Rumor Transmission in Online Social Networks Under Nash Equilibrium of a Psychological Decision Game

Abstract

This paper investigates rumor transmission over online social networks, such as those via Facebook or Twitter, where users liberally generate visible content to their followers, and the attractiveness of rumors varies over time and gives rise to opposition such as counter-rumors. All users in social media platforms are modeled as nodes in one of five compartments of a directed random graph: susceptible, hesitating, infected, mitigated, and recovered (SHIMR). The system is expressed with edge-based formulation and the transition dynamics are derived as a system of ordinary differential equations. We further allow individuals to decide whether to share, or disregard, or debunk the rumor so as to balance the potential gain and loss. This decision process is formulated as a game, and the condition to achieve mixed Nash equilibrium is derived. The system dynamics under equilibrium are solved and verified based on simulation results. A series of parametric analyses are conducted to investigate the factors that affect the transmission process. Insights are drawn from these results to help social media platforms design proper control strategies that can enhance the robustness of the online community against rumors.

Appendix A. Derivation of \(R_e\)

We follow the idea mentioned in Diekmann et al. (1990) to derive $R_e(t)$ which the definition can be found in Sect. 3.2. The computation of $R_e(t)$ requires information only on compartments $H_{jk}(t)$, $I_{jk}(t)$ and $M_{jk}(t)$. For simplicity, we hereon omit the time argument for all variables in this appendix.

Based on Eq. (18), for all $j \in \{1, ...,J\}, k \in \{1, ...,K\}$, we have

$$\begin{aligned} \begin{aligned}&\dot{H}_{jk} = j(r_I\frac{\theta _I}{\theta } + r_M\frac{\theta _M}{\theta })S_{jk} - \beta H_{jk}, \\&\dot{I}_{jk} = \beta \gamma _{jk} q H_{jk} - \mu _{I}I_{jk} , \\&\dot{M}_{jk} = \beta \gamma _{jk} (1 - q) H_{jk} - \mu _{M}M_{jk}. \end{aligned} \end{aligned}$$

This can be written in the following vector form

$$\begin{aligned}{}[ \dot{H}_{11}, \cdots , \dot{H}_{JK}, \dot{I}_{11}, \cdots , \dot{I}_{JK}, \dot{M}_{11}, \cdots , \dot{M}_{JK}]^T = \mathcal {F} - \mathcal {V}, \end{aligned}$$

(28)

where $\mathcal {F} = [1 \cdot (r_I \frac{\theta _I}{\theta } + r_M \frac{\theta _M}{\theta })S_{11}, \cdots , J(r_I\frac{\theta _I}{\theta } + r_M\frac{\theta _M}{\theta })S_{JK}, \mathbf {0}_{1 \times JK}, \mathbf {0}_{1 \times JK}]^T$ represents the increment of hesitating users from all other compartments, while $\begin{aligned}&\mathcal {V} = [\beta H_{11}, \cdots , \beta H_{JK}, \mu _{I}I_{11}-\beta \gamma _{11} q H_{11}, \cdots , \mu _{I}I_{JK}-\beta \gamma _{JK}qH_{JK}, \mu _{M}M_{11}-\beta \gamma _{11}(1 - q)H_{11}, \cdots , \mu _{M}M_{JK}-\beta \gamma _{JK}(1 - q)H_{JK}]^T\end{aligned}$ represents the transmission of hesitating users into other compartments. Here $\mathbf {0}$ represents a zero vector/matrix. The Jacobian matrices of $\mathcal {F}$ and $\mathcal {V}$, denoted as F and V, can be computed respectively as follows:

$$\begin{aligned} {F} = \begin{bmatrix} \varvec{0}_{JK \times JK} &{} B \\ \varvec{0}_{2JK \times JK} &{} \varvec{0}_{2JK \times 2JK} \end{bmatrix}, \ {V} = \begin{bmatrix} D &{} \varvec{0}_{JK \times 2JK} \\ P &{} Q \end{bmatrix}, \end{aligned}$$

