Spreading dynamics of a 2SIH2R, rumor spreading model in the homogeneous network (2101.10731v1)

Abstract: In the era of the rapid development of the Internet, the threshold for information spreading has become lower. Most of the time, rumors, as a special kind of information, are harmful to society. And once the rumor appears, the truth will follow. Considering that the rumor and truth compete with each other like light and darkness in reality, in this paper, we study a rumor spreading model in the homogeneous network called 2SIH2R, in which there are both spreader1(people who spread the rumor) and spreader2(people who spread the truth). In this model, we introduced discernible mechanism and confrontation mechanism to quantify the level of people's cognitive abilities and the competition between the rumor and truth. By mean-field equations, steady-state analysis and numerical simulations in a generated network which is closed and homogeneous, some significant results can be given: the higher discernible rate of the rumor, the smaller influence of the rumor; the stronger confrontation degree of the rumor, the smaller influence of the rumor; the large average degree of the network, the greater influence of the rumor but the shorter duration. The model and simulation results provide a quantitative reference for revealing and controlling the spread of the rumor.

This paper introduces the 2SIH2R model for simulating the spread of rumors and competing truths within a homogeneous social network (Wang et al., 2021 ). It addresses limitations in prior models by incorporating mechanisms for individual cognitive ability (discernibility) and the direct competition between rumor and truth (confrontation).

The 2SIH2R Model

The model divides the population into six states:

1. I (Ignorant): Unaware of the rumor or the truth.
2. S1 (Spreader1): Spreading the rumor.
3. S2 (Spreader2): Spreading the truth.
4. H (Hesitant1): Heard the rumor but is undecided and does not spread it immediately.
5. R1 (Stifler1): Heard the rumor but stopped spreading it (or never spread it).
6. R2 (Stifler2): Heard the truth and stopped spreading it (or never spread it).

Key features and mechanisms include:

• Discernible Mechanism: When an Ignorant (I) encounters a rumor spreader (S1), they might:
  • Become a rumor spreader (S1) with probability (1-f(m))λ1.
  • Become hesitant (H) with probability f(m)η.
  • Remain ignorant or become a stifler (R1) otherwise.
  • Here, m represents the inherent discernibility of the rumor, and f(m) is a function mapping this to the population's ability to recognize it (assumed positively correlated). λ1 is the rumor spreading rate, and η is the potential spreading rate (rate of becoming hesitant). Higher discernibility (f(m)) reduces direct transmission to S1 and increases hesitation (H).
• Confrontation Mechanism: When a rumor spreader (S1) encounters a truth spreader (S2), the rumor spreader (S1) converts to a truth stifler (R2) with probability α (confrontation rate), reflecting the truth overcoming the rumor upon interaction.
• Hesitation Dynamics: Hesitant individuals (H) can spontaneously become rumor spreaders (S1) with rate θ1 or truth spreaders (S2) with rate θ2.
• Stifling Mechanisms:
  • Spreaders (S1, S2) can become stiflers (R1, R2) upon meeting others of the same spreader type (rate β1, β2) or spontaneously (forgetting rate γ1, γ2).
  • Rumor stiflers (R1) can spontaneously become truth stiflers (R2) with rate ω, representing eventual realization or acceptance of the truth.

Mathematical Formulation and Analysis

The model dynamics are described using mean-field equations, assuming a homogeneous network where each node has an average degree <k>. This results in a system of ordinary differential equations (ODEs) governing the density of each population state over time:

