# Robustness of Double-Layer Group-Dependent Combat Network with Cascading Failure
## 2 Double-Layer Group-Dependent Combat Network
We denote the node of the communication net as node C.  
With regard to functional net, the combat units and relations among them in CSOS are usually abstracted into the nodes and edges of the complex network, namely, \"O, P, D, A\" nodes and the corresponding edges [20]. The \"O, P, D, A\" nodes represent the intelligence obtaining unit, intelligence processing unit, decision and command unit, and attack or damage unit, respectively, and more details on structural abstraction can be found in [20]. There are complex interaction relations among different units, forming a network structure of heterogeneous components, multipoint interaction, multidomain fusion, and dynamic evolution, so the heterogeneous information network is used for modeling. The functional net of CSOS with heterogeneous information is defined as follows:  
**Definition 2**. : _Suppose \(G_{\rm G}=(V,W_{a},E,W_{b};\varphi,\psi;V_{\rm G},E_{\rm G})\) is the functional net of CSOS, where the node set is \(V=\{v_{1},v_{2},\cdots,v_{\rm N_{G}}\}\), and \(N_{\rm G}\) is the number of nodes. \(W_{a}=\{w_{i}|v_{i}\}\) represents the service attributes of the functional node, including the service category, processing load, and affordable capacity undertaken by the node. \(E=\{e_{1},e_{2},\cdots,e_{\rm M_{c}}\}\) is the edge set, in which \(M_{\rm G}\) is the number of edges. \(W_{b}=\{w_{jk}|e_{jk}\}\) is an attribute of the edge, indicating the interaction strength of service information. Both nodes and edges have characteristics of type. The type set of nodes is \(V_{\rm G}\), and there is a mapping function \(\varphi:V\to V_{\rm G}\) that satisfies \(\varphi(v_{i})\in V_{\rm G}\), while type set of edges is \(E_{\rm G}\) with mapping function \(\psi:E\to E_{\rm G}\) satisfying \(\psi(e_{i})\in E_{\rm G}\). If \(|V_{\rm G}|>1\) or \(|E_{\rm G}|>1\), then the functional net of CSOS is called heterogeneous combat functional net (HCFN)._  
Similarly, for the sake of simplicity, this paper assumes that the service interaction strength of the edge of the HCFN is within the acceptable range, so the attribute characteristics of the edge will not be considered. In addition, the type set of nodes is \(V_{\rm G}=\{\rm O,P,D,A\}\), and the type set of directed edge is \(E_{\rm G}=\{\)O-O, O-P, P-P, P-D, D-D, D-A\(\}\). It is notable that the functional net is unidirectionally dependent on the communication net to achieve combat utility. Although the obtained and processed intelligence information and command orders are all circulated in the functional net, the information exchanges among the nodes are all based on communication units in practice. Apart from the communication net and the functional net, there is another important network called the dependent net. To build the dependent relation between CSOS, the following two assumptions need to be specified:  
(1) The node in CSOS only has a single function, that is, node C of the communication net can only perform information transmission, node O of the functional net can only perform information acquisition, node P can only perform information processing, node D can only perform commands and decisions, and node A can only perform combat attack. (2) Although the communication net is abstracted from the actual equipment, it is assumed that the constraints of space and time are neglected when it undertakes communication missions. Therefore, the connection conditions will not be taken into consideration for how the functional net depends on the communication net. On the basis of the above assumptions, we can construct a double-layer heterogeneous dependent combat network (DHDCN) of CSOS when the functional net is relying on the communication net, which is defined as follows:  
**Definition 3**. : _Assuming that the communication net of CSOS is \(G_{\rm W}\), the corresponding functional net of is \(G_{\rm G}\). The coupling relation between two networks is \(E_{\rm D}=\{E_{\rm GW},E_{\rm WG}\}\), which means that the functional net relies on the communication net.