3.1. Stages in the Evolution of Public Opinion
3.1.1. Formation Period
The initial period of formation is a process in which public opinion constantly changes. Generally speaking, online public opinion primarily conveys potential public opinion, which materializes based on online expression and offline dissemination and diffusion, through which emotional focus is generated.
In an online environment, netizens are the main actors and play a vital role. This period is generally relatively short, often leading to missed opportunities for governance. Changes in social policies aggravate the escalation of social conflicts to a certain extent, and some social–psychological imbalances and social disorders occur frequently. Due to the popularity of the internet and the immediacy and openness of online platforms, public opinion spreads rapidly in cyberspace. Individuals not only express their will and opinions through online dialogue but also discuss views with the surrounding public offline, prompting more participation. The combination of online and offline means leads to an increase in the expression of public opinion. The desire to determine the truth behind an event tends to complicate its development; thus, network public opinion on emergencies is gradually formed, as shown in
Figure 1.
3.1.2. Diffusion Period
After its initial formation, internet public opinion enters a long period of development and evolution. The diffusion period is one of the critical development stages. During this stage, the intensity increases dramatically compared to the formation period, mainly due to the evolution of online public opinion as the number of participants increases (
Figure 2).
Considering the gradual increase in the number of participants, together with the involvement of the media and opinion leaders who play an essential role in the evolution of public opinion, and the appropriate measures taken by the government and other relevant departments, extensive relevant information is generated from public opinion. The participation of netizens, the promotion of related media, the leadership of opinion leaders, and the intervention and guidance of the government all significantly contribute to the evolution of online public opinion during the diffusion period.
Firstly, the broad group of netizens includes the main parties involved in emergencies, some online individuals facing physical or mental health risks, and some bystanders. Secondly, the media, including traditional media and self-media, use their technical advantages to integrate and report information, thus playing a leading role in disseminating public opinion in cyberspace [
37]. Thirdly, although mainstream values tend to materialize in internet public opinion during the diffusion stage, netizens often have different views due to a lack of knowledge or information due to the massive information generated. This highlights the guiding role of opinion leaders. Compared with traditional opinion leaders, internet opinion leaders in the new era are more diversified, and the relationship between netizens and opinion leaders is no longer a single mode of acceptance in the past but an emerging mode of interaction. Fourthly, regarding government control, the timely response of relevant government departments and their active guidance and understanding of public views are significant in ensuring the normal development of internet public opinion.
3.1.3. Explosion Period
This stage refers to the period when, after the initial formation and dissemination of online public opinion, multiple individuals, such as the involved parties, media, and opinion leaders, continue to pay attention to and discuss the event using their resources, which leads to a collision of various opinions and a rapid increase in the event’s impact, thus generating eruption forces similar to a volcano effect. This is also when expressing views is most likely to cause harm [
38]. Generally speaking, the eruption period is relatively short, but it can severely damage society [
39]. The process involves the diffusion of public opinion and the interaction of different individuals, leading to exponential growth in information and a full-scale eruption of public opinion. After entering this stage, the influence of public opinion rapidly escalates to the threshold value, and network clustering is significant. The original stable state of network groups to safeguard their interests deteriorates, thus escalating public opinion. During this phase, the participation of media, netizens, and opinion leaders reaches its maximum; views are exchanged through lively debates; and the existing conflicts rapidly intensify, leading to explosive growth in the expression of online public opinion, which is difficult to control, as shown in
Figure 3.
During the high-risk period of disseminating online public opinion on emergencies, the degree and scope of the influence of an emergency event reach their maximum after the formation and spread of public sentiments. At this stage, people’s attitudes, views, and emotions toward emergencies accumulate to the highest level [
40]. The evolution of resonance law during this phase involves an interactive phenomenon in which netizens’ emotions resonate with the primary heat of online public opinion on critical incidents, including the associated topics that netizens reflect on and their causes. These particular issues stimulate netizens’ subconscious minds and prompt the formation of relevant emotions. Regarding online public opinion issues, the heat and duration of public opinion risk evolution, the heat of netizens’ discussions, and the heat of online public opinion issues show the same resonance trend. After public opinion on emergencies undergoes the formation and diffusion periods, the heat of netizens’ issues stays within the normal fluctuation range. Nevertheless, public opinion risks are likely due to the abnormally high intensity of netizens’ sentiments. This leads to the rapid spread and eruption of public opinion risks. Eventually, as netizens’ sentiments on the incident increase to a relative saturation point, public opinion risks are maximized. The resonance between netizens’ sentiments and the heat of public opinion issues influences the evolution of the risks, which consequently reach a climax.
