necessary condition becomes$$\begin{array}{}\text{(19)}& \frac{d}{d\Theta }{\frac{1}{k}{\displaystyle \sum }_{k}kPk\frac{k\xi \rho \sigma +\varphi \Theta t}{\sigma \phi \chi +k\sigma \phi \rho \Theta t}}_{\Theta =0}\ge 1\text{.}\end{array}$$

Therefore,$$\begin{array}{}\text{(20)}& \frac{1}{k}{\displaystyle \sum }_{k}kPk\frac{k\xi \rho \sigma +\varphi }{\sigma \phi \chi }\ge 1\text{.}\end{array}$$

Therefore, the threshold $TV$ of Internet public opinion diffusion under the action of multiple participants is obtained as$$\begin{array}{}\text{(21)}& TV=\frac{\rho \xi \sigma +\varphi {\sum }_k{k}^{2}Pk}{\sigma \phi \chi {\sum }_{k}kPk}\text{.}\end{array}$$

When $TV=1$, it corresponds to the threshold of whether Internet public opinion spreads widely in multilayer networks. When $TV<1$, Internet public opinion is gradually disappearing in multilayer networks. When $TV>1$, Internet public opinion spreads with nonzero probability in multilayer networks, and the greater the $TV$ is, the higher the intensity of diffusion is, the more netizens will be affected.

5. Simulation Analysis of Evolution Characteristics of Internet Public Opinion Diffusion

5.1. Parameter Value Setting

Based on the Internet public opinion multilayer network and its diffusion model, this part numerically simulates and analyzes the evolution characteristics of Internet public opinion diffusion mechanism under the interaction of multisubject association and multilayer network with the help of MATLAB R2014a software. In this part, the parameter values of netizen attention and netizen recognition are set according to Abeles et al. [6] and Dinh et al. [7]. According to Cacciatore [10] and Chen et al. [35], the parameter values of media coverage intensity and media emotion tendency are set. According to Jacobsen [14] and Jang and Kim [12], the parameter values of government intervention intensity and perfection of government’s early warning mechanism are set, and the parameter values of platform openness and platform audit strength are set according to D'andrea et al. [15] and Liu et al. [17]. The parameter values of other network indicators in this model are mainly set according to Wang et al. [33], Xia et al. [36], and Chen et al. [37, 38]. See Table 1 for specific parameter benchmark values.

Table 1

Benchmark values of parameters.

Parameter	Describe	Benchmark values	Value ranges
${\mu }_{ON}$(or ${\mu }_{OF}$)	The probability of random connection between newly added susceptible netizen and existing infectious netizen in the online public opinion association network (or the offline public opinion association network). The value of this parameter is closely related to the structural characteristics of the public opinion association network.	0.4	[0, 1]
${\epsilon }_{ON}$(or ${\epsilon }_{OF}$)	The increasing number of new susceptible netizen in the online public opinion association network (or offline public opinion association network). The value of this parameter is closely related to the intensity of the public opinion diffusion process.	100	Positive integer
${\theta }_{ON}$(or ${\theta }_{OF}$)	The increasing number of association edges of new susceptible netizen in the online public opinion association network (or offline public opinion association network). The value of this parameter is closely related to the range of public opinion diffusion process.	100	Positive integer
$k$	The degree of existing netizens, that is, the number of connected sides of existing netizens.	600	Positive integer
$N$	Total number of netizens in internet public opinion multilayer network.	1000	Positive integer
$\alpha$	Netizen attention reflects the enthusiasm of netizen to pay attention to hot issues on the Internet.	0.6	[0, 1]
$\beta$	Netizen recognition reflects the ability of netizens to judge hot issues on the Internet.	0.2	[0, 1]
$\delta$	Media coverage intensity reflects the sustainability of media reports.	0.2	[0, 1]
$\lambda$	Media emotion tendency reflects the different attitudes of the media when reporting hot issues on the Internet.	0.2	[0, 1]
$\gamma$	Government intervention intensity reflects the government influence on Internet public opinion.	0.3	[0, 1]
$\kappa$	Perfection of government’s early warning mechanism. It reflects the perfection of the government’s early warning mechanism for Internet public opinion monitoring.	0.5	[0, 1]
$\tau$	Platform openness reflects the free participation of all kinds of people and the free release of all kinds of information.	0.7	[0, 1]
$\eta$	Platform audit strength reflects the audit strength of bad and illegal content and highlights the bottom line rule awareness of the platform.	0.4	[0, 1]
$\rho$	Probability of susceptible netizen changing to infectious netizen.	0.4	[0, 1]
$\chi$	Probability of susceptible netizen directly changing to recovery netizen.	0.2	[0, 1]
$\phi$	Probability of infectious netizen changing to recovery netizen.	0.2	[0, 1]
$\varphi$	Probability that some recovery netizen may change into susceptible netizen again.	0.2	[0, 1]
$\xi$	Probability of new netizens entering the multilayer network of Internet public opinion.	0.3	[0, 1]
$\sigma$	Probability of quitting some netizens in Internet public opinion multilayer network.	0.1	[0, 1]

