# Robustness of Double-Layer Group-Dependent Combat Network with Cascading Failure
## 2 Double-Layer Group-Dependent Combat Network
When the dependency node \(v^{j}_{\rm G}\to v^{j}_{\rm W}\), let \(E_{\rm D}(v^{j}_{\rm G},v^{j}_{\rm W})=1\) (for the convenience of calculation, \(E_{\rm D}(v^{j}_{\rm W},v^{j}_{\rm G})\) is also equal to \(1\)). The dependency nodes and edges form a dependent net as \(G_{\rm D}\). All networks above together constitute the DHDCN of CSOS, which is denoted as_  
\[G=(G_{\rm G},G_{\rm D},G_{\rm W}). \tag{1}\]_The adjacency matrix of DHDCN is expressed as_  
\[S=\left[ \begin{array}{cc}S_{\text{G}}&S_{\text{GW}}\\ S_{\text{WG}}&S_{\text{W}}\end{array} \right], \tag{2}\]  
_where \(S_{\text{G}}\) represents the adjacency matrix of the functional net; correspondingly, \(S_{\text{W}}\) and \(S_{\text{WG}}(S_{\text{GW}})\) are the adjacency matrices of the communication net and interdependent net._  
It is worth noting that the abovementioned dependencies between heterogeneous coupling networks can be the type of one-to-one, one-to-many, and many-to-one [21]. A form of one-to-many dependency is called group dependency; therefore, a DHDCN with a one-to-many dependency is referred to as a double-layer group-dependent combat network (DGCN). **Example 1**. : _According to the network model described in Definition 3, the network structure of the DGCN for CSOS is given for a certain combat scenario, as shown in Figure 1. In the figure, the functional net is composed of nodes and edges corresponding to four combat units including \(O\), \(P\), \(D\), and \(A\), and the communication net is composed of nodes corresponding to different communication support units. Based on the dependent net, the functional net and the communication net form asymmetric coupling relations._  
The widely studied combat cycle model takes the loop composed of the target node T, search node S, decision node D, and influence node I as a measure of combat capability [22]. Inspired by this work, the functional net of CSOS in this paper also presents the flow of information when relying on the communication net's communication nodes. The information flow model based on communication nodes is shown in Figure 2. As we can see from the figure, the red solid lines represent the entire information flow of combat links, and the gray dotted lines indicate the information transmission through the communication net. According to the dependency rule, the information flow of the most typical combat link \"O \(\rightarrow\) P \(\rightarrow\) D \(\rightarrow\) A\" will be transformed into \"O \(\rightarrow\) C \(\rightarrow\) P \(\rightarrow\) C \(\rightarrow\) D \(\rightarrow\) C \(\rightarrow\) A\", and other types of combat link can also be obtained similarly. The typical combat link's dynamics follow the rule of \"information obtain-intelligence processing-command and decision-combat response (attack)\" to form a kill link, and the more concrete concept of it is described in Section 4.1. Once the communication node in Figure 2 is broken, the combat link will be destroyed and CSOS will lose its combat capability. Therefore, the coupling relation between the functional subnet and communication subnet is tighter and more important in the DGCN of CSOS. Figure 1: Structure diagram of a DGCN of CSOS.