3.1.4. Declining Period
After the first three stages, online public opinion gradually enters a period of decline, which eventually ends after the duration of the outbreak is over. However, the end of public opinion does not mean the outbreak is over. During this period, netizens’ attention to emergencies gradually decreases or is shifted by new public opinion hotspots resulting from the practical guidance of the main parties involved in the emergency response. In this stage, risks exhibit a gradually decreasing trend. This means that while resources, energy, and information are constantly devoted to the topic related to emergencies, the responses to emergencies and online public opinion are also continually outputting matter, power, and information. Under multiple parties’ continuous exchange and transformation, public opinion risks gradually weaken and enter a receding state, but new threats associated with public opinion may also appear. However, this means that the heat of public opinion decreases and enters a latent state, as shown in
Figure 4.
When public opinion reaches an outbreak point, the outbreak duration is usually relatively short. The government will cooperate with the official media and relevant online governmental organizations to maintain the regular operation of social order. The official press will release positive information, eliminate false public opinion, and guide the general public in forming correct public opinion values. Over time, the truth behind the incident will gradually become more apparent, and the government will add facts. With the supervision of relevant departments and the dissemination of positive reports by official media, the blind extreme emotions of netizens are eliminated, public opinion’s attention will be significantly weakened, the number of participants will gradually decrease, and public opinion will decline. The influence of online public opinion will be reduced considerably during this period, and the number of discussions on related topics will also decrease significantly. Although some potentially relevant comments remain, the overall impact will be insignificant. Moreover, as new events emerge, public opinion regarding the original incident will give way to new discussions. However, it is worth noting that although public opinion on the event enters a recession period, and the number of remaining topics has little impact, it has not entirely disappeared, so it is necessary to pay attention to the “backlash effect” of public opinion. The original event only reaches an equilibrium to a certain extent; thus, considering the emergence of new triggers is necessary. With the occurrence of a secondary crisis related to the event itself or the reoccurrence of a similar event, secondary views are likely generated, thus shaping secondary public opinion more rapidly and efficiently.
A system diagram regarding the characteristics of the evolutionary process of online public opinion on emergencies is presented in
Figure 5.
3.2. Modeling
The evolutionary process of public opinion on emergencies is characterized by its rapidity, dynamism, and uncertainty. Information spreads rapidly on social media and online platforms, leading to rapid formation and changes in public opinion. This process is influenced by various factors, including the development of the event itself, media reports, public sentiments, and online interactions, highlighting the significant unpredictability associated with public opinion. At the same time, public opinion shows a clear tendency to become emotional and polarized, and different groups may react differently to the same event. In addition, the filtering, distortion, and emotional dissemination of information often affect the public’s perception and judgment of events. Overall, online public opinion triggered by emergencies is complicated, volatile, and challenging, reflecting the complexity of information dissemination and social interaction in contemporary online society.
3.2.1. Model Design
Discrete events are commonly designed using state-based formalisms, like state diagrams and Petri nets [
41]. Petri nets are a mathematical modeling tool for describing and analyzing multiple concurrent processes in a system.
In the early 1980s, scholars proposed a Stochastic Petri Net model based on the original Petri net by introducing time parameters, further improving the theory and application of Petri nets [
42]. A Stochastic Petri Net (SPN) is a random delay time model that indicates the association between each implementable transition and its implementation in a Petri net. The currently popular method is the continuous-time SPN proposed by Molly, where the implementation time of each transition follows an exponential distribution [
43].
Considered a variant of Petri nets, they are more suitable for modeling systems with uncertainty and variability, such as those encountered with online public opinion. These models are often regarded as a particular type of directed graph, with the following main features:
- (1)
Places: These are the state tokens of a Petri net, usually represented by circles. Places can contain any number of tokens to indicate the system’s state.
- (2)
Transitions: Events representing changes in the system’s state are usually presented as rectangles or bars. When a transition occurs, the markers in the positions connected to it change according to specific rules.
