Modeling and simulation on the spreading dynamics of public opinion information in temporal group networks

Abstract

In the new media environment, the constantly emerging social platforms further expand the channels for the propagation of public opinion. Under the framework of complex network theory and faced the needs of management practice, modeling the spreading dynamics of public opinion in temporal group networks is of great significance for understanding its spreading law and improving the governance system of cyberspace and the development of network science. Through analyzing the changes of group networks topology and the spreading rules of public opinion, the spreading model of public opinion in temporal group networks was proposed by coupling the two dynamic processes, and the spreading thresholds of public opinion in static and temporal group networks were derived respectively. Then, the spreading characteristics of public opinion under different network topology, as well as the influence of important parameters on public opinion spreading process were discussed with the help of simulation experiments. The research results indicated that the propagation of public opinion in static and temporal group networks exhibits both similar trends and differentiated characteristics; compared with Spreader, the propagation of public opinion in temporal group networks is more sensitive to Ignorant’s behavior; both groups’ and netizens’ active probability have significant influence on public opinion propagation, but netizens’ affects more. Based on the relevant results, this paper proposed a series of countermeasures such as, grading social platforms, strengthening relationship management between them and introducing time management systems, so as to promote the formation of a good network ecosystem and the modernization of the national governance system.

Introduction

With the help of intelligent technologies such as language processing and algorithm distribution, online social networks (OSN), as the carrier of public opinion information (referred as public opinion) propagation, have taken the lead in enjoying technical dividends, and their structure, form and quantity are showing a dramatic trend. Such as, as of the end of 2022, there were approximately 300000 social communication APPs monitored in the Chinese market, with a cumulative user scale of more than 1 billion¹. Along with this, the number and density of opinions expressed by 1.097 billion netizens have also exploded², further intensifying the impact of public opinion propagation on economic development and social stability.

The new media environment driven by artificial intelligence and big data, have endowed public opinion with unique spreading characteristics, and also put forward new and higher requirements for its governance, forcing the regulatory to make corresponding changes in governance methods and strategies. Specifically, omnimedia technology has greatly improved the efficiency of public opinion propagation, while the temporal changes in social networks topology caused by the social activities of billions of netizens and their flow between different platforms have led to the unpredictability of public opinion propagation, thereby exacerbating the complexity of public opinion supervision. For example, the constantly emerging social platforms not only provide conditions for the cross-platform propagation of public opinion, but also realize traffic transfer between social platforms. In view of this, facing the social platforms with dynamic changes in internal and external topologies, research on modeling of public opinion propagation in temporal group networks is of great significance for expanding the research field of spreading dynamics theory, improving public opinion governance system and further purifying the cyberspace.

Literature review

Network analysis and modeling was widely used in diverse fields and can be a powerful framework for identifying system structural principles and analyzing dynamic processes. According to the development of complex network theory, since 2000, as the main tool to describe the dynamic processes, such as epidemic spreading, rumor diffusion, information propagation, etc., the academic community has achieved rich research in terms of network types, network topology modeling and dynamic processes, greatly promoting the rapid development of network science. Among them, the achievements related to network modeling and spreading dynamics are the most abundant, especially the dynamics characteristics under different network topologies, which have attracted widespread discussion in academia. For example, ER random network³, WS/NW small world network^4,5, BA scale free network⁶, weighted network⁷, group (community) network⁸, directed network⁹, multi-layer networks^10,11, etc. These network topologies can be regarded as static networks. While to date, many systems undergo topological changes over time and temporal networks, which incorporate time into conventional network models, are therefore more accurate representations of such dynamic systems. Then, research on temporal networks has gradually aroused academic interest, such as types, topological evolution and dynamic characteristics. The details are described below.

The temporal or time-varying network is a kind of network containing time information, in which temporal edges appear and disappear over time. In terms of temporal network types, Holme Petter^12,13 was the first to conduct systematic research on them, summarizing node contact sequences (as shown in Figure 1(a)) and connected edge interval graphs (as shown in Figure 1(b)) two ways to describe the dynamic changes in temporal network topology. Such as contact sequences, a pair of nodes is represented by a quadruplet $(i,j,t,{\delta }_t)$, ${\delta }_t$is contact time duration. Then, taking social platforms as an example, Perra et al¹⁴took the lead in conducting temporal network modeling and proposed activity-driven (AD) temporal network model, considering that the netizens’ social activities are the main caused of changes in network topology. Following the similar idea, Perra’ team proposed a temporal population model by defining population activities¹⁵. However, some scholars have expressed different opinions, stating that order correlations in temporal networks lead to causality structures, which significantly deviate from what is expected based on paths in the corresponding time-aggregated networks^16,17. And then a Causality-Driven (CD) temporal network model was proposed by Scholtes et al¹⁸, considering the causal relationship between connected edges with in the platform. As another manifestation of temporal network, memory network is a representation of a slightly more information data sets than contact sequences, and it assumes there is a recorded walk process on a set of nodes, like flight passengers on multi-hop itineraries^13,19. Inspired by the above opinion, Williams et al²⁰ proposed a memory-driven (MD) temporal network model accordingly.

Fig. 1.

Contact sequences (a) and interval graphs (b) of temporal networks.