dI/dt = -<k>( (1-f(m))λ1 + f(m)η + λ2 ) S1(t)I(t) - <k>λ2 S2(t)I(t) // Simplified from paper's Eq. 1, combining transitions dS1/dt = (1-f(m))λ1<k>S1(t)I(t) + θ1H(t) - α<k>S1(t)S2(t) - β1<k>S1(t)(S1(t)+R1(t)+H(t)) - γ1S1(t) dS2/dt = λ2<k>S2(t)I(t) + θ2H(t) - β2<k>S2(t)(S2(t)+R2(t)) - γ2S2(t) dH/dt = f(m)η<k>S1(t)I(t) - θ1H(t) - θ2H(t) dR1/dt = (1 - (1-f(m))λ1 - f(m)η)<k>S1(t)I(t) + β1<k>S1(t)(S1(t)+R1(t)+H(t)) + γ1S1(t) - ωR1(t) dR2/dt = (1-λ2)<k>S2(t)I(t) + α<k>S1(t)S2(t) + β2<k>S2(t)(S2(t)+R2(t)) + γ2S2(t) + ωR1(t)

(Note: The equations presented here combine some terms for clarity compared to the paper's explicit formulation, but represent the same dynamics).

The paper derives conditions (spreading thresholds) under which rumors and/or truths can spread widely. A key finding is the general condition for spread when both rumor and truth start spreading simultaneously (assuming large N): (1 - f(m))λ1 + λ2 > (α + β1 + β2 + γ1 + γ2) / <k> - f(m)η If this condition isn't met, neither the rumor nor the truth will achieve significant penetration in the network. Simplified thresholds are also derived for scenarios where only a rumor or only the truth initially exists.

Simulation Results and Practical Implications

Numerical simulations on a generated homogeneous network (N=10000, <k>=8) illustrate the model's behavior:

• Higher Discernibility (m): Reduces the peak prevalence and final size of the rumor (less S1 and R1), while increasing the final size of truth stiflers (R2).
  • Application: Strategies that increase the public's ability to discern fake news (e.g., media literacy campaigns, fact-checking labels) can significantly curb rumor impact.
• Higher Confrontation (α): Reduces the peak and final size of the rumor (less S1 and R1) and increases the final size of truth stiflers (R2).
  • Application: Actively disseminating verified truth to counter specific rumors is an effective intervention strategy. The stronger the "confrontation" (persuasiveness/reach of truth vs. rumor), the better the outcome.
• Higher Average Degree (<k>): Increases the speed and peak prevalence of the rumor but shortens the overall duration of the spreading event.
  • Application: In highly connected online platforms (high <k>), rumors can spread extremely quickly and widely, requiring rapid response mechanisms. However, the faster burnout might also be observed.
• R1 to R2 Transition (ω): Even a small, non-zero rate ω dramatically shifts the final state from rumor stiflers (R1) to truth stiflers (R2).
  • Application: Suggests that even if people initially fall for or become indifferent to a rumor, mechanisms allowing for later correction or acceptance of truth are highly impactful in the long run.

Implementation Considerations

• Computational Approach: The mean-field ODE formulation makes simulation computationally efficient, suitable for large populations. Standard ODE solvers can be used.