- (3)
Tokens: These are entities placed at different locations to indicate some resource or state of the system. The distribution of tokens defines the current state of the Petri net.
- (4)
Input Arcs: These are arrows pointing from a location to a change, indicating the conditions or resources required for the change to occur.
- (5)
Output Arcs: These are arrows pointing from a change that occurs to a location, indicating the results or resources generated after the change.
- (6)
Weights: These are labeled on the arcs to indicate the number of markers moving from position to variation or vice versa. Weights in simple Petri nets are usually 1, but weights can represent more complex relationships in more complicated models.
A Markov chain is a stochastic process in which the future state of a system is only related to the current state and not to the past state. To compensate for the limitations of Stochastic Petri Nets, they can be transformed into Markov chain models, which utilize the theory and methods of Markov chains to analyze the long-term behavior and stability of the system, thereby providing a deeper understanding and more accurate prediction. This conversion process includes creating a state table and transition matrix, as well as determining transition probability, to provide more reliable analysis results.
- (1)
State table: describes the possible states defined in the Stochastic Petri Net model.
- (2)
Transition matrix: represents the possible transitions between states. For any state sequence, a Markov chain is used to calculate the probability of each state. In a Stochastic Petri Net model, state transitions occur with a deterministic probability. By transforming the Stochastic Petri Net model into a Markov chain, we can obtain a probability model that guides the system’s state transition.
Through the transformation of Stochastic Petri Net isomorphic Markov chains, the system behavior can be more accurately estimated, and the complexity present in the system can be better explained.
Based on the above, the specific steps of model design are as follows:
Step 1: An SPN model is developed according to
Figure 5 (systematic diagram of the evolutionary process of online public opinion on emergencies.), and the time delays are associated with the corresponding variations.
Step 2: An isomorphic Markov chain is constructed. The set of reachable identifiers R(M) is determined and an isomorphic Markov chain is constructed for each arc of the SPN model, given the variation utilization rate corresponding to that arc.
Step 3: The Markov chain is solved, and the SPN model is systematically analyzed based on the steady-state probability of the Markov chain. Since the SPN model is isomorphic to the time-continuous Markov chain, the steady-state probability of the SPN model can be obtained when the constructed Markov chain has a smooth distribution.
3.2.2. Constructing the SPN Model
(1) SPN model
Definition: An SPN is a directed graph consisting of six elements, i.e.,
SPN = (
P,
T,
F,
W,
M,
λ), where (
P,
T,
F,
W,
M,
λ) is a
P/T system and
λ is the set of variation implementation rates, if, and only if, (1)
; (2)
; (3)
; (4)
is the initial identifier; (5)
; (6) the time interval between the feasible implementation time and the implementation time
is a continuous random variable and follows a distribution function
. The actual parameter
is the average implementation rate of transition
t considering values of
. As shown in
Figure 6.
①
denotes the position in the system, with 11 positions in total, as shown in
Table 1;
②
denotes the finite set of transitions in the system, with a total of 9 transition numbers, as shown in
Table 1;
③ denotes the set of directed arcs in the system, where denotes the set of variable input arcs, and denotes the set of variation output arcs. indicates that there can be forbidden arcs in the system, which only exist in arcs pointing from a location to a change;
④ is the arc weight function;
⑤ is the identity of the Stochastic Petri Net and a vector in which is the number of markers in the first position, and is the initial marker, which indicates the initial state of the system.
(2) SPN model validation
In Stochastic Petri Nets (SPNs), the association matrix A is a critical mathematical tool for representing a system’s relationship between locations and transitions. The association matrix is usually defined as , where is a positive association matrix (representing arcs from locations to transitions), and is a negative association matrix (representing arcs from transitions to locations).
The correlation matrix
B of the SPN model is calculated according to Equation (1), and the results are shown in
Table 2.
Here, denotes the elements of the input matrix I; denotes the elements of the output matrix B; and F represents the flow relationships between repositories and variations.
Solving the system of linear equations reveals non-negative integer values; thus, the following conclusions are drawn: (1) Specific resources in the model (e.g., information, user attention) can maintain a homeostatic state in the evolutionary process of online public opinion. This reflects an equilibrium or repetitive pattern in opinion formation; (2) Self-organization is an essential feature in the evolution of online public opinion. The non-negative integer solution indicates that the system can reach a state of self-equilibrium without external regulation; (3) This points to the stable pattern of information flow or influence distribution in online public opinion. Some important nodes (e.g., individuals, communities, or media) play a vital role in the evolution of public opinion.
