# Research on the Robustness of Interdependent Networks under Localized Attack
## 2 Models
### Failure Probability Degradation Model of a Localized Attack  
A localized attack is often used to simulate the impact of natural disasters on interdependent systems. In a localized attack scenario, nodes are chosen from a certain area, starting from a root node,  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Step**} & \multicolumn{2}{c}{**Subnetwork A**} & \multicolumn{2}{c}{**Subnetwork B**} \\ \cline{2-4}  & **Remaining Fraction** & **Giant Component** & **Remaining Fraction** & **Giant Component** \\ \hline
1 & \(\psi^{\prime}_{n}=1-p\) & \(\psi_{n}=\psi^{\prime}_{n}P_{A}(\psi^{\prime}_{n})\) & \(\phi^{\prime}_{1}=1-q_{B}[1-p_{A}(\psi^{\prime}_{n})p]\) & \(\phi_{1}=\phi^{\prime}_{n}P_{B}(\phi^{\prime}_{1})\) \\
2 & \(\psi^{\prime}_{2}=(1-p)\big{\{}1-q_{A}[1-p_{B}(\phi^{\prime}_{1})]\big{\}}\) & \(\psi^{\prime}_{2}=\psi^{\prime}_{2}P_{A}(\psi^{\prime}_{2})\) & \(\cdots\cdots\) & \(\cdots\cdots\) \\ \(\cdots\) & \(\cdots\) & \(\cdots\) & \(\cdots\) \\ \(n\) & \(\psi^{\prime}_{n}=(1-p)\big{\{}1-q_{A}[1-p_{B}(\phi^{\prime}_{n-1})]\big{\}}\) & \(\psi_{n}=\psi^{\prime}_{n}P_{A}(\psi^{\prime}_{n})\) & \(\phi^{\prime}_{n}=1-q_{B}[1-P_{A}(\psi^{\prime}_{n})p]\) & \(\phi_{n}=\phi^{\prime}_{n}P_{B}(\phi^{\prime}_{n})\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Cascading failures of interdependent networks.  
Figure 1: Cascading failures of interdependent networks [4]. Red and blue nodes represent nodes of subnetwork \(A\) and subnetwork \(B\), respectively. Solid circles denote functional nodes, hollow circles denote failure nodes. (**a**) One node in sub network \(A\) is chosen to be attacked; (**b**) the attacked node in \(A\) and its dependency counterpart in \(B\) fail, and all of the links of the failure nodes, including connectivity and dependency links, are removed. Then \(A\) breaks into three clusters, and only nodes in cluster \(a_{13}\) remain functional according to the assumption; (**c**) because of the dependency and larger cluster, cluster \(b_{21}\), \(b_{22}\), and \(b_{23}\) in \(B\) are eliminated; and (**d**) failure of cluster \(b_{23}\) in turn leads to failure of some nodes in \(A\), and cascading failures stop because no further failure occurs. Finally, only nodes in cluster \(a_{34}\) and \(b_{24}\) remain functional in the interdependent networks after only one node in subnetwork \(A\) is attacked.  
then its nearest neighbors, then the next nearest neighbors, as shown as Figure 2, until the fraction of affected nodes reach a certain value, denoted by \(p\). The traditional localized attack (TLA) assumes that all of these nodes will fail. However, the fact is not like that. It is a basic fact that destructive force usually decays with distance, and there is a distance threshold, \(L_{0}\), beyond which the probability of node failure is 0. Additionally, for earthquakes, the closer to the epicenter, the slower the destructive force increases [46]. Simple and without generality, we assume the failure probability of a node at the epicenter is 1. Thus, we can define a new failure probability degradation model of a localized attack (PDLA), as shown as Figure 3, and the failure probability of one node with distance \(l\) from the attack center is as follows:  
\[p(l)=\left\{\begin{array}{cc}1-\frac{l^{2}}{(L_{0}+1)^{2}},&0\leq l\leq L_{ 0}\\ 0&,&l>L_{0}\end{array}\right.,\;l=0,1,2,\;\ldots \tag{2}\]  
where \(L_{0}\) represents the furthest distance from the affecting natural disaster. In this article, the length of the shortest path between nodes is regarded as distance for simplicity.