In order to construct a theoretical model to accurately describe the topological changes of temporal networks, scholars have carried out relevant research from the node level and edge level. At the node level, based on the survey data of netizens’ social activities, Perra et al¹⁴ defined a probability distribution function for netizens’ social activities that follows power law distribution $F(a)=B{a}^{-\gamma }$. It is assumed that nodes enter the active state with probability $a_i(t)$to engage in social interaction with other users, which in turn leads the topology evolution. In Rocha et al’ model²¹, whenever a vertex becomes active, it is randomly connected to another active vertex and the corresponding edge remains available during one time step and is destroyed afterwards combined priority queuing model²²and nonhomogeneous Poissonian process including periodic activity patterns²³. Based on this, considering the role of the memory effect of netizens’ social activities in the evolution of network topology, Sun et al²⁴and Williams et al²⁰ proposed temporal network models driven by strong memory and weak memory, respectively. At the edge level, Scholtes et al¹⁸ defined the probability distribution of the activity of all edges by introducing an active data set of network edges $t(e_i)\in \lambda (e)$. It is assumed that all edges ($e_i$) are in an active state at a specific time and participate in network topology evolution. With the help of Markov process, Wang et al²⁵ proposed a general temporal network evolution model in their research, where all edges are randomly disconnected or reconnected with probability. According to the literature review, it is the mainstream research paradigm of temporal network topology modeling to focus on social activities at the node level and analyze the evolutionary mechanism of temporal network topology.

Based on the above definition and topology evolution rules, the research results of dynamic behavior on temporal networksare the most abundant. Such as, introducing SIS dynamic model into temporal networks, Perra et al¹⁴took the lead in initiating research on the temporal networks dynamics and derived the SIS epidemic spreading threshold. Then, Perra’s team (Liu et al)¹⁵extended the above conclusion to the temporal population networks and derived the epidemic spreading threshold in temporal population networks with any size. Based on activity-driven (AD) temporal network model and the memory effect of users’ social activities, Tizzani et al²⁶and Zhong et al²⁷compared epidemic spreading process in temporal networks with and without memory, and the result showed that the introduction of memory effect significantly changes its spreading threshold. As two major features of social network topology, temporality and multiplexity are proposed to be combined to describe network topology. For example, focusing on the dynamic characteristics in temporal multiplex networks, Yang et al²⁸proposed an epidemic spreading model in temporal multiplex networks and the research results showed the more significant multiplexity and temporality of networks, the lower the spreading threshold of epidemic dynamic process. In addition to epidemic spreading, some scholars also try to describe information propagation with the help of temporal network model. Such as, Wang et al²⁹discussed the public opinion propagation process in temporal networks with Facebook network data and the results have shown that compared with static networks, the propagation of public opinion in temporal networks has faster speed and wider range. In the application of temporal network dynamics process, such as the identification of key nodes and edges, scholars have also achieved rich research results. Such as, the algorithm MLI based on network embedding and mechanism learning were proposed to identify critical nodes in temporal networks and experimental results show that the proposed method outperform these well-known methods in identifying critical nodes under spreading dynamic³⁰. Similarly, focused on the identification of key edges in temporal networks, they proposed a method for identifying the minimum number of edges that need to be moved in the scenario where the epidemic spreading range is minimized in temporal networks^31–33.

It can be seen that scholars have carried out lots of research on temporal networks and the related results not only promote the development of network science, but also provide theoretical foundation for this paper. However, it is found that there is still room for further research in this field.

• Research on temporal network dynamics mainly focused on epidemic spreading, and few scholars have paid attention to the co-evolution of topological changes of OSN and public opinion propagation. Compared with epidemic spreading, the propagation of public opinion under the new media environment is more limited by information itself and carrier network topology, which has stronger randomness and uncertainty. Considering the significant impact of public opinion propagation on economic development and social stability, research on the above issues is more valuable.

• Existing temporal network models mostly focused on the topological changes within a single group, and few models involve the topological changes between multiple groups. In recent years, online social platforms have continued to emerge, providing differentiated channels for the propagation of public opinion. And the stable and varying bridging relationships between them provide conditions for netizens’ mobility and traffic transfer. For this issue, scholars have also attempted to use multi-layer networks to describe the cross-network propagation of public opinion, but as the number of social platforms increases, the applicability of multi-layer network spreading models gradually decreases.

In addition, network topology research has experienced a trend from simple to complex, single layer to multi layer, static to dynamic since 2000, according to the development of complex network theory. Mason A. Porter, a renowned scholar in network science who won the ER award at the network science conference, summarized four hot topics for future network science research in 2020, among which temporal networks and dynamical processes on network ranking first and second respectively. Under the framework of the development trend of complex network theory, combined with the needs of management practice, this paper planned to couple public opinion propagation and temporal group network topology changing two kinds of dynamic processes, to construct a dynamic model of public opinion propagation in temporal group networks and solve public opinion spreading threshold. With the help of simulation, the spreading characteristics of public opinion in temporal group networks were discussed, and suggestions are provided for the guidance and management of public opinion propagation.

Model introduction

Temporal group networks model

The total number of netizens in China’s online social platforms is defined as Y, which is randomly distributed among X groups (online social platforms). For any time t, the size of netizens in group i is $Y_i(t)$. It is worth noting that there are intersections among netizens on many groups, that is, the same user may be active on multiple groups at the same time. However, considering the huge user scale of online social platforms and the fact that each type of groups has its main active users, it is assumed that the number of netizens in such groups satisfies: $Y={\sum }_{i=1}^X{Y}_i(t)$.

With the development of intelligent technology, the types of online social platforms have increased rapidly, but the strength of relationship among netizens on different groups shows obvious differences. Combined with the statistical data of actual social platforms and previous research conclusions^29,34, this paper divided online social platforms into open and closed OSN, and used BA scale-free, NW small-world and ER random network models respectively to simulate the degree distribution of netizens in each groups. Based on the adjacency matrix $d_{\mathit{ij}}^k$($k=1,2\dots X;i,j=1,2\dots Y_k$) within each group, a social network of netizens $U^k$ within the group is formed.

