In the era of new media, the public can use social platforms to obtain information triggered by major emergencies worldwide, but due to the diverse forms, large quantities and rich content of network information, it is impossible for the public to grasp all the information. Therefore, the public can watch the audio and video of video bloggers, live broadcasts of anchors, and information organized by self-media to obtain rapid access to relevant information. This paper defines video bloggers, anchors, self-media outlets and other people who have a certain degree of discourse power and are able to influence and shape the opinions of others through their own speech or behavior as opinion leaders. The members of the public who can use online social media platforms to obtain information are defined as netizens. Based on this, this paper defines opinion leaders and netizens as the main actors in the process of disseminating uncertain information after major emergencies. Both of them are limited rational participants with learning ability; in the case of incomplete information, they cannot judge whether the choices they make can maximize the benefits the first time, but due to their learning ability, they can make choices toward the strategy of maximizing the benefits in the process of learning continuously.

After major emergencies occur, opinion leaders need to integrate and sort out the relevant information related to major emergencies and propose their own views for netizens to adopt to gain attention. Since opinion leaders have the right to speak, which leads to their remarks being able to guide netizens’ thoughts, the gaming strategy of opinion leaders can include positive guidance (by investigating the relevant events and releasing the real information about the events) or negative guidance (by rumor mongering, releasing unconfirmed and false information, stirring up the netizens’ emotions, and other means to make profits). The netizens’ game strategies include adopting or not adopting the opinions published by opinion leaders. Upon combinations of the above game strategies, the resultant scenarios encompass: (positive guidance, adoption), (negative guidance, adoption), (positive guidance, nonadoption), and (negative guidance, nonadoption).

This paper is based on the literature [32]. The following assumptions are made for this evolutionary game model:

(1) The probability that an opinion leader chooses posotive guidance or negative one is x and 1-x; the probability that a netizen chooses to adopt or not is y and 1-year, respectively.

The above parameters C ₁, C ₂, C ₃, C ₄, I ₁, I ₂, I ₃, I ₄, L ₁, L ₂, M ₁, M ₂, π, λ are all greater than 0, where 0 ≤ x ≤ 1 , 0 ≤ y ≤ 1 , 0 ≤ π ≤ 1 , 0 < λ < 1 , I 4 > I 3 .

Based on the assumptions above, we can examine that when netizens adopt the positive guidance of opinion leaders, the total profit of the opinion leader is composed of their own investigation cost C ₁ and the profit obtained from netizens’ adoption I ₁, the gain of opinion leaders is − C 1 + I 1 ; the total profit of netizens is composed of their own time and effort cost C ₃ , the cost of judging the statements released by opinion leaders (1-λ) C₄ and the profit from obtaining true information from opinion leaders, as well as the satisfaction and sense of participation gained from participating in the discussion of uncertain information λM ₁ +I ₃ -(1-λ) C ₄ -C _3. In this way, when netizens adopt the negative guidance of opinion leaders, the total profit of the opinion leader is I ₁ +I ₂ -πL ₂ and the total profit of netizens is -λM ₂ +I ₄ -C ₃ -(1-λ) C ₄ . When netizens do not adopt the positive guidance of opinion leaders, the gain of opinion leaders is − C 1 − L 1 , and the gain of netizens is − C 2 . When netizens do not adopt the negative guidance of opinion leaders, the gain of opinion leaders is − π L 2 − L 1 , and the gain of the netizen is − C 2 . The gain matrix is shown in Table 1.

Combining the benefit matrix in Table 1, let the expected benefit of opinion leaders choosing positive guidance be U ₁₁, the expected benefit of opinion leaders choosing negative guidance be U ₁₂, and the average benefit of opinion leaders be U 1 ¯ ; then, U ₁₁, U ₁₂, and U 1 ¯ , respectively: U 11 = y − C 1 + I 1 + 1 − y − C 1 − L 1 = y I 1 + L 1 − C 1 + L 1 (1) U 12 = y I 1 + I 2 − π L 2 + 1 − y − π L 2 − L 1 = y I 1 + I 2 + L 1 − π L 2 + L 1 (2) U 1 ¯ = x U 11 + 1 − x U 12 = − x y I 2 + x π L 2 − C 1 + y I 1 + I 2 + L 1 − π L 2 + L 1 (3)

Based on Eqs 1–3, it can be obtained that, the replication dynamic equation for opinion leaders’ choice of positive guidance strategy is: F x = d x d t = x U 11 − U 1 ¯ = x 1 − x U 11 − U 12 = x 1 − x − y I 2 + π L 2 − C 1 (4)

Similarly, let the expected benefit of the netizen’s choice to adopt be U ₂₁, the expected benefit of the netizen’s choice not to adopt be U ₂₂, and the average benefit of the netizen be U 2 ¯ ; then, U ₂₁, U ₂₂, and U 2 ¯ , respectively: U 21 = x λ M 1 + I 3 − 1 − λ C 4 − C 3 + 1 − x − λ M 2 + I 4 − 1 − λ C 4 − C 3 (5) U 22 = x − C 2 + 1 − x − C 2 = − C 2 (6) U 2 ¯ = y U 21 + 1 − y U 22 = x y λ M 1 + I 3 − 1 − λ C 4 − C 3 + 1 − x y − λ M 2 + I 4 − 1 − λ C 4 − C 3 − 1 − y C 2 = x y λ M 1 + I 3 + λ M 2 − I 4 + y − λ M 2 + I 4 − 1 − λ C 4 − C 3 + C 2 − C 2 (7)