5.2. Multilayer Network Structure and Internet Public Opinion Diffusion

Existing studies have shown that complex association networks may present random network structure characteristics or scale-free network structure characteristics [39], or they may also present a mixture of two types of network structure characteristics at the same time [40]. In addition, different association network structures can reflect the different association forms and tightness of netizens, and according to the Internet public opinion diffusion probability in different association network structures, it can explain which network structure can better promote or inhibit the spread of Internet public opinion. Figure 2 shows the evolution characteristics of Internet public opinion diffusion caused by different subject factors under different Internet public opinion multilayer network structures. Specific analysis shows those as follows.

[figure(s) omitted; refer to PDF]

First, as can be seen from Figure 2, with the continuous improvement of the random connection probability in the online public opinion association network and offline public opinion association network, the diffusion intensity of Internet public opinion shows a monotonous downward trend. Moreover, the greater the probability of random connection in the two-tier network is, the stronger the influence of different subject factors on the diffusion of Internet public opinion is. This reflects that it is easier to trigger Internet public opinion diffusion through the establishment of associations among netizens in a preferred way. This is consistent with the practical understanding; that is, netizens often purposefully choose to follow the attitudes and views expressed by some influential people so as to form resonance, which will accelerate the spread of hot issues. However, the way of random connection is just the opposite. Because views and attitudes may be different and random, they may not resonate quickly, which inhibits the further spread of hot issues.

Second, when considering the influence of netizens' factors on the Internet public opinion diffusion independently, it can be seen from Figures 2(a)–2(d) that with the increasing attention of netizens, the intensity of Internet public opinion diffusion also gradually increases, and the two show a positive correlation. As can be seen from Figures 2(e)–2(h), with the increasing awareness of netizens, the intensity of Internet public opinion diffusion also gradually decreases, and the two show a negative correlation. This reflects that when considering such subjects as netizens, the scope and influence of Internet public opinion can be reduced by effectively reducing the netizen attention or improving the netizen recognition.

Third, when considering the influence of media factors on the Internet public opinion diffusion independently, it can be seen from Figures 2(i)–2(l) that with the continuous improvement of media coverage intensity, the intensity of Internet public opinion diffusion also gradually increases, and the two show a positive correlation. As can be seen from Figures 2(m)–2(p), with the continuous change of media emotion tendency, the more positive or negative media emotion is, the higher the intensity of Internet public opinion diffusion is, and the more objective media emotion is, the lower the intensity of Internet public opinion diffusion is, and the two show a positive “U” correlation. This reflects that when considering subjects such as the media, the scope and influence of Internet public opinion can be reduced by reducing the media coverage intensity or maintaining the objectivity of media emotion.

Fourth, when considering the influence of government factors on Internet public opinion diffusion independently, it can be seen from Figures 2(q)–2(t) that with the continuous improvement of government intervention intensity, the intensity of Internet public opinion diffusion gradually decreases, and the two show a negative correlation. Similarly, it can be seen from Figures 2(u)–2(x) that with the continuous improvement of the perfection of government’s early warning mechanism, the intensity of Internet public opinion diffusion also gradually decreases, and the two show a negative correlation. This shows that the effective intervention and early warning mechanism of the government is a powerful means to curb the spread of public opinion.

Fifthly, when considering the influence of Internet platform factors on Internet public opinion diffusion independently, it can be seen from Figures 2(y)–2(ab) that with the continuous improvement of platform openness, the intensity of Internet public opinion diffusion gradually increases, and the two show a positive correlation. As can be seen from Figures 2(ac)–2(af), with the continuous enhancement of platform audit, the intensity of Internet public opinion diffusion gradually decreases, and the two show a negative correlation. This shows that appropriately reducing the platform openness or improving the platform audit strength can effectively curb the spread of public opinion.

In addition, when the probability of netizens random connection in different layers changes interactively, Internet public opinion diffusion characteristics are shown in Figure 3. It can be seen from Figure 3 that when the probability of netizens random connection in different layers increases synchronously, the Internet public opinion diffusion intensity shows a gradual downward trend. This is consistent with the conclusion obtained in Figure 2. At the same time, it can also be seen from Figure 3 that when only considering the impact of multilayer network structure on Internet public opinion diffusion, the intensity of Internet public opinion diffusion has always maintained a high level. On the one hand, it reflects the internal solidity of Internet public opinion diffusion, and on the other hand, it shows that different subject factors need to be specifically considered to adjust the intensity of Internet public opinion diffusion. This will be the next key research content.