For the relationships between groups, generally spreading, there are fixed cooperations for many social platforms. Such as, news in NetEase can be directly forwarded to WeChat or Sina; information in Sina can also be directly forwarded to WeChat or QQ. In addition, with the increasing variety of online social platforms, temporary relationships may also be established between them, that is, social relationships between groups can be seen as dynamically changing over time. The topology relationship between groups at any time is defined as the adjacency matrix $D(u,v)(u,v=1,2\dots X)$. At time t, if there exits interaction between group u and group v, $D(u,v)=1$, information can be spread across two groups; otherwise $D(u,v)=0$, and a temporal group networks model $G_t$ is formed. Then, the temporal changes in the connectivity between group networks were described with the help of the activity-driven (AD) model.

Based on the research findings^14,15 and the active time distribution of online social platforms, a power law distribution of group active probability ($a_u\in [{\zeta }_a,1],u=1,2\dots X$, $F(a)=B_g{a}^{-{\gamma }_a}$) was introduced to represent the tendency of establishing social relationship between groups. The larger the $a_i$, the higher the possibility of establishing connections with other groups. Therefore, the topological evolution process of temporal group networks can be defined as:

• (1) At initial time $t_0$, X groups are randomly connected with probability p.

• (2) At time t, group k become active with probability $a_k$ and establishes social relationships with the other m groups randomly. An inactive group cannot establishes social relationships with other groups, but can accepts connections from active groups passively.

• (3) The maintenance time of the new connections between groups is $\mathrm{\Delta }t$. If $D(u,v)=1$, netizens and information in group u and v can move between the two groups.

• (4) At time $t+\mathrm{\Delta }t$, all edges between groups are randomly deleted, continue Step (2).

In order to describe the movement process of netizens between groups, the active probability of netizens ($b_h\in [{\zeta }_b,1]$) obeying the power law distribution ($F(b)=B_n{b}^{-{\gamma }_b}$) was introduced. Meanwhile, $p_{\mathit{uv}}^h$ was defined as the probability of user h moving from group u to group v. Obviously, $p_{\mathit{uv}}^h$ can be expressed as:

$$\begin{array}{c}\hfill p_{\mathit{uv}}^h=\left\{\begin{array}{ccc}\hfill 0,& & D(u,v)=0\hfill \\ \hfill {\displaystyle \frac{b_h\ast \mathrm{\Delta }t}{k_u}},& & D(u,v)=1\hfill \end{array}\right)\end{array}$$ 1

In Eq.(1), $k_u$ is degree of group u. Obviously, the greater the degree of group u, the lower the probability of user h moving to group v.

Public opinion propagation model

Considering the similar dynamic mechanism between the propagation of public opinion and epidemic spreading^35,36, this paper intends to use the classic SIR spreading model to describe public opinion propagation process. Firstly, all users were divided into the following three categories during public opinion propagation process. Ignorant: users who have not yet been exposed to public opinion; Spreader: users who are spreading public opinion in OSN and Recovered: netizens who are not interested in public opinion propagation. Based on the above definition, the spreading rules of public opinion can be expressed as:

• When Ignorant come into contact with Spreader, the former will become the latter with probability $\beta$, taking into account factors such as their social relationships, netizens’ perceived value, etc.

• Considering the timeliness of public opinion, social reinforcement effect and netizens’ interest transfer behavior, Spreader may quit public opinion propagation process with probability $\mu$.

• When netizens become Recovered, their state will not change. The propagation process of public opinion is shown in Figure 2.

Fig. 2.

The propagation model of public opinion.

As a classic model in spreading dynamics, scholars have conducted many in-depth discussion on the SIR model and achieved relatively unified conclusions. Such as, there exits a spreading threshold $R_0$ when describing the public opinion propagation process using this model. If $R_0>1$, public opinion can spread in OSN and the density of Spreader will stabilize at an equilibrium state. At this point, the density of Spreader is $\alpha \approx \frac{2(R_0-1)}{R_0^2}$¹⁵. On the contrary, the number of Spreader will exponentially decline and gradually disappear from social networks. In addition, network topology also significant affects public opinion propagation, and compared with homogeneous networks ($R_0=\frac{\beta }{\mu }$), public opinion is more likely spread in heterogeneous networks ($R_0=\frac{\beta <k^2>}{\mu <k>}$)^11,13,36.

Combining the temporal group networks model with the general conclusion of spreading dynamics, it can be concluded that when public opinion propagation is in a steady state, the density of Spreader within a group with degree k can be expressed $\alpha Y_k$. These spreaders will exit the spreading process with probability $\mu$, so their maintenance time in the propagation state can be expressed as $\alpha Y_k{\mu }^{-1}$. Within time $\mathrm{\Delta }t$, if there is a connection between group k and a group with degree $k^{\prime }$, the number of public opinion spreaders who transfer from group k to group $k^{\prime }$ ($q_{k\to k^{\prime }}$) can be expressed as:

$$\begin{array}{c}\hfill q_{k\to k^{\prime }}=p_{k{k}^{\prime }}\alpha Y_k{\mu }^{-1}={\displaystyle \frac{b\mathrm{\Delta }t\alpha Y_k}{k\mu }}\end{array}$$ 2

In temporal group networks, the flow process of netizens between groups can be described as Fig. 3. Considering the randomness of public opinion propagation and the non uniqueness of the initial spreaders, the probability of public opinion explosion can be expressed as $\mathrm{\Phi }=1-R_0^{-n}$^24,36.

Fig. 3.

Users’ moving process in group networks.

Modeling on public opinion propagation in group networks

According to the above literature review, it can be seen that scholars have conducted extensive and in-depth research on dynamic process in single group and static group networks, and achieved relatively consistent research conclusions. Such as, the propagation of public opinion mainly depends on the spreading threshold, and the diffusion of public opinion under different topologies has differentiated spreading characteristics^29,34,36. While for the propagation of public opinion in temporal group networks, considering the dynamic interaction between groups and the movement of netizens among various groups, in addition to the spreading threshold, its propagation will also be affected by the activity of groups and netizens. Therefore, for the propagation of public opinion in temporal group networks, it is also important to identify the threshold of groups and netizens’ active probability. Such as, even if $R_0>1$, but the platform and its users have a low active probability, the propagation of public opinion may be restricted within the single group and cannot spread to the entire cyberspace.

