## 4 Robustness Analysis of the ADN against the CCF
### Robustness Analysis
#### 4.1.2 Unintentional Random Initial Failure Rate and Robustness of the ADN  
From the Equation (5), Equation (7) can be further obtained.  
\[\gamma\ =\ 1-\left[1-(1-\theta_{P})\times\left[1-(1-\gamma\times f_{I}(\gamma))^{ \langle K_{2}\rangle}\right]\times f_{P}((1-\theta_{P})\times\left[1-(1-\gamma \times f_{I}(\gamma))^{\langle K_{2}\rangle}\right]\right)^{\langle\Omega_{1}\rangle} \tag{7}\]  
The solution of Equation (7) can be obtained by the graphic method of two curves as follows in Equation (8).  
\[\left\{\begin{array}{l}\lambda\ =\ \gamma\\ \lambda\ =\ 1-\left[1-(1-\theta_{P})\times\left[1-(1-\gamma\times f_{I}(\gamma))^{ \langle K_{2}\rangle}\right]\times f_{P}((1-\theta_{P})\times\left[1-(1-\gamma \times f_{I}(\gamma))^{\langle K_{2}\rangle}\right]\right)\right]^{\langle \Omega_{1}\rangle}\end{array}\right. \tag{8}\]  
By solving Equation (8), we can get that it has a trivial solution \(\gamma=0\), \(\lambda=0\). That is, an intersection point of the two curves is the original point. The meaning of this trivial solution is that for any \(\Theta_{P}\), the expected proportions \(R_{IF}(\infty)\) and \(R_{PF}(\infty)\) of normal operational nodes in the PN and CN are zero. There is another solution which corresponds to the critical value \(\Theta_{P\_critical}\) of \(\Theta_{P}\). When the two curves in Equation (8) are tangent, that is \(\partial\lambda/\partial\gamma=1\). We can get the equation set for solving \(\Theta_{P\_critical}\) as follows in Equation (9).  
\[\left\{\begin{array}{l}1\ =\ [1-\lambda\times f_{P}(\lambda)]^{\langle \Omega_{1}\rangle-1}\times\left[\frac{\partial\lambda}{\partial\gamma}\times \left(f_{P}(\lambda)+\lambda\times\frac{\partial f_{P}(\lambda)}{\partial \gamma}\right)\right]\\ \gamma\ =\ 1-\left[1-(1-\theta_{P\_critical})\times\left[1-(1-\gamma\times f_{I}( \gamma))^{\langle K_{2}\rangle}\right]\times f_{P}((1-\theta_{P\_critical}) \times\left[1-(1-\gamma\times f_{I}(\gamma))^{\langle K_{2}\rangle}\right] \right)\right]^{\langle\Omega_{1}\rangle}\end{array}\right. \tag{9}\]  
where these equations are nonlinear and the critical value \(\Theta_{P\_critical}\) of \(\Theta_{P}\) can be solved by numerical methods.  
The relationship between the final expected proportions \(R_{IF}(\infty)\) and \(R_{PF}(\infty)\) of remaining normal operational nodes and the unintentional random initial failure rate \(\Theta_{P}\) is obtained by solving the nonlinear Equation (5). From Equation (5), it can be seen that the greater the unintentional random initial failure rate \(\Theta_{P}\) is, the smaller \(\gamma\) and \(\lambda\). When the value of the unintentional random initial failure rate \(\Theta_{P}\) is greater than the critical value \(\Theta_{P\_critical}\), the solution of Equation (7) is only the trivial solution \(\gamma=0\), \(\lambda=0\). Therefore, the second conclusion of our analysis is that the robustness of the ADN is negatively correlated with the unintentional random failure rate \(\Theta_{P}\). Once the value of \(\Theta_{P}\) is greater than the critical value \(\Theta_{P\_critical}\), the whole ADN will collapse totally.