Based on Eqs 5–7, it can be obtained that, the replication dynamic equation for netizens’ choice of adoption strategy is: F y = d y d t = y U 21 − U 2 ¯ = y 1 − y U 21 − U 22 = y 1 − y x λ M 1 + λ M 2 + I 3 − I 4 − λ M 2 + I 4 − 1 − λ C 4 − C 3 + C 2 (8)

Due to limitations, the above Eq. 4 considers only the expected and overall benefits of opinion leaders adopting positive guidance strategies and does not take into account the difference between the benefits of opinion leaders adopting positive guidance and the benefits of adopting negative guidance; therefore, the above Eq. 4 adjusted to: d x d t = x − y I 2 + π L 2 − C 1 (9)

The above Eq. 8 considers only the expected and overall benefits of adopting an adoption strategy by netizens and does not take into account the difference between the benefits of adopting and the benefits of not adopting by netizens; therefore, the above Eq. 8 adjusted to: d y d t = y x λ M 1 + λ M 2 + I 3 − I 4 − λ M 2 + I 4 − 1 − λ C 4 − C 3 + C 2 (10)

Considering that in the real world, opinion leaders and netizens are influenced by uncertain factors caused by their own or external environment, making their decision-making behavior highly uncertain. Therefore, to better understand the influence of the random perturbation term on the decision-making behavior of the game subjects, this section combines stochastic analysis theory and introduces Gaussian white noise into the replication dynamic equation of the evolutionary game between opinion leaders and netizens Eqs 9, 10 in the game, i.e., d x t = x t − y I 2 + π L 2 − C 1 d t + σ x t d w t (11) d y t = y t x λ M 1 + λ M 2 + I 3 − I 4 − λ M 2 + I 4 − 1 − λ C 4 − C 3 + C 2 d t + σ y t d w t (12) where w t is a one-dimensional standard Brownian motion; d w t is Gaussian white noise; when t > 0 , the step size h > 0 , and its increment Δ w t = w π t + h − w t obeys the normal distribution N 0 , h ; σ refers to the intensity of the random perturbation; and when σ is 0, the model is not affected by random disturbing factors.

The above equation is a one-dimensional Itô stochastic differential equation containing a Gaussian random disturbance term, which represents the replicated dynamic equations for opinion leaders and netizens subjected to random perturbations.

Compared with the evolutionary game model between opinion leaders and netizens constructed in literature [32], the stochastic evolutionary game model between opinion leaders and netizens constructed in this paper not only takes into account the impact of known factors on both parties when making behavioral choices but also considers the influence of uncertainties caused by themselves or the external environment. After introducing Gaussian white noise, the stochastic evolutionary game model differs from the ordinary evolutionary game model in the calculation of equilibrium solutions. The ordinary evolutionary game is mathematically expressed based on ordinary differential equations, while the stochastic evolutionary game model is mathematically expressed based on stochastic differential equations.

Analyze the existence and stability of equilibrium solutions for stochastic evolutionary game models separately. Firstly, evaluate the existence of equilibrium solutions. Lemma 1 below is a sufficient condition for the solutions of stochastic differential equations to satisfy existence and uniqueness.

Lemma 1

There is a stochastic process x = x t , t ≥ 0 satisfying the Itô differential equation: d x t = f t , x t d t + g t , x t d w t , ∀ 0 ≤ t ≤ T (13)

If f t , x , g t , x : 0 , T × R → R satisfies all of the following conditions:

(1) The measurability condition, i.e., f t , x , g t , x is binary measurable, f t , x 1 2 , g t , x ∈ L T × R 2

(2) The Lipschitz condition, i.e., the existence of a constant H such that f t , x − f t , y + g t , x − g t , y ≤ H x − y , where ∀ 0 ≤ t ≤ T , x , y ∈ R ;

(3) Linear growth bounded condition: There exists a positive constant P such that f t , x + g t , x ≤ P 1 + x , where ∀ 0 ≤ t ≤ T , x ∈ R ;

(4) Initial conditions: x t 0 on F t 0 is measurable and E x 2 t 0 < ∞ .

Then, there exists a unique process x = x t , t ≥ 0 satisfying the Itô differential Eq. 13, and x(t) is adaptive; with respect to F t 0 being measurable, then E x 2 t 0 < ∞ and ∀ 0 ≤ t ≤ T .

Proposition 1

A stochastic differential equation has a unique solution under x , y ∈ 0 , 1 , t ∈ 0 , T .

Proof:

(1) Measurability condition

Rewrite the equations as d x t = f i x , y d t + g i x , y d w t , i = 1 , 2 , Among them, f 1 x , y = x − y I 2 + π L 2 − C 1 g 1 x , y = σ x f 2 x , y = y x λ M 1 + λ M 2 + I 3 − I 4 − λ M 2 + I 4 − 1 − λ C 4 − C 3 + C 2 g 2 x , y = σ y .

Clearly, f 1 t , x , g 1 t , x , f 2 t , x and g 2 t , x are continuous at 0 , 1 × 0 , 1 , and condition (1) holds.