[figure(s) omitted; refer to PDF]

5.3. Single Subject Factor and Internet Public Opinion Diffusion

Figure 4 shows the evolution characteristics of Internet public opinion diffusion intensity caused by different main factors when the multilayer network structure of Internet public opinion is determined. The specific analysis shows that first, as shown in Figure 4, the influence relationship between different subject factors and the Internet public opinion diffusion intensity is consistent with the research conclusion obtained in Figure 3. This shows that the model and results constructed in this paper are robust. Second, according to Figure 4, among the different main factors, government intervention intensity has the greatest impact on the Internet public opinion diffusion intensity, followed by platform audit strength, netizen attention, perfection of government’s early warning mechanism, netizen recognition, platform openness, media emotion tendency, and media coverage intensity. This points out the direction and focus for regulating Internet public opinion diffusion from different subjects. Third, it can be seen from the different subgraphs in Figure 4 that there are great differences in the interaction effects among different subject factors, and then there are great differences in the impact on the Internet public opinion diffusion intensity. For example, it can be seen from Figures 4(a)–4(g) that compared with the interaction effects of netizen attention and other factors, the interactive effect of netizen attention and netizen recognition has a more significant impact on the Internet public opinion diffusion intensity. As can be seen from Figures 4(h)–4(n), the interactive effect of netizen recognition and the government intervention intensity has a more significant impact on the Internet public opinion diffusion intensity. Therefore, in order to more clearly reflecting the impact of the interaction among different subject factors on the Internet public opinion diffusion, this paper will focus on the next part.

[figure(s) omitted; refer to PDF]

5.4. Interaction Effect of Multisubject Factors and Internet Public Opinion Diffusion

Figure 5 reflects the change of Internet public opinion diffusion intensity caused by the interaction among multiple subject factors when the multilayer network structure of Internet public opinion is determined. Specific analysis shows those as follows.

[figure(s) omitted; refer to PDF]

First, as can be seen from Figures 5(a), 5(b), 5(f), 5(h), 5(l), and 5(q), when netizens attention interacts with netizen recognition, media coverage intensity, and platform openness, netizen recognition interacts with media coverage intensity and platform openness, and media coverage intensity interacts with platform openness, Internet public opinion diffusion intensity shows a monotonous upward trend. This reflects that when multiple subject factors interact, to effectively control Internet public opinion diffusion intensity, it should be realized through comprehensive adjustment means synchronously reducing netizens attention, improving netizen recognition, controlling platform openness, and reducing media coverage intensity. However, further analysis shows that it is difficult to eliminate Internet public opinion diffusion by adjusting the interaction between these factors, especially when adjusting the factors of a single subject. Therefore, multisubject factors need to be coordinated to achieve a better inhibition effect.

Second, as can be seen from Figures 5(g), 5(k), 5(m), and 5(aa), when netizen attention interacts with platform audit strength, netizen recognition interacts with perfection of government’s early warning mechanism and platform audit strength, and Internet public opinion diffusion intensity shows a monotonous downward trend. This reflects that when netizens, Internet platforms, and the government interact, it can effectively reduce the intensity and scope of Internet public opinion diffusion, especially the participation of Internet platforms can improve the situation of Internet public opinion diffusion. However, once the media participate, it will increase the complexity of the diffusion and evolution of Internet public opinion. Therefore, when controlling and regulating the spread of Internet public opinion, we should focus on the influence mechanism played by the platform and the media.

Third, as shown in Figures 5(e), 5(p), 5(r), 5(z), and 5(ab), when netizen attention interacts with perfection of government’s early warning mechanism, media coverage intensity interacts with perfection of the government’s early warning mechanism and platform audit strength, and perfection of the government’s early warning mechanism interacts with platform openness and platform audit strength respectively, Internet public opinion diffusion intensity increases first and then decreases. This highlights the special impact of perfection of the government’s early warning mechanism on the three main factors such as netizens, media, and Internet platform. Specific analysis can be found such that when perfection of the government’s early warning mechanism is low, it will not be able to effectively control the spread of Internet public opinion. However, when perfection of the government’s early warning mechanism is gradually improved, the effect of government regulation is gradually apparent, and Internet public opinion diffusion intensity is also significantly reduced. Therefore, when formulating strategies to regulate the spread of Internet public opinion, we need to attach great importance to the improvement of the perfection of the government’s early warning mechanism.