  import numpy as np from scipy.integrate import odeint import matplotlib.pyplot as plt # Define the model parameters (example values) N = 10000 k_avg = 8.0 m = 0.3 f_m = 0.7 * m # Example function f(m) = 0.7m lambda1 = 0.7 lambda2 = 0.7 eta = 0.8 theta1 = 0.5 theta2 = 0.3 beta1 = 0.3 beta2 = 0.3 gamma1 = 0.1 gamma2 = 0.1 alpha = 0.5 omega = 0.0 # Base case: No R1->R2 transition # Define the system of ODEs def dSdt(y, t, k_avg, f_m, lambda1, lambda2, eta, theta1, theta2, beta1, beta2, gamma1, gamma2, alpha, omega): I, S1, S2, H, R1, R2 = y dIdt = -k_avg * S1 * I * ((1 - f_m) * lambda1 + f_m * eta) - k_avg * S2 * I * lambda2 dS1dt = (1 - f_m) * lambda1 * k_avg * S1 * I + theta1 * H - alpha * k_avg * S1 * S2 - beta1 * k_avg * S1 * (S1 + R1 + H) - gamma1 * S1 dS2dt = lambda2 * k_avg * S2 * I + theta2 * H - beta2 * k_avg * S2 * (S2 + R2) - gamma2 * S2 dHdt = f_m * eta * k_avg * S1 * I - (theta1 + theta2) * H # Calculate dR1/dt based on the paper's logic (Eq 5) # Transition from Ignorant: (1 - (1-f(m))λ1 - f(m)η) <k> S1 I dR1dt_from_I = (1 - (1-f_m)*lambda1 - f_m*eta) * k_avg * S1 * I dR1dt = dR1dt_from_I + beta1 * k_avg * S1 * (S1 + R1 + H) + gamma1 * S1 - omega * R1 # Calculate dR2/dt based on the paper's logic (Eq 6) # Transition from Ignorant: (1-λ2)<k>S2 I dR2dt_from_I = (1-lambda2) * k_avg * S2 * I dR2dt = dR2dt_from_I + alpha * k_avg * S1 * S2 + beta2 * k_avg * S2 * (S2 + R2) + gamma2 * S2 + omega * R1 # Ensure conservation (optional check) # if abs(dIdt + dS1dt + dS2dt + dHdt + dR1dt + dR2dt) > 1e-9: # print("Warning: Sum of derivatives is not zero") return [dIdt, dS1dt, dS2dt, dHdt, dR1dt, dR2dt] # Initial conditions: 1 S1, 1 S2, rest I S1_0 = 1.0 / N S2_0 = 1.0 / N I_0 = 1.0 - S1_0 - S2_0 H_0 = 0.0 R1_0 = 0.0 R2_0 = 0.0 y0 = [I_0, S1_0, S2_0, H_0, R1_0, R2_0] # Time points t = np.linspace(0, 50, 500) # Solve ODEs sol = odeint(dSdt, y0, t, args=(k_avg, f_m, lambda1, lambda2, eta, theta1, theta2, beta1, beta2, gamma1, gamma2, alpha, omega)) # Plot results (similar to Fig 2 in paper) plt.figure(figsize=(10, 6)) plt.plot(t, sol[:, 1], label='Spreader1 (S1)') plt.plot(t, sol[:, 2], label='Spreader2 (S2)') plt.plot(t, sol[:, 4], label='Stifler1 (R1)') plt.plot(t, sol[:, 5], label='Stifler2 (R2)') plt.plot(t, sol[:, 0], label='Ignorant (I)') plt.plot(t, sol[:, 3], label='Hesitant (H)') plt.xlabel("Time (T)") plt.ylabel("Densities") plt.title("2SIH2R Model Simulation") plt.legend() plt.grid(True) plt.show() print(f"Final R1 density: {sol[-1, 4]:.4f}") print(f"Final R2 density: {sol[-1, 5]:.4f}")

• Parameter Estimation: A major practical challenge is estimating the model parameters (λ1, λ2, η, θ1, θ2, β1, β2, γ1, γ2, α, ω, and the function f(m)) for a specific real-world scenario. This often requires fitting the model to empirical data on information spread, surveys on belief changes, or using expert judgment.
• Network Homogeneity: The model assumes a homogeneous network (uniform average degree). Real social networks are often heterogeneous (e.g., scale-free). Applying this model directly to highly heterogeneous networks might yield inaccurate results. Future work mentioned in the paper aims to address this by extending the model to heterogeneous networks.
• Closed Population: The model assumes a closed population (N is constant). This is suitable for analyzing spread within a specific community but doesn't account for users joining or leaving.

In summary, the 2SIH2R model provides a more nuanced framework for studying rumor and truth co-evolution by incorporating cognitive factors (discernibility) and direct competition (confrontation). Its practical value lies in simulating the potential impact of interventions like media literacy campaigns (f(m)) or active truth dissemination (α, λ2). While parameter estimation and the homogeneity assumption present challenges for direct real-world application, the model offers valuable qualitative insights and a basis for more complex simulations.