3.2.3. SPN Model Isomorphic to a Markov Chain
The specific transformation process is as follows:
Step 1: To obtain a reachable graph according to the SPN model, the reachable identification set and the identification state
of each repository are determined. Each identification state of the SPN model of the evolutionary process of online public opinion on emergencies is expressed as follows:
This process mainly involves identifying all the states in the SPN model and mapping them to the states of the Markov chain while defining the transfer probabilities between them, which is accomplished by analyzing the variation and labeling distributions in the SPN model to construct a state transfer matrix that reflects all possible state transfers and their probabilities. In this way, the SPN model is transformed into a Markov chain to obtain an isomorphic Markov chain for this model, which facilitates Markov chain analysis for predicting and evaluating the system behavior, as shown in
Figure 7. The following text will refer to it as the SPN-MC model for short.
The SPN-MC model demonstrates excellent performance in simulating online public opinion regarding critical incidents. The reachability of all states and the smooth transitions between neighboring states confirm that the model is structurally free of anomalies and meets the stability and reliability requirements. Together, these features confirm the validity and practicality of the model in modeling online public opinion on critical incidents.
3.3. Indicator Selection and Calculation Methodology
Based on the above, in a continuous-time Stochastic Petri Net, the time interval between the implementable transition and the implementation of a change t obeys the exponential distribution function , where is the average implementation rate of a change , and the time of each position transition is a random variable. It obeys an exponential distribution, in which the average implementation rate of a change is . Considering eight states in the SPN-MC model, namely , where directed arcs denote the transitions from one state of online public opinion on incidents to another, and denotes the average implementation rate of the state transitions. Let be the probability that the SPN-MC model of the evolutionary process of online public opinion on emergencies is in state .
In this study, we mainly focused on the analysis of the stage, node, and duration of the evolutionary process of online public opinion, so we selected the stable probability of reachable marking, the busy probability of locations, the utilization rate of variation, the marking flow rate of variation, and the average execution time of the system for calculation.
(1) Stability probability of reachable markings
All equilibrium state equations are first listed according to the isomorphic Markov chain as follows:
where
; thus, the stabilization probability of each reachable marking
can be determined, which is the basis of the system model performance indicator.
This indicator can describe the characteristics and trends of the evolution of public opinion in different types of emergencies. It can reflect the characteristics of event development, public response, information dissemination, and emergency response under various emergencies.
(2) Busy probability of places
determines locations
with
, which is the probability of a token; then, the busy probability of locations is expressed as follows:
(3) Variable utilization rate
Let
be the variation utilization rate; then,
is equal to the sum of the stable probabilities of transforming
. The variable utilization rate is computed as follows:
where
is the set of
, through which the set of all reachable identifiers is determined.
The probability of the places being busy and the variation utilization rate are relevant indicators for the degree of variation in the model. It indicates how often or to what extent a particular change is implemented during the simulation. A higher variant utilization rate suggests that the variant plays a more critical or frequent role in the model dynamics, and processes with high variant utilization rates are also the focus of specification and optimization. The variation utilization rate distribution can reflect each key node’s activity and importance. This helps to understand the key aspects of online public opinion regarding critical incidents and the focus of subsequent optimization.
(4) Marked flow rate of variation
The marker flow rate for the variation
is the average number of markers per unit of time flowing into
with position
per unit time
. The formula is expressed as follows:
where
is the average implementation rate of the change.
The marker flow rate of variation is the rate at which a marker moves between a particular variant and the repository in the SPN-MC model. It reflects the efficiency or frequency with which a specific variant processes markers in the model, with a high marker flow rate indicating that the variant is more active or critical in the model dynamics. It also indicates the importance of processing at each essential stage in the evolution of public opinion on emergencies. These data can help identify the critical links in public opinion management and the areas that should be focused on and optimized in different scenarios.
(5) Average system implementation time
The average execution time of the system is determined as follows:
where
denotes the average number of markers in the position set at steady state.
The difference in the average execution time of the system reflects the efficiency and complexity of the model processing public opinion toward different events. The results can also be compared with accurate case data to verify the model’s effectiveness.