(2) Lipschitz condition

Due to ∀ x , x * ∈ 0 , 1 : f 1 x , y − f 1 x * , y = − y I 2 + π L 2 − C 1 x − x * ≤ π L 2 − C 1 x − x * (14) g 1 x , y − g 1 x * , y = σ x − σ x * ≤ σ x − x * (15) where π L 2 − C 1 and σ are both positive constants. Assume H = ⁡ max π L 2 − C 1 , σ : Based on Eqs 14, 15, it can be obtained that, max f 1 x , y − f 1 x * , y , g 1 x , y − g 1 x * , y ≤ H x − x * , therefore, when y ∈ 0 , 1 , the Eq. 11 satisfies the local Lipschitz condition. Similarly, Eq. 12 also satisfies the local Lipschitz condition. Thus, condition (2) holds.

(3) Linear growth bounded condition

For any x ∈ 0 , 1 : f 1 x , y 2 = − y I 2 + π L 2 − C 1 x 2 ≤ π L 2 − C 1 2 x 2 ≤ π L 2 − C 1 2 1 + x 2 (16) g 1 x , y 2 = σ x 2 ≤ σ 2 1 + x 2 (17) where π L 2 − C 1 2 and σ 2 are both positive constants, such that P = ⁡ max π L 2 − C 1 2 , σ 2 , then: Based on Eqs 16, 17, it can be obtained that, max f 1 x , y , g 1 x , y ≤ P 1 + x 2 . Therefore, the equation satisfies the linear growth bounded condition. Similarly, the Eq. 12 also satisfies the linear growth bounded condition. Thus, condition (3) holds.

(4) Initial conditions

The initial condition (4) clearly holds.

Thus, Proposition 1 is proved.

From Proposition 1, we can see that there is an equilibrium solution in the stochastic evolution game model of opinion leaders and netizens; i.e., under the premise of no Gaussian white noise interference, opinion leaders and netizens can reach a consensus with each other at the beginning and choose a strategy that meets their own interests. However, in reality, with the passage of time and changes in the internal or external environment, opinion leaders and netizens may be affected by random interference, which changes their decision-making behavior. For example, after a period of time after a major emergency, as relevant investigations are conducted, an increasing number of details are reported, and netizens are able to obtain more information, which makes netizens quickly choose whether to adopt or not to adopt the comments released by opinion leaders. This random factor plays an important role in the original decision-making behavior, and the random factor plays a crucial role in the selection of the final stabilization strategies of opinion leaders and netizens. Therefore, fully considering random factors is more conducive to understanding the decision-making behaviors of opinion leaders and netizens in real society.

Next, the stability of the equilibrium solution is analyzed based on Lemma 2 below:

Lemma 2

Let the stochastic differential equation x = x t , t ≥ 0 satisfy the solution of the Itô differential equation initial value problem: d x t = f t , x t d t + g t , x t d w t , ∀ t ≥ 0 , x t 0 = x 0 (18)

There exist continuous differentiable functions V t , x with positive constants c₁ and c₂ such that c 1 x p ≤ V t , x ≤ c 2 x p .

(1) If there exists a positive constant γ such that L V t , x ≤ − γ V t , x , t ≥ 0 , then the Eq. 18 of the zero solution of the pth order moment index is stable and holds E x t , x 0 p ≤ c 2 c 1 x 0 p e − γ t , t ≥ 0 .

Among others, L V t , x = V t t , x + V x t , x f t , x + 1 2 g 2 t , x V x x t , x .

Proposition 2

For the Eq. 11, when c 1 = c 2 = 1 , p = 1 , γ = 1 , V t , x = x 2 , L V t , x = 2 x 2 − y I 2 + π L 2 − C 1 + σ 2 x 2 .

(1) The exponential stability condition for the zero solution moment of the stochastic evolution equation satisfying the opinion leader is DL₁.

When y ≥ − 1 − σ 2 − 2 π L 2 + 2 C 1 − 2 I 2 and − π L 2 + C 1 + I 2 ≥ 1 + σ 2 2 , DL_1.

(2) The zero-solution moment-exponential instability condition for a stochastic evolutionary equation satisfying the opinion leader is DL₂.

When y ≤ 1 − σ 2 + 2 C 1 − 2 π L 2 − 2 I 2 and π L 2 − C 1 ≥ 1 − σ 2 2 , DL_2.

Proof: For Eq. 11, when c 1 = c 2 = 1 , p = 1 , γ = 1 , V t , x = x 2 , L V t , x = 2 x 2 − y I 2 + π L 2 − C 1 + σ 2 x 2 .

(1) When the zero-solution moment exponent is stabilized, the equation is required to satisfy 2 x 2 − y I 2 + π L 2 − C 1 + σ 2 x 2 ≤ − x 2 . Because x ∈ 0 , 1 and − I 2 < 0 , 2 − y I 2 + π L 2 − C 1 + σ 2 ≤ − 1 ; that is, y ≥ − 1 − σ 2 − 2 π L 2 + 2 C 1 − 2 I 2 ; and because y ∈ 0 , 1 , − 1 − σ 2 − 2 π L 2 + 2 C 1 − 2 I 2 ≤ 1 , the solution is − π L 2 + C 1 + I 2 ≥ 1 + σ 2 2 . Based on this, we can obtain Eq. 11 The exponential stabilization condition for the zero-solution moment of Eq. y ≥ − 1 − σ 2 − 2 π L 2 + 2 C 1 − 2 I 2 and − π L 2 + C 1 + I 2 ≥ 1 + σ 2 2 .

