## 4 Slp-macgt Model Design
### Game Process between Sensor Nodes and Attackers
#### 4.5.3 Replication Dynamic Equation of Tripartite Evolutionary Game  
A key concept in game theory is equilibrium analysis. By studying the Nash equilibrium point of the game, insight can be gained into the interactions between the source node and the attacker and possible results. At the equilibrium point, no party can achieve better results by changing its strategy, which helps balance the trade-off between privacy protection and information access. Therefore, the payoff of the source node choosing a different strategy is  
\[\left\{\begin{array}{l}G(X_{1})=yz(R_{X1}-R_{Y1}-R_{Z1}-C_{X1})+(1-y)z(R_{X1} -R_{Y2}-R_{Z1}-C_{X1})\\ +y(1-z)(R_{X1}-R_{Y1}-R_{Z2}-C_{X1})+(1-y)(1-z)(R_{X1}-R_{Y2}-R_{Z2}-C_{X1})\\ G(X_{2})=yz(R_{X2}-R_{Y1}-R_{Z1}-C_{X2})+(1-y)z(R_{X2}-R_{Y2}-R_{Z1}-C_{X2})\\ +y(1-z)(R_{X2}-R_{Y1}-R_{Z2}-C_{X2})+(1-y)(1-z)(R_{X2}-R_{Y2}-R_{Z2}-C_{X2})\\ \end{array}\right., \tag{3}\]  
The average revenue of the source node is as follows: \(\overline{G(X)}=xG(X_{1})+(1-x)G(X_{2})=\sum\limits_{i}P_{x}G(X_{i})\). The replication dynamic equation of the source node is  
\[F(x)=\frac{dx(t)}{dt}=x\Big{(}G(X_{i})-\overline{G(X)}\Big{)}. \tag{4}\]  
The payoff for attacker A in choosing a different strategy is  
\[\left\{\begin{array}{l}G(Y_{1})=xz(R_{Y1}-R_{X1}-C_{Y1})+x(1-z)(R_{Y1}-R_{X1 }-C_{Y1})\\ +(1-x)z(R_{Y1}-R_{X2}-C_{Y1})+(1-x)(1-z)(R_{Y1}-R_{X2}-C_{Y1})\\ G(Y_{2})=xz(R_{Y2}-R_{X1}-C_{Y1})+x(1-z)(R_{Y2}-R_{X1}-C_{Y1})\\ +(1-x)z(R_{Y2}-R_{X2}-C_{Y1})+(1-x)(1-z)(R_{Y2}-R_{X2}-C_{Y1})\\ \end{array}\right., \tag{5}\]  
The average revenue of attacker A is as follows: \(\overline{G(Y)}=yG(Y_{1})+(1-y)G(Y_{2})=\sum\limits_{i}P_{y}G(Y_{i})\). The replication dynamic equation of attacker A is as follows:  
\[F(y)=\frac{dy(t)}{dt}=y\Big{(}G(Y_{i})-\overline{G(Y)}\Big{)}. \tag{6}\]  
The payoff for attacker B in choosing a different strategy is  
\[\left\{\begin{array}{l}G(Z_{1})=xy(R_{Z1}-R_{X1}-C_{Z1})+x(1-y)(R_{Z1}-R_{X1 }-C_{Z1})\\ +(1-x)y(R_{Z1}-R_{X2}-C_{Z1})+(1-x)(1-y)(R_{Z1}-R_{X2}-C_{Z1})\\ G(Z_{2})=xy(R_{Z2}-R_{X1}-C_{Z2})+x(1-y)(R_{Z2}-R_{X1}-C_{Z2})\\ +(1-x)y(R_{Z2}-R_{X2}-C_{Z2})+(1-x)(1-y)(R_{Z2}-R_{X2}-C_{Z2})\\ \end{array}\right., \tag{7}\]  
The average revenue of attacker B is as follows: \(\overline{G(Z)}=zG(Z_{1})+(1-z)G(Z_{2})=\sum\limits_{i}P_{z}G(Z_{i})\). The replication dynamic equation of attacker B is as follows:  
\[F(z)=\frac{dz(t)}{dt}=z\Big{(}G(Z_{i})-\overline{G(Z)}\Big{)}. \tag{8}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline Strategy combination & \(\Big{(}X_{i},Y_{i},Z_{k}\Big{)}\) \\ \hline Payoff of source node gain & \(R_{Xi}-R_{Yj}-R_{Zk}-C_{Xi}\) \\ Payoff of attacker A gain & \(R_{Yj}-R_{Xi}-C_{Yj}\) \\ Payoff of attacker B gain & \(R_{Zk}-R_{Xi}-C_{Zk}\) \\ \hline Where \(i,j,k=1\)_or_\(2\). & \\ \end{tabular}
\end{table}
Table 2: Game gain matrix.  
Combining the above Equations (4), (6) and (8) into Equation (9), the replication dynamic equation system is established as follows:  
\[Y=\begin{bmatrix}F(x)\\ F(y)\\ F(z)\end{bmatrix}=f(Y,t)=0. \tag{9}\]  
Solving the equation yields: \(Y_{1}=\begin{bmatrix}0\\ 0\\ 0\end{bmatrix}\), \(Y_{2}=\begin{bmatrix}1\\ 0\\ 0\end{bmatrix}\), \(Y_{3}=\begin{bmatrix}0\\ 1\\ 0\end{bmatrix}\), \(Y_{4}=\begin{bmatrix}0\\ 0\\ 1\end{bmatrix}\), \(Y_{5}=\begin{bmatrix}1\\ 1\\ 0\end{bmatrix}\), \(Y_{6}=\begin{bmatrix}1\\ 0\\ 1\end{bmatrix}\), \(Y_{7}=\begin{bmatrix}0\\ 1\\ 1\end{bmatrix}\), and \(Y_{8}=\begin{bmatrix}1\\ 1\\ 1\end{bmatrix}\).