(29)

where

$$\begin{aligned} \begin{aligned}&B = \\&{\left[ \begin{array}{ccccc} 1 \cdot \frac{1 \cdot r_I}{\theta \langle k \rangle }S_{11} &{} \cdots &{} 1 \cdot \frac{1 \cdot r_M}{\theta \langle k \rangle }S_{11} &{} \cdots &{} 1 \cdot \frac{K \cdot r_M}{\theta \langle k \rangle }S_{11}\\ \vdots &{} \ddots &{} \vdots &{} \ddots &{} \vdots \\ 1 \cdot \frac{1 \cdot r_I}{\theta \langle k \rangle }S_{1K} &{} \cdots &{} 1 \cdot \frac{1 \cdot r_M}{\theta \langle k \rangle }S_{1K} &{} \cdots &{} 1 \cdot \frac{K \cdot r_M}{\theta \langle k \rangle }S_{1K} \\ \vdots &{} \ddots &{} \vdots &{} \ddots &{} \vdots \\ j \cdot \frac{1 \cdot r_I}{\theta \langle k \rangle }S_{JK} &{} \cdots &{} j \cdot \frac{1 \cdot r_M}{\theta \langle k \rangle }S_{JK} &{} \cdots &{} j \cdot \frac{K \cdot r_M}{\theta \langle k \rangle }S_{JK} \end{array} \right] } , \end{aligned} \end{aligned}$$

$$\begin{aligned} D = \beta \cdot E_{J \times K} , Q = {\left[ \begin{array}{cc} \mu _I \cdot E_{J \times K} &{} \mathbf {0}_{JK \times JK} \\ \mathbf {0}_{JK \times JK} &{} \mu _M \cdot E_{J \times K} \end{array} \right] } , \end{aligned}$$

$$\begin{aligned} P = {\left[ \begin{array}{ccc} -\beta \gamma _{11}q &{} \cdots &{} 0 \\ 0 &{} \cdots &{} 0 \\ \vdots &{} \ddots &{} \vdots \\ 0 &{} \cdots &{} -\beta \gamma _{JK}q \\ -\beta \gamma _{11}(1-q) &{} \cdots &{} 0 \\ 0 &{} \cdots &{} 0 \\ \vdots &{} \ddots &{} \vdots \\ 0 &{} \cdots &{} -\beta \gamma _{JK}(1-q) \end{array} \right] } . \end{aligned}$$

In the above, E represents the identity matrix. Following Diekmann et al. (1990), we call $FV^{-1}$ the next generation matrix for the model, and it can be calculated as:

$$\begin{aligned} FV^{-1} = {\left[ \begin{array}{cc} -BQ^{-1}PD^{-1} &{} BQ^{-1} \\ \mathbf {0}_{2JK \times JK} &{} \mathbf {0}_{2JK \times 2JK} \end{array} \right] } . \end{aligned}$$

(30)

It is well known from Lyapunov stability theorem that the effective reproduction number $R_e$ is the spectral radius of matrix $FV^{-1}$; i.e.:

$$\begin{aligned} R_e = \rho (FV^{-1}). \end{aligned}$$

(31)

From Eq. (29), $FV^{-1}$ is an upper triangular block matrix, whose eigenvalues are given by those of its only nonzero diagonal block submatrix $-BQ^{-1}PD^{-1}$. Note that

$$\begin{aligned} -BQ^{-1}PD^{-1} = \frac{1}{\langle k \rangle }{\left[ \begin{array}{ccc}1 \cdot p_{11}\theta ^0 \\ \vdots \\ 1 \cdot p_{1K}\theta ^0 \\ 2 \cdot p_{21}\theta ^1 \\ \vdots \\ J \cdot p_{JK}\theta ^{J-1} \end{array} \right] } \cdot \ {\left[ \begin{array}{ccc} 1 \cdot \gamma _{11}(\frac{r_Iq}{\mu _{I}} + \frac{r_M(1 - q)}{\mu _{M}}) \\ \vdots \\ K \cdot \gamma _{1K}(\frac{r_Iq}{\mu _{I}} + \frac{r_M(1 - q)}{\mu _{M}}) \\ 1 \cdot \gamma _{21}(\frac{r_Iq}{\mu _{I}} + \frac{r_M(1 - q)}{\mu _{M}}) \\ \vdots \\ K \cdot \gamma _{JK}(\frac{r_Iq}{\mu _{I}} + \frac{r_M(1 - q)}{\mu _{M}}) \end{array} \right] }^T . \end{aligned}$$

(32)

Clearly, rank ($-BQ^{-1}PD^{-1}$) = 1. Hence,

$$\begin{aligned} R_e = {\mathrm {tr}(-BQ^{-1}PD^{-1})} = \frac{1}{\langle k \rangle } \sum _{j,\ k} jkp_{jk} \theta ^{j - 1} \gamma _{jk} \cdot (\frac{r_I q}{\mu _{I}} + \frac{r_M (1 - q)}{\mu _{M}}). \end{aligned}$$

(33)