# Robustness of Double-Layer Group-Dependent Combat Network with Cascading Failure
## 2 Double-Layer Group-Dependent Combat Network  
For the convenience of reading, a glossary of the notations used in this paper is presented in Table 1.  
To be as real as possible, the combat network is modeled from two aspects. One is the communication net in terms of physical combat equipment, and the other is the functional net in terms of logic. For the combat communication net, each piece of equipment is deployed in a dispersed manner in physical locations and undertakes the task of energy and material transmission dominated by information flow. Therefore, combat equipment is the basis for communication and information interaction. An information grid network that achieves high data sharing, efficient information interaction, dynamic port access, and flexible combination requirements is the result of the deep fusion of all types of equipment. The definition of the communication net is given as follows:  
**Definition 1**. : _Suppose \(G_{\text{W}}=(V,W_{a},E,W_{b})\) is the communication net of combat system-of-system (CSSO), in which the node set is \(V=\{v_{1},v_{2},\cdots,v_{N_{\text{W}}}\}\), \(v_{i}\) represents the combat equipment with communication function in the communication net, and \(N_{\text{W}}\) is the number of nodes. \(W_{a}=\{w_{i}|v_{i}\}\) represents the attributes of the node, including its initial load and bearable capacity. The set of edges is \(E=\{e_{1},e_{2},\cdots,e_{N_{\text{W}}}\}\), where \(e_{i}=\{e_{jk}|v_{j}\times v_{k}\}\) represents the communication relations among different nodes, and \(M_{\text{W}}\) is the number of edges. The existence of edge is mainly determined by infrastructure deployment, affiliation, and mission requirements. It is relatively fixed in general and will be flexible and changeable when carrying out missions. \(W_{b}=\{w_{jk}|e_{jk}\}\) is the attribute of the communication edge, such as bandwidth and delay. \begin{table}
\begin{tabular}{c c} \hline \hline
**Symbol** & **Meaning** \\ \hline \(G\) & The complex network, including the communication network \(G_{\text{W}}\), the functional network \(G_{\text{G}}\) and the dependent network \(G_{\text{D}}\) \\ \(V\) & The nodes set of the complex network, the specific node in it is \(v_{i}\) \\ \(E\) & The edges set of the complex network, the specific edge in it is \(e_{i}\) \\ \(N\) & The number of the nodes set, including \(N_{\text{O}}\), \(N_{\text{P}}\), \(N_{\text{D}}\), \(N_{\text{A}}\), \(N_{\text{W}}\) and \(N_{\text{G}}\) \\ \(S\) & The adjacent matrix of the network \\ \(p_{\text{XX}}\) & The probability of two different nodes can connect with each other \\ \(L_{i}\) & The load of the node \(v_{i}\) \\ \(C_{i}\) & The capacity of the node \(v_{i}\) \\ \(k_{i}\) & The degree of the no \(v_{i}\) \\ \(\zeta_{i}\) & The communication connectivity index of node \(v_{i}\) \\ \(S_{\text{links}}\) & The amount of total intelligence effectiveness links \\ \(S_{\text{huge}}\) & The largest component scale of the combat network \\ \(\alpha\) & The proportion parameter indicating the preference for \(S_{\text{links}}\) and \(S_{\text{huge}}\) \\ \(f\) & The nodes failure ratio for the total network \\ \(\tau\) & The endurance threshold of connectivity index \\ \(\kappa(\kappa_{\text{W}}/\kappa_{\text{G}})\) & The power parameter of node load \\ \(\lambda(\lambda_{\text{W}}/\lambda_{\text{G}})\) & The linear parameter of node capacity \\ \(\gamma(\gamma_{\text{W}}/\gamma_{\text{G}})\) & The power parameter of node capacity \\ \(\delta(\delta_{\text{W}}/\delta_{\text{G}})\) & The bearing range of node when it is overloaded \\ \hline \hline \end{tabular}
\end{table} 
Table 1: List of symbols. For simplicity, it is assumed that the communication net is fixed and undirected and \(W_{b}=1\), so the initial perturbation of the network topology and the performance on edges will not be considered.