Proposition 2 suggests that the stabilizing strategy of opinion leaders will be influenced by their own factors as well as by random factors. When condition DL₁ is satisfied, opinion leaders will eventually choose the strategy of negative guidance; i.e., over time, no matter what the initial state is, opinion leaders will eventually reach a stable state under the strategy of negative guidance after continuously adjusting their decision-making behavior. When condition DL₂ is met, the opinion leader will eventually choose the strategy of positive guidance; that is, over time, regardless of the initial state, the opinion leader will eventually reach a stable state under the strategy of positive guidance after constantly adjusting his or her decision-making behavior.

From condition DL₁ and condition DL₂ of Proposition 2, it is clear that opinion leaders are more inclined to choose a strategy that is in their own interest. That is, opinion leaders adopt the strategy of negative steering when the expected benefit of choosing negative steering is greater than the expected benefit of choosing positive steering; conversely, they adopt the strategy of positive steering. This finding is consistent with the actual situation because the decision-making behavior of opinion leaders is influenced by factors such as penalties from regulatory agencies and the traffic generated by the attention of netizens, and which strategy is adopted is determined by the expected benefit of that strategy. However, random factors from internal or external sources can also have an impact on the final decision-making behavior of opinion leaders. For example, to smear the public image of China, hostile forces force opinion leaders to make negatively guided choices by means of coercion and enticement when major emergencies occur. In addition, to help the government reduce the impact of major emergencies as soon as possible, opinion leaders always choose to guide them positively, regardless of their own interests. All of these random factors may affect the decision-making behavior of opinion leaders, causing the final stabilization strategy to fluctuate or change.

Proposition 3

For Eq. 12, when c 1 = c 2 = 1 , p = 1 , γ = 1 , V t , y = y 2 , L V t , y = 2 y 2 x λ M 1 + λ M 2 + I 3 − I 4 − λ M 2 + I 4 − 1 − λ C 4 − C 3 + C 2 + σ 2 y 2 .

(1) The exponential stability condition for the zero-solution moment of the stochastic evolutionary equation that satisfies the netizen is D W 1 ∪ D W 2 .

DW₁ when λ M 1 + λ M 2 + I 3 − I 4 > 0 , x ≤ 2 λ M 2 − 2 I 4 + 2 1 − λ C 4 + 2 C 3 − 2 C 2 − 1 + σ 2 2 λ M 1 + λ M 2 + I 3 − I 4 and λ M 2 − I 4 + 1 − λ C 4 + C 3 − C 2 ≥ 1 2 1 + σ 2 .

DW₂ when λ M 1 + λ M 2 + I 3 − I 4 < 0 , x ≥ 2 λ M 2 − 2 I 4 + 2 1 − λ C 4 + 2 C 3 − 2 C 2 − 1 + σ 2 2 λ M 1 + λ M 2 + I 3 − I 4 and 1 − λ C 4 + C 3 − C 2 − λ M 1 − I 3 ≥ 1 2 1 + σ 2 .

(2) The zero-solution moment-exponential instability condition for the stochastic evolutionary equations that satisfy the netizens is D W 3 ∪ D W 4 .

DW₃ when λ M 1 + λ M 2 + I 3 − I 4 > 0 , x ≥ 1 − σ 2 + 2 λ M 2 − 2 I 4 + 2 1 − λ C 4 + 2 C 3 − 2 C 2 2 λ M 1 + λ M 2 + I 3 − I 4 and λ M 1 + I 3 − 1 − λ C 4 − C 3 + C 2 ≥ 1 2 1 − σ 2 .

DW₄ when λ M 1 + λ M 2 + I 3 − I 4 < 0 , x ≤ 1 − σ 2 + 2 λ M 2 − 2 I 4 + 2 1 − λ C 4 + 2 C 3 − 2 C 2 2 λ M 1 + λ M 2 + I 3 − I 4 and − λ M 2 + I 4 − 1 − λ C 4 − C 3 + C 2 ≥ 1 2 1 − σ 2 .

Proof: For Eq. 12, when c 1 = c 2 = 1 , p = 1 , γ = 1 , V t , y = y 2 , L V t , y = 2 y 2 x λ M 1 + λ M 2 + I 3 − I 4 − λ M 2 + I 4 − 1 − λ C 4 − C 3 + C 2 + σ 2 y 2 .

(1) When the zero-solution moment exponent is stabilized, the equation is required to satisfy 2 y 2 x λ M 1 + λ M 2 + I 3 − I 4 − λ M 2 + I 4 − 1 − λ C 4 − C 3 + C 2 + σ 2 y 2 ≤ − y 2 . Since y ∈ 0 , 1 , 2 x λ M 1 + λ M 2 + I 3 − I 4 − λ M 2 + I 4 − 1 − λ C 4 − C 3 + C 2 + σ 2 ≤ − 1 .