Modeling on public opinion propagation in static group networks

In order to clearly describe public opinion propagation process in group networks, ${\mathrm{\Omega }}_k^t$ is introduced to represent the number of public opinion spreader in the group with degree k at time t. Combined with the spreading rules of public opinion and the moving mechanism of netizens between group networks, ${\mathrm{\Omega }}_k^t$ can be expressed as:

$$\begin{array}{c}\hfill {\mathrm{\Omega }}_k^t=\sum _{k^{\prime }}{\mathrm{\Omega }}_{k^{\prime }}^{t-1}(k^{\prime }-1)(1-R_0^{-q_{k^{\prime }\to k}})P(k|k^{\prime })(1-{\displaystyle \frac{{\mathrm{\Omega }}_k^{t-1}}{X_k}})\end{array}$$ 3

For Eq.(3), the explanation is as follows. Each group with degree $k^{\prime }$ will seed the infection in $k^{\prime }-1$ groups (the total number minus the one from which the group got seeded). When the system is in a steady state, the number of users in group with degree $k^{\prime }$ can be expressed as: $Y_{k^{\prime }}=\frac{Y{k}^{\prime }}{X<k>}$. $q_{k^{\prime }\to k}$ means the number of spreader that may be moved from group with degree $k^{\prime }$ to group with degree k. $1-R_0^{-q_{k^{\prime }\to k}}$ represents the probability of public opinion explosion in group k. When $R_0\sim 1$, it can be approximated as $1-R_0^{-q_{k^{\prime }\to k}}\sim q_{k^{\prime }\to k}(R_0-1)$. The conditional probability $P(k|k^{\prime })$ measures the probability that each of the $k^{\prime }$ neighbors has degree k. $1-\frac{{\mathrm{\Omega }}_k^{t-1}}{X_k}$ means the density of Ignorant in group k at time $t-1$. At time $t=0$, the density of Ignorant is approximately expressed as $1-\frac{{\mathrm{\Omega }}_k^{t-1}}{X_k}\sim 1$.

Therefore, Eq.(3) can be simplified as:

$$\begin{array}{c}\hfill {\mathrm{\Omega }}_k^t={\displaystyle \frac{\alpha b\mathrm{\Delta }t(R_0-1)}{\mu }}\sum _{k^{\prime }}{\mathrm{\Omega }}_{k^{\prime }}^{t-1}(k^{\prime }-1){\displaystyle \frac{Y_{k^{\prime }}}{k^{\prime }}}P(k|k^{\prime })\end{array}$$ 4

Besides, in the definition of Eq.(3), we only consider the inflow process of spreaders into the group, ignoring the outflow process. First, for the whole social networks, the inflow and outflow of public opinion spreaders are consistent. Second, if the number of inflowing nodes is greater than 1 (${\mathrm{\Omega }}_k^t>1$ ), it means that public pinion can be spread across group networks. Therefore, the threshold of public opinion propagation in static group networks and netizens’ active probability can be solved as follows ($\mathrm{\Delta }t=1$).

$$\begin{array}{c}\hfill {\mathbb{R}}_0^1={\displaystyle \frac{<k^2>-<k>}{<k{>}^2}}{\displaystyle \frac{Y\alpha b(R_0-1)}{X\mu }};b>b_{}^{\ast }={\displaystyle \frac{<k{>}^2}{<k^2>-<k>}}{\displaystyle \frac{X\mu }{Y\alpha (R_0-1)}}\end{array}$$ 5

In static group networks, its topology structure significantly affects the spreading process of public opinion. Specifically, for group networks with significant heterogeneity $\frac{<k{>}^2}{<k^2>-<k>}\sim 0$, even a lower active probability can still promote the propagation of public opinion among groups. The above conclusions are consistent with those of spreading dynamics in single group networks^34,36.

Modeling on public opinion propagation in temporal group networks

This section will launch the modeling of public opinion propagation in temporal networks from the following two scenarios.

$m=1$

The number of spreader in the group with active probability a at time t is defined as ${\mathrm{\Omega }}_a^t$, and it can be expressed as:

$$\begin{array}{c}\hfill {\mathrm{\Omega }}_a^t=a{X}_a\mathrm{\Delta }t\sum _{a^{\prime }}{\mathrm{\Omega }}_{a^{\prime }}^{t-1}(1-R_0^{-q_{k^{\prime }\to k}})(1-{\displaystyle \frac{{\mathrm{\Omega }}_a^{t-1}}{X_a}}){\displaystyle \frac{1}{X}}+X_a\sum _{a^{\prime }}{a}^{\prime }\mathrm{\Delta }t{\mathrm{\Omega }}_{a^{\prime }}^{t-1}(1-R_0^{-q_{k^{\prime }\to k}})(1-{\displaystyle \frac{{\mathrm{\Omega }}_a^{t-1}}{X_a}}){\displaystyle \frac{1}{X}}\end{array}$$ 6

In Eq.(6), the first item represents the number of spreader that flowing into the group a from the external groups when the active group a actively establishes connection with the external groups. The second item means the number of spreader that flowing into the group a from the external groups when the inactive group a passively establishes connection with the external groups. Then, Eq.(6) can be simplified based on the above conditions.

$$\begin{array}{c}\hfill {\mathrm{\Omega }}_a^t={\displaystyle \frac{a{X}_{a}b\alpha (R_0-1)}{X\mu }}\sum _{a^{\prime }}{\mathrm{\Omega }}_{a^{\prime }}^{t-1}{Y}_{a^{\prime }}+{\displaystyle \frac{X_{a}b\alpha (R_0-1)}{X\mu }}\sum _{a^{\prime }}{a}^{\prime }{\mathrm{\Omega }}_{a^{\prime }}^{t-1}{Y}_{a^{\prime }}\end{array}$$ 7

