# Robustness of Double-Layer Group-Dependent Combat Network with Cascading Failure
## 4 Simulation Experiments on Robustness  
### Robustness Evaluation Index  
The general definition of the robustness of the combat network is the ability of a system to maintain its original functions, characteristics, and organizational structure under external disturbances [40]. In this paper, the robustness of DGCN for CSOS refers to the  
Figure 6: Failure process under different attack modes. ability of a system to continue maintaining combat performance after some nodes failed in the network. Due to the existence of a functional net in DGCN, the performance evaluation can be more real. Here we define the robustness evaluation index from two aspects:  
The first aspect of the robustness measure is based on combat network topology. The scale of the maximal component is selected as an index of the damage effect of the combat network. One individual component is a subgraph with connectivity and isolation in the network. As a result, the maximal component is a subgraph with the largest node scale in the network, which is denoted as \(S_{\text{huge}}\). With the increase of the scale of the maximal component in the combat network, the interconnections among combat network nodes become closer and the information flow efficiency of the combat network becomes higher. In this paper, we adopt the approach of node shrinking to iteratively calculate the scale of the maximal component [41]. The initial maximal component of DGCN is  
\[S_{\text{huge}}(G)=N_{\text{G}}+N_{\text{W}}, \tag{16}\]  
where \(N_{\text{G}}\) and \(N_{\text{W}}\) are the node number of the functional net and the communication net, respectively. The second aspect of the robustness evaluation index is based on the operational capability of the combat network. The number of kill links can be used to describe the operational capability of a combat network [42]. In the case of a DGCN of CSOS, the number of combat effectiveness links (CELKs) \(S_{\text{links}}\) is introduced to measure the operational capability. According to the idea of [PERSON]'s OODA cycle [43], the combat network exerts operational capability by forming a CELK of \"intelligence obtaining-intelligence processing-commanding and decision-attack and damage\" around the target, namely, the OODA kill link. For a more general CELK, the mutual coordination among the intelligence obtaining node O, the intelligence processing node P, and the commanding and decision node D should also be taken into consideration. The flow of generalized CELK with a target can be demonstrated as the generalized combat effectiveness loop (CELP) (see Figure 7). Since the nodes cooperation in generalized CELK may lead to an infinite long link, seven types of CELKs commonly used in practice are selected as the basis for quantity calculation [44]. The detailed description of those CELKs is indicated in Table 2.  
Figure 7: Generalized CELP diagram. The accessibility matrix \(\hat{S}\) of the entire combat network is calculated according to the adjacency matrix \(\hat{S}\) of the functional net:  
\[ \begin{split}(S+\mathbf{I})^{(1)}&\
eq(S+\mathbf{I})^{(2)} \
eq\cdots\\ &\
eq(S+\mathbf{I})^{(r)}=(S+\mathbf{I})^{(r+1)}=\tilde{\mathbf{S}}\,,\end{split}  \tag{17}\]  
where \(I\) is the identity matrix. Equation (17) is the power Boolean operation on \((S+\mathbf{I})\), and \(r+1\) is the times of power multiplication. The connectivity between intelligence obtaining and attack/damage nodes can be known based on the accessibility matrix. For any node \(O_{\mathrm{i}}\) and node \(A_{\mathrm{j}}\), if \(\tilde{S}(i,j)=1\), then \(O_{\mathrm{i}}\) can reach \(A_{\mathrm{j}}\) according to a path with practical meaning. Assuming that \(S(j,i)=1\), then the number of CELPs is the trace of the product of the corresponding nodes' accessibility matrices, which is also the number of CELKs. Because of the dependency of functional net on the communication net, the information transmission must pass through the communication node, as shown in Figure 2. The CELK should also correspondingly consider the role of communication nodes.