In the case when λ M 1 + λ M 2 + I 3 − I 4 > 0 , x ≤ 2 λ M 2 − 2 I 4 + 2 1 − λ C 4 + 2 C 3 − 2 C 2 − 1 + σ 2 2 λ M 1 + λ M 2 + I 3 − I 4 , since x ∈ 0 , 1 , 2 λ M 2 − 2 I 4 + 2 1 − λ C 4 + 2 C 3 − 2 C 2 − 1 + σ 2 2 λ M 1 + λ M 2 + I 3 − I 4   ≥ 0 , and λ M 2 − I 4 + 1 − λ C 4 + C 3 − C 2 ≥ 1 2 1 + σ 2 is solved.

In the case when λ M 1 + λ M 2 + I 3 − I 4 < 0 , x ≥ 2 λ M 2 − 2 I 4 + 2 1 − λ C 4 + 2 C 3 − 2 C 2 − 1 + σ 2 2 λ M 1 + λ M 2 + I 3 − I 4 , since x ∈ 0 , 1 , 2 λ M 2 − 2 I 4 + 2 1 − λ C 4 + 2 C 3 − 2 C 2 − 1 + σ 2 2 λ M 1 + λ M 2 + I 3 − I 4   ≤ 1 , and 1 − λ C 4 + C 3 − C 2 − λ M 1 − I 3 ≥ 1 2 1 + σ 2 is solved.

Based on this, one can obtain Eq. 12 The exponential stability condition for the zero solution moment is:

When λ M 1 + λ M 2 + I 3 − I 4 > 0 , x ≤ 2 λ M 2 − 2 I 4 + 2 1 − λ C 4 + 2 C 3 − 2 C 2 − 1 + σ 2 2 λ M 1 + λ M 2 + I 3 − I 4 and λ M 2 − I 4 + 1 − λ C 4 + C 3 − C 2 ≥ 1 2 1 + σ 2 ; alternatively, when λ M 1 + λ M 2 + I 3 − I 4 < 0 , x ≥ 2 λ M 2 − 2 I 4 + 2 1 − λ C 4 + 2 C 3 − 2 C 2 − 1 + σ 2 2 λ M 1 + λ M 2 + I 3 − I 4 and 1 − λ C 4 + C 3 − C 2 − λ M 1 − I 3 ≥ 1 2 1 + σ 2 .

(2) When Eq. 12 The zero solution moment exponent is unstable, the equation needs to satisfy 2 y 2 x λ M 1 + λ M 2 + I 3 − I 4 − λ M 2 + I 4 − 1 − λ C 4 − C 3 + C 2 + σ 2 y 2 ≥ y 2 . Since y ∈ 0 , 1 , 2 x λ M 1 + λ M 2 + I 3 − I 4 − λ M 2 + I 4 − 1 − λ C 4 − C 3 + C 2 + σ 2 ≥ 1 .

In the case when λ M 1 + λ M 2 + I 3 − I 4 > 0 , x ≥ 1 − σ 2 + 2 λ M 2 − 2 I 4 + 2 1 − λ C 4 + 2 C 3 − 2 C 2 2 λ M 1 + λ M 2 + I 3 − I 4 , because x ∈ 0 , 1 , 1 − σ 2 + 2 λ M 2 − 2 I 4 + 2 1 − λ C 4 + 2 C 3 − 2 C 2 2 λ M 1 + λ M 2 + I 3 − I 4   ≤ 1 , and λ M 1 + I 3 − 1 − λ C 4 − C 3 + C 2 ≥ 1 2 1 − σ 2 is solved.

In the case when λ M 1 + λ M 2 + I 3 − I 4 < 0 , x ≤ 1 − σ 2 + 2 λ M 2 − 2 I 4 + 2 1 − λ C 4 + 2 C 3 − 2 C 2 2 λ M 1 + λ M 2 + I 3 − I 4 , because x ∈ 0 , 1 , 1 − σ 2 + 2 λ M 2 − 2 I 4 + 2 1 − λ C 4 + 2 C 3 − 2 C 2 2 λ M 1 + λ M 2 + I 3 − I 4   ≥ 0 , and − λ M 2 + I 4 − 1 − λ C 4 − C 3 + C 2 ≥ 1 2 1 − σ 2 is solved.

Based on this, one can obtain Eq. 12 The exponential instability condition for the zero solution moment is:

When λ M 1 + λ M 2 + I 3 − I 4 > 0 , x ≥ 1 − σ 2 + 2 λ M 2 − 2 I 4 + 2 1 − λ C 4 + 2 C 3 − 2 C 2 2 λ M 1 + λ M 2 + I 3 − I 4 and λ M 1 + I 3 − 1 − λ C 4 − C 3 + C 2 ≥ 1 2 1 − σ 2 ; alternatively, when λ M 1 + λ M 2 + I 3 − I 4 < 0 , x ≤ 1 − σ 2 + 2 λ M 2 − 2 I 4 + 2 1 − λ C 4 + 2 C 3 − 2 C 2 2 λ M 1 + λ M 2 + I 3 − I 4 and − λ M 2 + I 4 − 1 − λ C 4 − C 3 + C 2 ≥ 1 2 1 − σ 2 .

Proposition 3 suggests that netizens’ stabilizing strategies are influenced by their own factors as well as by random factors. When D W 1 ∪ D W 2 is satisfied, netizens will eventually choose the strategy of nonadoption; i.e., over time, regardless of the initial state, after continuously adjusting their own decision-making behavior, they will eventually reach a stable state under the strategy of nonadoption. When D W 3 ∪ D W 4 is met, netizens will eventually choose the strategy of adoption; i.e., over time, regardless of the initial state, netizens will eventually reach a stable state under the strategy of adoption after continuously adjusting their decision-making behavior.