Introducing $\mathrm{\Theta }={\displaystyle \frac{b\alpha (R_0-1)}{\mu }}$, Eq.(7) can be expressed as:

$$\begin{array}{c}\hfill {\mathrm{\Omega }}_a^t=aF(a)\mathrm{\Theta }\sum _{a^{\prime }}{\mathrm{\Omega }}_{a^{\prime }}^{t-1}{Y}_{a^{\prime }}+F(a)\mathrm{\Theta }\sum _{a^{\prime }}{a}^{\prime }{\mathrm{\Omega }}_{a^{\prime }}^{t-1}{Y}_{a^{\prime }}\end{array}$$ 8

In order to find a close form, we therefore introduce a set quantities:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\tau }^t={\sum }_a{\mathrm{\Omega }}_a^t{Y}_a\hfill \\ {\upsilon }^t={\sum }_{a}a{\mathrm{\Omega }}_a^t{Y}_a\hfill \\ {\varphi }_n={\sum }_a{a}^{n}F(a)Y_a\hfill \end{array}\right)\end{array}$$ 9

By multiplying both sides of Eq.(8) by $Y_a$ and summing, we obtain

$$\begin{array}{c}\hfill \int {\mathrm{\Omega }}_a^t{Y}_a=\int aF(a)Y_a\mathrm{\Theta }\sum _a{\mathrm{\Omega }}_a^{t-1}{Y}_a+\int Y_{a}F(a)\mathrm{\Theta }\sum _{a}a{\mathrm{\Omega }}_a^{t-1}{Y}_a\end{array}$$ 10

That is: ${\tau }^t={\varphi }_1\mathrm{\Theta }{\tau }^{t-1}+<Y>\mathrm{\Theta }{\upsilon }^{t-1}$.

Then, by multiplying both sides of Eq.(8) by $a{Y}_a$ and summing, we obtain

$$\begin{array}{c}\hfill \int {\mathrm{\Omega }}_a^{t}a{Y}_a=\int a^{2}F(a)Y_a\mathrm{\Theta }\sum _a{\mathrm{\Omega }}_a^{t-1}{Y}_a+\int a{Y}_{a}F(a)\mathrm{\Theta }\sum _{a}a{\mathrm{\Omega }}_a^{t-1}{Y}_a\end{array}$$ 11

That is: ${\tau }^t={\varphi }_2\mathrm{\Theta }{\tau }^{t-1}+{\varphi }_1\mathrm{\Theta }{\upsilon }^{t-1}$.

In the continuous time limit, the above two equations can be written in a differential form:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle \frac{d{\tau }^t}{\mathit{dt}}}={\tau }^t-{\tau }^{t-1}=({\varphi }_1\mathrm{\Theta }-1){\tau }^{t-1}+<Y>\mathrm{\Theta }{\tau }^{t-1}\hfill \\ {\displaystyle \frac{d{\upsilon }^t}{\mathit{dt}}}={\upsilon }^t-{\upsilon }^{t-1}={\varphi }_2\mathrm{\Theta }{\tau }^{t-1}+({\varphi }_1\mathrm{\Theta }-1){\upsilon }^{t-1}\hfill \end{array}\right)\end{array}$$ 12

The Jacobian matrix of this set of linear differential equations can be expressed as:

$$\begin{array}{c}\hfill J_1=\left[\begin{array}{cc}{\varphi }_1\mathrm{\Theta }-1& <Y>\mathrm{\Theta }\\ {\varphi }_2\mathrm{\Theta }& {\varphi }_1\mathrm{\Theta }-1\end{array}\right]\end{array}$$ 13

The propagation of public opinion in temporal group networks requires the maximum eigenvalue of the Jacobian matrix is greater than 0, that is, ${\mathrm{\Lambda }}_{\mathit{max}}(J_1)={\varphi }_1\mathrm{\Theta }-1+\mathrm{\Theta }\sqrt{{\varphi }_2<Y>}>0$, therefore, the threshold of public opinion propagation in temporal group networks and netizens’ active probability can be solved as follows.

$$\begin{array}{c}\hfill {\mathbb{R}}_0^2={\displaystyle \frac{(R_0-1)\alpha b}{\mu }}\ast [{\varphi }_1+\sqrt{{\varphi }_2<Y>}];b>b_{}^{\ast }={\displaystyle \frac{\mu }{(R_0-1)\alpha }}{\displaystyle \frac{1}{{\varphi }_1+\sqrt{{\varphi }_2<Y>}}}\end{array}$$ 14

$m>1$

In the scenario $m>1$, the solution idea of threshold is similar to the previous section, but the representation of ${\mathrm{\Omega }}_a^t$ is different.

$$\begin{array}{c}\hfill {\mathrm{\Omega }}_a^t=ma{X}_a\mathrm{\Delta }t\sum _{a^{\prime }}{\mathrm{\Omega }}_{a^{\prime }}^{t-1}(1-R_0^{-q_{k^{\prime }\to k}})(1-{\displaystyle \frac{{\mathrm{\Omega }}_a^{t-1}}{X_a}}){\displaystyle \frac{1}{X}}+X_{a}a\sum _{a^{\prime }}{a}^{\prime }\mathrm{\Delta }t{\mathrm{\Omega }}_{a^{\prime }}^{t-1}(1-R_0^{-q_{k^{\prime }\to k}})(1-{\displaystyle \frac{{\mathrm{\Omega }}_a^{t-1}}{X_a}}){\displaystyle \frac{1}{X}}\end{array}$$ 15

