## Appendix B Derivation of Mapping \(\mathbf{f_{P}}\) and \(\mathbf{f_{I}}\)
\[\mathit{Prob}_{P}(\frac{\mathit{E}\mathit{t}\mathit{t}^{\prime}_{\mathit{P}2 }}{\mathit{E}\mathit{t}\mathit{t}\mathit{t}_{\mathit{P}1}})\ =\ \frac{1}{\langle d_{P}\rangle}\sum\limits_{d_{P}\ =\ 0}^{\infty}\ [(d_{P}+1)\times \mathit{Prob}_{P}(d_{P}+1)]\times(1-\mathit{Prob}_{P}(\mathit{E}\mathit{t} \mathit{t}_{\mathit{P}4})).\] (A14)  
Secondly, the number of neighbor nodes of a node \(n_{P1}\) in the CN of the ADN is \(D_{I1}=d_{I}-O_{1}\). In order to calculate the conditional probability \(\mathit{Prob}_{I}(\mathit{E}\mathit{t}\mathit{t}_{\mathit{13}}/[\mathit{ Neigh}_{I}(n_{P1})])\), we need to know the conditional probability \(\mathit{Prob}_{I}(\mathit{E}\mathit{t}\mathit{t}_{\mathit{13}}/D_{I2}=d^{ \prime}_{I})\) which is equal to Equation (A15). \[\mathit{Prob}_{I}(\frac{\mathit{E}\mathit{t}\mathit{t}_{\mathit{13}}}{D_{I2} \ =\ d^{\prime}_{I}})\ =\ \mathit{Prob}_{I}\Bigg{(}d^{\prime}_{I}\times\Bigg{(}\sum\limits_{d \ =\ 1}^{\infty}\frac{F_{I}(d)\times\mathit{W}_{I}(n_{P1},n_{P1})}{\sum\limits_{d \ =\ 1}^{\infty}\frac{F_{I}(d)\times\mathit{W}_{I}(n_{P1},n_{P1})}{\mathit{W}_{I}(n_{ P1},n_{P1})}\times\mathit{Prob}_{I}(d)\Bigg{)}+F_{I}(d_{I})>\alpha_{I}\times F_{I}(d_{I}) \Bigg{)}\] (A15)  
where \(\sum\limits_{d\ =\ 1}^{\infty}\frac{F_{I}(d)\times\mathit{W}_{I}(n_{P1},n_{P1})}{ \sum\limits_{d\ =\ 1}^{\infty}\frac{F_{I}(d)\times\mathit{W}_{I}(n_{P1},n_{P1})}{ \mathit{W}_{I}(n_{P1},n_{P1})}\times\mathit{Prob}_{I}(d)}\) is the expected data traffic flow redistributed to the node \(n_{P1}\) after the failure of the neighbor node \(n_{P1}\) connected to the node \(n_{P1}\). Then there is Equation (A16). \[\mathit{Prob}_{I}(\frac{\mathit{E}\mathit{t}\mathit{t}_{\mathit{13}}}{D_{I2}\ =\ d^{\prime}_{I}})\ =\ \Bigg{\{}\ \frac{1,d^{\prime}_{I}>\frac{(\alpha_{I}-1)\mathit{F}_{I}(d_{I})}{ \sum\limits_{d\ =\ 1}^{\infty}\frac{F_{I}(d)\times\mathit{W}_{I}(n_{P1},n_{P1})}{ \sum\limits_{d\ =\ 1}^{\infty}\frac{F_{I}(d)\times\mathit{W}_{I}(n_{P1},n_{P1})}{ \mathit{W}_{I}(n_{P1},n_{P1})}\times\mathit{Prob}_{I}(d)}}{1-\mathit{W}_{I}(n_{ P1},n_{P1})}\times\mathit{Prob}_{I}(d)}\ \.\] (A16)The probability of selecting a node from the neighbor nodes set \(Neigh_{l}(n_{li})\) of the node \(n_{li}\) in the CN is shown in Equation (A17). \[\text{Prob}_{l}(\frac{D_{I2}\ =\ d_{I}^{\prime}}{D_{I1}\ =\ d_{I}-O_{1}})\ =\ \frac{ \left( \begin{array}{c}d_{I}-O_{1}\\ d_{I}^{\prime}\end{array} \right)}{2^{d_{I}-O_{1}}},\] (A17)  
where  
\[2^{d_{I}-O_{1}}\ =\ \big{(} \begin{array}{c}d_{I}-O_{1}\\ 0\end{array} \big{)}+\left( \begin{array}{c}d_{I}-O_{1}\\ 1\end{array} \right)+\ldots+\left( \begin{array}{c}d_{I}-O_{1}\\ d_{I}-O_{1}\end{array} \right)\big{)}.\] (A18)  
Therefore, under the condition that the number of neighbor nodes set is \(D_{I1}=d_{I}-O_{1}\), the probability of an event \(Evt_{I3}\) occurring is shown in Equation (A19). \[\text{Prob}_{l}(\frac{Evt_{I3}}{D_{I1}\ =\ d_{I}-O_{1}})\ =\ \sum_{d^{\prime}\ =\ 0}^{d_{I}-O_{1}} Prob_{l}(\frac{Evt_{I3}}{D_{I2}\ =\ d^{\prime}})\times\text{Prob}_{l}(\frac{D_{I2}\ =\ d^{\prime}}{D_{I1}\ =\ d_{I}-O_{1}}).\] (A19)  
After further simplifications, the results are obtained as shown in Equation (A20). \[\text{Prob}_{l}(\frac{Evt_{I3}}{D_{I1}\ =\ d_{I}-O_{1}})\ =\ \frac{1}{2^{d_{I}-O_{1}}} \sum_{d_{I}^{\prime}\ =\ \frac{1}{2^{d_{I}-O_{1}}}}^{d_{I}-O_{1}} \sum_{ \begin{subarray}[]{c}d_{I}-O_{1}\\ d_{I}^{\prime}\end{subarray} }^{\infty}\left( \begin{array}{c}d_{I}-O_{1}\\ d_{I}^{\prime}\end{array} \right).\] (A20)  
Furthermore, it is assumed that \(NON_{I}\times\Theta_{I}\) nodes fail in the CN. A node \(n_{li}\) does not belong to the connected clusters depending on the connected clusters containing generators in the PN due to the following four situations:  
1. This node \(n_{li}\) has been removed due to random failures, etc. 2. This node \(n_{li}\) exists but does not belong to the connected clusters depending on the connected clusters containing generators in the PN. 3. This node \(n_{li}\) has been removed due to the excessive data traffic flow. 4.