From the condition and the condition of Proposition 3, it is clear that netizens are more inclined to choose a strategy that is in their own interest. That is, when the expected benefit of not adopting is greater than the expected benefit of adopting, netizens will choose to not adopt; in contrast, they will adopt the strategy of adopting. Because netizens’ decision-making behavior is influenced by the degree to which they pay attention to major emergencies and by their psychological recognition of opinion leaders, which strategy to adopt is determined by the expected benefit of the strategy. Moreover, random factors from internal or external sources may also have an impact on netizens’ final decision-making behavior. For example, at the early stage of major emergencies, the investigation of major emergencies by relevant departments has just begun, and specific real information has not yet been released in time; in these cases, netizens can only pay attention to major emergencies through the remarks released by opinion leaders. In this case, netizens can only pay attention to the comments of opinion leaders. Or for other reasons, the attention of netizens may be drawn to other events and they may give up paying attention to major emergencies. All of these random factors may affect the decision-making behavior of netizens, which may lead to fluctuations or shifts in the final stabilization strategy.

Based on Proposition 3 in Proposition 4, the final evolutionary stability strategies of the random evolutionary game between opinion leaders and netizens can be described by the following four scenarios:

(1) When D L 1 ∩ D W 1 ∪ D W 2 is satisfied, there exists a unique evolutionary stable strategy, ESS(0,0), for the system; i.e., the opinion leaders and netizens will ultimately choose the strategy of negative guidance or nonadoption over time. In this case, although the opinion leader chooses to negatively guide and does not provide netizens with true information related to major emergencies, netizens ultimately choose not to adopt the statements posted by the opinion leader. This prevents netizens from being swayed by the negative guidance released by opinion leaders, limits the spread of panic, and reduces the burden on the relevant authorities to maintain the smooth operation of society.

Among the above four scenarios, the strategy most conducive to the society stablity is (positive guidance, adoption). In this scenario, opinion leaders and netizens will cooperate together. Opinion leaders publish positive guidance statements, and netizens choose to believe the truthful information released by opinion leaders. Both parties work together to clarify uncertain information, which can greatly reduce the adverse effects caused by the dissemination of uncertain information. The strategy least conducive to the stable operation of society is (negative guidance, adoption). In this situation, opinion leaders, for some purpose, publish negative guidance statements on social platforms, inciting panic among netizens who believe in negative guidance statements, which will directly exacerbate the adverse effects caused by the dissemination of uncertain information.

Since Eqs 11, 12 is a nonlinear Itô stochastic differential equation, it is impossible to find its analytical solution directly, so it needs to be solved numerically; in the next part of this paper, we use the stochastic Taylor expansion and Itô’s formula to solve Eqs 11, 12 In the next part of this paper, the stochastic Taylor expansion and Itô’s formula are applied to numerically solve Eq.

For the following Itô stochastic differential equation: d x t = f t , x t d t + g t , x t d w t (19) where t ∈ t 0 , T , x t 0 = x 0 , x 0 ∈ R and w t are standard Brownian motions obeying a normal distribution N 0 , t and d w t obeys a normal distribution N 0 , Δ t .

Let the step size h = T − t 0 N , t n = t 0 + n h , where N is the number of samples, t₀ is the initial time point, t_n is the nth sample time point, and n ∈ 0 , 1 , 2 , … , N . The stochastic Taylor expansion of Eq. 19 the stochastic Taylor expansion of Eq: x t n + 1 = x t n + P 0 f x t n + P 1 g x t n + P 11 K 1 g x t n + P 00 K 0 f x t n + R (20) where R is the remainder term of the expansion, P 0 = h , P 1 = Δ w n , P 11 = 1 2 Δ w n 2 − h , P 00 = h 2 2 , K 1 = ∂ g x ∂ x , K 0 = ∂ f x ∂ x + ∂ 2 g x 2 ∂ x 2 .

Based on the Formula 20, the random Taylor expansions for Eqs 11, 12 are: x t n + 1 = x t n + h x t n − y I 2 + π L 2 − C 1 + Δ w n σ x t n + 1 2 Δ w n 2 − h σ 2 x t n + h 2 2 − y I 2 + π L 2 − C 1 2 x t n + R 1 (21) y t n + 1 = y t n + h y t n x λ M 1 + λ M 2 + I 3 − I 4 − λ M 2 + I 4 − 1 − λ C 4 − C 3 + C 2 + Δ w n σ y t n + 1 2 Δ w n 2 − h σ 2 y t n + h 2 2 x λ M 1 + λ M 2 + I 3 − I 4 − λ M 2 + I 4 − 1 − λ C 4 − C 3 + C 2 2 y t n + R 2 (22) where R₁ and R₂ are the remainder of the expansion of Eqs 21, 22.