Then, Eq.(15) can be simplified as:

$$\begin{array}{c}\hfill {\mathrm{\Omega }}_a^t=maF(a)\mathrm{\Theta }\sum _{a^{\prime }}{\mathrm{\Omega }}_{a^{\prime }}^{t-1}{Y}_{a^{\prime }}+F(a)\mathrm{\Theta }\sum _{a^{\prime }}{a}^{\prime }{\mathrm{\Omega }}_{a^{\prime }}^{t-1}{Y}_{a^{\prime }}\end{array}$$ 16

By multiplying both sides of Eq.(16) by $Y_a$ and $a{Y}_a$ respectively, and summing, we obtain:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle \frac{d{\tau }^t}{\mathit{dt}}}={\tau }^t-{\tau }^{t-1}=(m{\varphi }_1\mathrm{\Theta }-1){\tau }^{t-1}+<Y>\mathrm{\Theta }{\tau }^{t-1}\hfill \\ {\displaystyle \frac{d{\upsilon }^t}{\mathit{dt}}}={\upsilon }^t-{\upsilon }^{t-1}=m{\varphi }_2\mathrm{\Theta }{\tau }^{t-1}+({\varphi }_1\mathrm{\Theta }-1){\upsilon }^{t-1}\hfill \end{array}\right)\end{array}$$ 17

The Jacobian matrix of this set of linear differential equations can be expressed as:

$$\begin{array}{c}\hfill J_2=\left[\begin{array}{cc}m{\varphi }_1\mathrm{\Theta }-1& <Y>\mathrm{\Theta }\\ m{\varphi }_2\mathrm{\Theta }& {\varphi }_1\mathrm{\Theta }-1\end{array}\right]\end{array}$$ 18

Therefore, the threshold of public opinion propagation in temporal group networks and netizens’ active probability can be solved as follows.

$$\begin{array}{c}\hfill \begin{array}{cc}& {\mathbb{R}}_0^3={\displaystyle \frac{\alpha b(R_0-1)}{\mu }}{\displaystyle \frac{{\varphi }_1(m+1)\sqrt{4m{\varphi }_2}<Y>+{(m-1)}^2{\varphi }_2^2}{2}}\hfill \\ \hfill & b>b_{}^{\ast }={\displaystyle \frac{\mu }{\alpha (R_0-1)}}{\displaystyle \frac{2}{{\varphi }_1(m+1)\sqrt{4m{\varphi }_2}<Y>+{(m-1)}^2{\varphi }_2^2}}\hfill \end{array}\end{array}$$ 19

It can be seen from Eq.(14) and Eq.(19) that, the threshold of public opinion propagation is inversely proportional to the immune probability ($\mu$), and proportional to parameters such as the propagation range of public opinion ($\alpha$), the active probability of netizens (b) and groups (a), and the number of added edges (m), which is in line with the expectation.

Simulation experiment and discussion

Based on the above public opinion propagation model in static and temporal group networks, this section plans to simulate the spreading process of public opinion, summarize its spreading law and characteristics, and propose suggestion for guiding public opinion propagation.

Construction of experimental platform

According to the above analysis, this section intends to design four scenarios to compare the spreading characteristics of public opinion.

Scenario I(Simple group): All netizens (Y) are randomly distributed with in X groups and there are no connections between groups.

Scenario II(Static group): All netizens (Y) are randomly distributed with in X groups and there are fixed connections between groups.

Scenario III(Dynamic group$_{}^{m=1}$): All netizens (Y) are initially randomly distributed with in X groups, and the connections between groups are in a temporal changing. Each active group is only connected to one other group within a time step.

Scenario III(Dynamic group$_{}^{m>1}$): All netizens (Y) are initially randomly distributed with in X groups, and the connections between groups are in a temporal changing. Each active group is connected to $m>1$ other groups within a time step.

The definition of experimental parameters follows the following criteria. The number of netizens and groups are scaled down according to the 54th Statistical Report on the Development of the Internet in China¹, the Guide to China’s Social Media Platforms in 2023². The active probability distributions of groups and netizens are based on questionnaire surveys and netizens social data. The definition of public opinion spreading parameters is based on research results^29,34,36and survey data of netizens’ spreading willingness. The topological structure within groups are randomly established base on ER random network³, NW small world network^4,5and BA scale free network⁶. At the initial time, one user is randomly selected as initial public opinion spreader. Last but not the least, aims to reduce the error caused by the randomness of experimental results, each experiment was conducted 100 times independently and the average of the 100 experiment results is selected as the final result for further analysis.

Comparison of public opinion propagation process

In order to compare the propagation process of public opinion under the four scenarios, the density of Spreader (S(t)) and Recovered (R(t)), as well as the number of infected groups (G(t)), were taken as observation indicators, and the experimental results were shown in Figure 4- 5.

Fig. 4.

The changes of S(t) (a) and R(t) (b) in group networks.

Fig. 5.

The changing trend of the number of infected groups (G(t)) in group networks.

Figure 4 presents the density changing curves of public opinion spreader and recovered in OSN under four scenarios respectively. Overall, S(t) and R(t) exhibit similar trends in four types of group network topologies: S(t) shows an “inverted U” trend and R(t) presents an “S-shaped” trend. In addition, public opinion also presents differentiated spreading characteristics under the four scenarios.

(1) In the initial spreading stage, compared with temporal group networks (Dynamic), public opinion spreads rapidly in static group networks (Simple and Static). It can be seen from Figure 4 that once public opinion appears ($t\in [1,10]$), it spread rapidly in static group networks, and over time, the density of Spreader (S(t)) and Recovered (R(t)) increase obviously. While in temporal networks, the propagation range of public opinion is still limited to the initial value and has not diffused widely in cyberspace. Such as at $t=5$, S(t) in Static group networks is as high as 2%; while it in Dynamic$_{}^{m>1}$ group networks is less than one in ten thousand.