Fourth, as can be seen from Figures 5(d), 5(j), 5(o), 5(s), 5(w), and 5(y), when government intervention intensity interacts with netizen attention, netizen recognition, media coverage intensity, media emotion tendency, perfection of government’s early warning mechanism, platform openness, and platform audit strength, although it has little impact on Internet public opinion diffusion intensity, it will induce the emergence characteristics of other factors, resulting in the sudden enhancement or disappearance of Internet public opinion diffusion intensity. The “anomaly” shows that the government intervention in Internet public opinion is a “double-edged sword,” which needs to set the intervention intensity specifically according to the characteristics of different subjects and their factors, rather than generalizing. Therefore, the determination of government intervention intensity needs to be particularly cautious.

6. Conclusion

The effective regulation of Internet public opinion diffusion from multiple subjects is conducive to resolving public opinion crisis and maintaining social harmony and stability. This paper deeply analyzes the diffusion mechanism of Internet public opinion from the cross perspective of multilayer network and multisubject association, constructs the multilayer network of Internet public opinion and its diffusion model, and then systematically analyzes the evolution characteristics of Internet public opinion diffusion from three aspects: multilayer network structure, single subject factor, and multisubject factor. The main conclusions of the study are as follows.

In terms of multilayer network structure, when building the model, this paper expands the research of Internet public opinion network from single layer to multiple layers through online and offline association attributes and finds some conclusions different from previous studies specifically, and it is easier to trigger Internet public opinion diffusion by establishing connections among netizens in a preferred way. Netizen attention, media coverage intensity, and platform openness are positively correlated with Internet public opinion diffusion intensity. Netizen recognition, government intervention intensity, perfection of government’s early warning mechanism, and platform audit strength are negatively correlated with Internet public opinion diffusion intensity. There is a positive “U” correlation between media emotion tendency and Internet public opinion diffusion intensity. In addition, when the probability of netizens random connection in different layers increases synchronously, Internet public opinion diffusion intensity shows a gradual downward trend.

In terms of multisubject association, when building the model, this paper not only relies on Internet public opinion multilayer network but also comprehensively considers the complex impact of multiple interaction effects of different participants on Internet public opinion diffusion, making the Internet public opinion diffusion model and its evolution characteristics more realistic. On the one hand, according to the single subject factor, this paper finds that among the different main factors, government intervention intensity has the greatest impact on Internet public opinion diffusion intensity, followed by platform audit strength, netizen attention, perfection of government’s early warning mechanism, netizen recognition, platform openness, media emotion tendency, and media coverage intensity. There are great differences in the interaction effects between different subject factors, and then there are great differences in the impact on Internet public opinion diffusion intensity. On the other hand, according to the multisubject factor, we can see that simultaneously reducing netizen attention, improving netizen recognition, controlling platform openness and media coverage intensity can achieve the purpose of eliminating the spread of Internet public opinion, but the realization conditions are relatively harsh. When netizens, Internet platform, and government interact, it can effectively reduce the intensity and scope of Internet public opinion diffusion. Once the media participates, it will increase the complexity of the evolution of Internet public opinion diffusion. The perfection of government’s early warning mechanism has a special impact on three main factors: netizens, media, and platform. In addition, the government intervention in Internet public opinion is a “double-edged sword,” which needs to set the intervention intensity according to the characteristics of different subjects and their factors.

This study can provide a theoretical reference for formulating strategies to adjust Internet public opinion diffusion from the cross perspective of multilayer network and multisubject association. However, in the research process, this paper found that information factors may also be an important aspect affecting Internet public opinion diffusion. Therefore, this paper will add the perspective of information economics to further explore the evolutionary characteristics of how different information transparency and communication rate affect Internet public opinion diffusion.

Authors’ Contributions

Defei Hou conceptualized the study and proposed the methodology. Chen Liu wrote the original draft and analyzed the study. Yuanqing Li performed the calculation and supervised the study. Defei Hou, Chen Liu, and Yuanqing Li contributed equally to this work. They are co-first authors.

Acknowledgments

This work was supported by Humanities and Social Science Planning Foundation of the Ministry of Education of China (grant number 22YJC790128), Soft Science Research Program General Project of Jiangsu Province (grant number BR2022057), General Project of Philosophy and Social Science Research in Colleges and Universities of Jiangsu Province (grant numbers 2021SJA0836 and 2022SJYB0215), Youth Program of Jiangsu Health Vocational College (grant number JKD202015), and Social Science Application Research Excellent Program Project of Jiangsu Province (grant numbers 22SYB-059S and 22SYA-010).

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