In real-world applications, the model can be simulated numerically using Euler’s method or Milstein’s method, which involves partial term interception of the stochastic Taylor expansion and subsequent numerical solution. In this paper, the numerical solution is based on Milstein’s method, taking Eq. 20 For example, the intercept format of the Milstein method is shown in Eq. 23: x t n + 1 = x t n + h f x t n + Δ w n g x t n + 1 2 Δ w n 2 − h g ′ x t n g x t n (23)

To better analyze the influence of random interference factors on opinion leaders and netizens in making decision-making behavioral choices, i.e., to analyze the evolutionary stabilization process of the game system after the introduction of Gaussian white noise. The other parameters in the model are randomly assigned, and the evolutionary processes in the absence of random interference factors ( σ = 0 ) and the presence of random interference factors ( σ = 1 ) are compared and analyzed.

(1) Analysis of the stochastic evolutionary process of opinion leaders’ and netizens’ choices (negative guidance, nonadoption).

The relevant parameters of the model are assigned as follows: I ₁ = 10, I ₂ = 3, I ₃ = 2, I₄ = 4, L₁ = 8, L₂ = 10, C₁ = 9, C₂ = 6, C₃ = 10, C₄ = 10, M₁ = 3, M₂ = 3, λ = 0.35, and π = 0.2. The initial values of the opinion leaders’ choice of the positive steering strategy and the netizens’ choice of the adoption strategy are set to x (0) = 0.5 and y (0) = 0.5, respectively, and the simulation step size is h = 0.01. The random evolutionary trend of opinion leaders and netizens can be obtained as shown in Figure 2 below. 0.01. Under the premise of ensuring that the values of the remaining parameters remain unchanged, the stochastic evolution trend graphs of opinion leaders and netizens at σ = 0 and σ = 1 can be obtained, as shown in Figure 2 below.

The blue line in Figure 2 represents the probability curve of opinion leaders choosing the positive guidance strategy, and the red line represents the probability curve of netizens choosing the adoption strategy. Figure 2 shows that after a period of evolution, opinion leaders and netizens gradually converge to 0. That is, no matter what the initial values of opinion leaders choosing the positive guidance strategy and netizens choosing the adoption strategy are, opinion leaders and netizens will ultimately choose the strategy of negative guidance or nonadoption over time, and opinion leaders tend to converge to 0 at a slightly faster rate than netizens. This is because, after major emergencies, opinion leaders can obtain more information from multiple sources and are able to adjust more quickly after gaming behavior has begun. This is because after a major emergency occurs, opinion leaders can obtain more information from multiple sources and make faster adjustments after the game behavior begins. When the benefits of negative guidance strategies are discovered due to positive guidance, opinion leaders can quickly choose negative guidance strategies.

By comparing Figures 2A, B, it can be found that at σ = 0 , opinion leaders and netizens evolve to a stable strategy faster because they are not interfered with by random factors in the decision-making process and need to combine each other’s expected benefits to make a strategy choice. Therefore, in an idealized situation without random interference, the two parties are able to quickly make choices that are in their own interest.

(2) Analysis of the stochastic evolutionary process of opinion leaders’ and netizens’ choices (positive guidance, adoption).

The relevant parameters of the model are assigned as follows: I ₁ = 10, I ₂ = 3, I ₃ = 2, I₄ = 4, L₁ = 8, L₂ = 20, C₁ = 6, C₂ = 6, C₃ = 6, C₄ = 4, M₁ = 3, M₂ = 3, λ = 0.5, and π = 0.5. The initial values of the opinion leaders’ choice of the positive steering strategy and the netizens’ choice of the adoption strategy are set as x (0) = 0.5 and y (0) = 0.5, and the simulation step size is h = 0.01. The random evolutionary trends of opinion leaders and netizens can be obtained as shown in Figure 3. 0.01. Under the premise of ensuring that the values of the remaining parameters remain unchanged, the stochastic evolution trend graphs of opinion leaders and netizens at σ = 0 and σ = 1 can be obtained, as shown in Figure 3 below.

The blue line in Figure 3 represents the probability curve of opinion leaders choosing the positive guidance strategy, and the red line represents the probability curve of netizens choosing the adoption strategy. Figure 3 shows that after a period of evolution, opinion leaders and netizens gradually converge to 1. That is, no matter what the initial values are for opinion leaders to choose the positive guidance strategy and for netizens to choose the adoption strategy, opinion leaders and netizens will ultimately choose the strategy of positive guidance adoption over time, and the speed at which opinion leaders converge to 1 is slightly faster than that of netizens.

By comparing Figures 3A, B, it can be observed that the speed at which opinion leaders and netizens evolve to a stable strategy is relatively slow. This is because both are subject to interference from internal or external random factors during the decision-making process, resulting in oscillations during the evolution to a stable state. Therefore, in a nonidealized situation, opinion leaders and netizens will be partially limited by random factors in their decision-making, and they will slow their decision-making speed when thinking about the pros and cons of random factors.

To analyze the influence of netizens’ psychological identity toward opinion leaders on the stochastic evolution process, other parameters in the model are randomly assigned, and the evolution process is simulated. The influence of changes in psychological identity on the decision-making behavior of opinion leaders and netizens should be observed.

The relevant parameters of the model were assigned as I ₁ = 10, I ₂ = 3, I ₃ = 2, I ₄ = 4, L ₁ = 8, L ₂ = 20, C ₁ = 6, C ₂ = 6, C ₃ = 6, C ₄ = 4, M ₁ = 3, M ₂ = 3, π = 0.5, and σ = 1. Setting the initial value of the opinion leader’s choice of the positive steering strategy, the netizen’s choice of the adoption strategy is set to x (0) = 0.5, y (0) = 0.5, and the simulation step size h = 0.01. Under the premise of ensuring that the values of the remaining parameters remain unchanged, we change the value of λ and set λ = 0.1, 0.5, and 0.8. We can construct a stochastic evolution trend graph of opinion leaders and netizens, as shown in Figure 4 below.