(2) In the middle and later spreading stages, compared with temporal group networks (Dynamic), the propagation of public opinion in static group networks is slightly weak. With the continuous propagation of public opinion, it can be seen from Figure 4 that S(t) and R(t) increase rapidly and it can be predicted that with the increase of m, the spreading gap between static and temporal group networks will gradually narrow. Such as $t=17$, S(t) in Dynamic group networks exceeds that in Simple group networks; and at $t=35$, S(t) in Static group networks has reached a turning point, and it in Dynamic group networks still shows an increasing trend. So as time goes on, the gap between the two scenarios will gradually narrow.

In addition, the changing curve of the number of infected groups (G(t)) shown in Figure 5 further confirms the above spreading characteristics. Except for the Simple group networks, the difference of G(t) between Dynamic and Static group networks gradually narrowed over time. At $t=43$, G(t) in Dynamic$_{}^{m>1}$ group networks exceeded that in Static group networks and still maintained an increasing trend.

For the above experimental results, we gave the following explanation. For the propagation of public opinion in static group networks, on the one hand, there are relatively fixed spreading sources within the group, such as in Simple group networks, cross-group propagation of public opinion cannot be realized, considering the fixed spreading sources and the spreading threshold is greater than 1, no matter whether the group is homogeneous or heterogeneous network, it provides conditions for the wide diffusion of public opinion in a short period time. On the other hand, there is a stable social relationship among the groups, so the initial groups containing spreaders can transmit spreading sources to the neighbor groups sustainably, which also promotes the large-scale diffusion of public opinion. While in temporal group networks, considering the movement mechanism of spreaders and the temporal changes in social relationships between groups, it is difficult for public opinion to spread quickly in cyberspace. With the continuous propagation of public opinion, the stable social relationships among static group networks also limit its further diffusion: public opinion can only spread in groups with connections, while it is difficult to penetrate some isolated groups with sparse social relationships. However, for temporal group networks, the temporal changes of social relationships make it possible for any two groups to establish connections. In particular, with the increase of m, public opinion in temporal group networks is more likely to traverse the entire network space compared with static group networks.

The above results have important implications for the theoretical research and management practice of public opinion propagation. On the one hand, in the theoretical study of network dynamics, as the carrier of public opinion propagation, the network topological structure within and among groups has a significant impact on the dynamic process, especially the temporal changes in connections between groups endow it with stronger random characteristics. On the other hand, for the guidance of public opinion propagation in social platforms, in addition to traditional means such as publicity and endorsement, reward and punishment, the regulator can also achieve their goals by adjusting the temporal changes in social relationships between groups. For example, if there exits negative information in a social platform, the regulators can cut off the connections between the platform and others at the first time to prevent its wide-diffusion. Then, negative public opinion within the platform can be centrally processed. While for the propagation of positive information, they can increase the social connections between the platform and external social platforms during the window period, so as to promote its rapid coverage of the entire network cyberspace.

Sensitivity analysis of experimental parameter $\beta$ and $\mu$

According to the theoretical results, in addition to group networks topology, the propagation of public opinion is also limited by the change of netizens’ state transition probability ($\beta$ and $\mu$). Identifying the key parameter that affect its diffusion is also great significance for guiding the evolution of public opinion. Taking the propagation of public opinion in Dynamic$_{}^{m>1}$ group networks as an example, the experimental parameters were adjusted as $\beta ,\mu \in [0,1]$ and the results are shown in Figure 6.

Fig. 6.

The variation process of Max(S(t)) (a) and Max(R(t)) (b) with $\beta$ and $\mu$.

Figure 6 shows the variation process of Max(S(t)) and Max(R(t)) with the probability $\beta$ and $\mu$. On the whole, both $\beta$ and $\mu$ significantly affect public opinion propagation process. With the increase of propagation probability ($\beta$) and the decrease of immune probability ($\mu$), the density of Spreader and Recovered showed a growth trend. Specifically, in the Upper Left (UL) corner, meaning $\beta$ is large and $\mu$ is small, the density of Spreader and Recovered is the largest; while in the Lower Right (LR) corner, representing $\beta$ is small and $\mu$ is large, it has the smallest propagation range of public opinion. The above conclusions are trivial and meet expectations.

In addition, it was found that they also have different reactions to the changes in $\beta$ and $\mu$, through analyzing the changing process of Max(S(t)) and Max(R(t)). First, comparing the propagation results in the Lower Left (LL) corner (where $\beta$ and $\mu$ are both small) and the Upper Right (UR) corner (where $\beta$ and $\mu$ are both large), it can be seen that Max(S(t)) and Max(R(t)) are significantly higher in the corresponding scenarios of the latter than the former. In other words, as long as the propagation probability ($\beta$) is large enough, no matter how high the immune probability ($\mu$) is, it can not hinder the wide diffusion of public opinion. Second, Max(S(t)) and Max(R(t)) show a obvious change along $\beta$’s direction, but a small change along $\mu$’s direction. For example, when $\beta >0.8$, there is almost no change in Max R(t) along $\mu$’s direction. Combining the above two findings, it can be concluded that compared to immune probability ($\mu$), public opinion propagation in temporal group networks is more sensitive to changes in propagation probability ($\beta$).

As for the reasons, according to the spreading rules of public opinion, parameters $\beta$ and $\mu$ describe the state changes of Ignorant and Spreader, and R(t) representing the spread range of public opinion are evolved from Spreader. Therefore, during its propagation process, if netizens convert into Spreader with a high probability, it will inevitably increase the coverage of public opinion, even if they quit the spreading process at the next moment. Compared with Spreader, the changes in the status of Ignorant have a more significant impact on public opinion propagation process.

The influence of groups’ and netizens’ active probability on public opinion propagation

Next, we will discuss the joint influence of groups’ and netizens’ active probability on public opinion propagation with the help of spectrograms. The parameters are adjust as ${\gamma }_a,{\gamma }_b\in [0.5,5]$, and the experimental results are shown in Figure 7.

Fig. 7.

The variation process of Max(S(t)) (a) and Max(R(t)) (b) with ${\gamma }_a$ and ${\gamma }_b$.

As shown in Figure 7, both groups’ and netizens’ active probability significantly affect public opinion propagation, and with the increase of parameters representing their active probability, the density of Spreader and Recovered shows a decline trend. In LL corner of Figure 6 (where ${\gamma }_a$ and ${\gamma }_b$ are both small), public opinion has largest spreading scope in cyberspace; on the contrary, in LR corner (where ${\gamma }_a$ and ${\gamma }_b$ are both large), it is difficult to diffuse in OSN.

In addition, the propagation of public opinion also shows differentiated characteristics with the changes in parameters. Comparing the results of public opinion propagation in UL (${\gamma }_a$ is small and ${\gamma }_b$ is large) and LR ($\gamma a$ is large and ${\gamma }_b$ is small) corner, we can see that public opinion propagation is slightly dominate in the latter. At the same time, compared with $\gamma a$, Max(S(t)) and Max(R(t)) showed a more significant change trend along $\gamma b$’s direction. Furthermore, the average range values of Max(S(t)) and Max(R(t)) under fixed ${\gamma }_a$ and ${\gamma }_b$ are calculated respectively, and it was found that the variation amplitude of Max(S(t)) and Max(R(t)) with ${\gamma }_b$ under constant ${\gamma }_a$ are greater than that of Max(S(t)) and Max(R(t)) with ${\gamma }_a$ under constant ${\gamma }_b$. The above findings show that compared with groups’ active probability, the propagation of public opinion is more dependent on netizens’ active probability.

In the definition of groups’ and netizens’ active probability ($F(x)=B{x}^{-\gamma }$), the larger $\gamma$, the more significant the heterogeneity in their active probability distribution. That is, the fewer groups and netizens with high active probability and the more groups and netizens with low active probability, and the lower their average active probability, as shown in Figure 8. Therefore, as ${\gamma }_a$ and ${\gamma }_b$ increase, the number of active groups and netizens decreases, leading to the suppression of public opinion propagation. In addition, considering that netizens are the backbone of public opinion propagation, when they have higher active probability, it can still promote the widespread propagation of public opinion even if the connections between group networks are sparse. On the contrary, in a dense temporal group networks, if netizens have lower active probability, it cannot guarantee the diffusion of public opinion in cyberspace. Compared with groups’ active probability, netizens’ has a more significant impact on public opinion propagation process in temporal group networks.

Fig. 8.

The changing trend of average active probability of groups and netizens with ${\gamma }_a$ and ${\gamma }_b$.

The above conclusions provide important reference for the governance of public opinion in cyberspace. First, the regulators can introduce time management system into social platforms to restrict netizens’ continuous use time, encourage them to use platforms with a healthy manner, which can compress the diffusion space of negative information. Then, the regulators can implement hierarchy for social platforms and grade them based on the content diffused within the platform. The connections between groups at the same level are allowed to be established, and social platforms at different levels are not allowed to establish connections. In particular, social platforms such as Tiktok, Sina and WeChat, which currently have a high number of active users, are not allowed to establish bridge relationships with other platforms privately, which can prevent the cross-platform propagation of negative information and achieve the purpose of purifying cyberspace.

Conclusions

The propagation of public opinion in cyberspace has received continuous attention from the academic and management practice due to its significant impact on economic development and social stability. Under the framework of the development trend of complex networks and the needs of management practice, this paper proposed a dynamic model coupling public opinion propagation process and group networks topology evolution, solved and compared the spreading threshold of public opinion in static and temporal group networks respectively. Then, the spreading characteristics of public opinion, the relationships between public opinion propagation and different factors were discussed with the help of simulation experiments. Lastly, targeted suggestions for the governance of public opinion were proposed based on experimental results. The detailed conclusion is as follows.

(1) The topology of group networks significantly affect public opinion propagation process and there exit spreading thresholds for both static and temporal group networks. (2) The propagation of public opinion in static and temporal group networks not only presents similar spreading trends, but also has differentiated spreading characteristics. Such as, in the initial spreading stage, public opinion has a rapid momentum in static group networks; while in the middle and later spreading stages, it has more potential in temporal group networks. (3) In temporal group networks, compared with Spreader, the propagation of public opinion is more affected by Ignorant’s state changing. (4) Both groups’ and netizens’ active probability have significant influence on public opinion propagation, but it depends more on netizens’ active probability. Based on research findings, this paper provided a series of suggestions such as introducing time management system, implementing hierarchy for social platforms and strengthening their connections management between platforms, so as to help the regulators actively respond public opinion propagation and achieve the purpose of purifying cyberspace.

Based on the current mainstream research results on temporal network topology changing^12–18and public opinion propagation^{25,27,29,34–36}, this paper coupled two kinds of dynamic processes mentioned above and conducted modeling on the public opinion propagation in temporal group networks. The research conclusions on the one hand, enrich the research materials of temporal networks, especially promoting the development of temporal networks topology modeling and network dynamics. On the other hand, as the carrier of public opinion propagation, the temporal characteristics of the topological structure of social platforms are more and more significant under the new media environment. This study is of great value for accurately grasping the propagation process of public opinion in social platforms and conducting public opinion guidance and management, and also plays an important role in the development of information science and the improvement of national information security and emergency management. While the co-evolution process of public opinion propagation and group networks evolution is a very complex dynamic process, and this study can also provide reference for the deep expansion of this field.

Data availability 

The data used in this research is from the 53rd Statistical Report on the Development of China’s Internet Network issued by China Internet Network Information Center (CNNIC) in Beijing; Annual Internet statistics released by the Ministry of Industry and Information Technology of the People’s Republic of China. The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.