The blue line in Figure 4A represents the probability curve of opinion leaders choosing the positive guidance strategy when the psychological identity λ = 0.1, the red line represents the probability curve when the psychological identity λ = 0.5, and the green line represents the probability curve when the psychological identity λ = 0.8. The three lines in Figure 4B are the probability curves of netizens choosing the adoption strategy. Figure 4A shows that the netizens’ psychological identity λ of the opinion leader does not affect the final strategy choice of the opinion leader; however, the smaller the value of psychological identity λ is, the faster the opinion leader equals 1. This is because, in reality, the less value a netizen places on the opinions of the opinion leader, the less likely it is that the opinion leader’s published speech will be adopted by the netizens. To increase their influence among netizens, opinion leaders need to make strategic choices as early as possible to attract netizens’ attention. The more content an opinion leader publishes, the more netizens will pay attention to it, and the more likely it is to increase netizens’ psychological identification with it.

Figure 4B shows that as netizens’ recognition of opinion leaders gradually increases, they gradually convert from converging to 0 to converging to 1; that is, netizens convert from not adopting the remarks released by opinion leaders to adopting the remarks released by opinion leaders. This is because the more netizens trust opinion leaders, the easier it is for them to believe their statements. However, the psychological recognition of opinion leaders by netizens is a double-edged sword. Due to their limited understanding, it is difficult to accurately distinguish whether the statements made by opinion leaders are true. This leads to the fact that when opinion leaders provide positive guidance, they can largely maintain the stable operation of society, but when they provide negative guidance, it is easy to exacerbate the spread of panic and greatly increase the difficulty of government work.

To analyze the influence of the government’s punishment strength on those who propagate and spread negative information on the stochastic evolution process, the other parameters in the model are randomly assigned values, and the evolution process is simulated. The influence of changes in punishment strength on the decision-making behavior of opinion leaders and netizens should be observed.

Since both π and L ₂ in the model can affect the punishment strength, for the convenience of analysis, only the government’s punishment probability π for those who propagate and spread negative information is selected for analysis. The relevant parameters of the model are assigned as follows: I ₁ = 10, I ₂ = 3, I ₃ = 2, I ₄ = 4, L ₁ = 8, L ₂ = 20, C ₁ = 6, C ₂ = 6, C ₃ = 6, C ₄ = 4, M ₁ = 3, M ₂ = 3, λ = 0.35, and σ = 1. The initial values of the opinion leaders’ choice of the positive guidance strategy and the netizens’ choice of the adoption strategy are set as x (0) = 0.5 and y (0) = 0.5, respectively, and the simulation step size is set as h = 0.01. Under the premise of ensuring that the values of the remaining parameters remain unchanged, changing the value of π and setting π = 0.2, 0.5, and 0.8, we can obtain the stochastic evolution trend graph of opinion leaders and netizens, which is shown in Figure 5 below.

The blue line in Figure 5A represents the probability curve of opinion leaders choosing the positive guidance strategy when the penalty probability is π = 0.2, the red line represents the probability curve when the penalty probability is π = 0.5, and the green line represents the probability curve when the penalty probability is π = 0.8. The three lines in Figure 5B are the probability curves of netizens choosing the adoption strategy. From Figure 5A, it can be seen that when the government’s punishment for propagandizing and spreading negative information gradually increases, the opinion leader gradually changes from 0 to 1, i.e., the opinion leader converts from negative guidance to positive guidance. This is because when the government’s punishment gradually exceeds the range that opinion leaders can bear, to avoid being punished, such as through banning, blocking or administrative punishment, opinion leaders will switch from negative guidance to positive guidance. Therefore, when there are major emergencies, the government should increase punishment for those who publicize and disseminate negative information to encourage opinion leaders to choose a positive guidance strategy.

Figure 5B shows that the government’s punishment strength for those who publicize and spread negative information does not affect netizens’ final strategy choice; however, the stronger the punishment is, the faster the netizens tend to 1. This is because, in netizens’ thinking, the more the government punishes opinion leaders for spreading negative statements, the more correct the statements spread by the possible opinion leaders will be; therefore, the netizens will adopt the strategy more quickly.

This study has certain limitations: 1) this paper assumes that the ability of opinion leaders to disseminate statements is the same and does not distinguish the ability of opinion leaders to disseminate statements, but in reality, the statements released by different opinion leaders are all different; 2) the vague relationship between opinion leaders and netizens has not been clearly distinguished; In the real world, opinion leaders can be natural persons or institutional accounts; 3)This paper uses MATLAB for numerical simulation of the model without combining real-world data.

In future work, we will conduct separate research on opinion leaders with different dissemination capabilities based on different network structures [33, 34] by combining complex networks with stochastic evolutionary game models. Furthermore, we will refine the differences between opinion leaders and netizens, incorporate more detailed decision-making behaviors, and include more accurate assumptions in the construction of the model. Additionally, we will try to combine real-world data and use real cases to simulate the model.