# Robustness of Double-Layer Group-Dependent Combat Network with Cascading Failure
## 3 Cascading Failure Model for DGCN
### Overload Failure Model
#### 3.3.4 Failure Status Judgment  
In the classical \"M-L\" model [4], a node has only two states, namely, the normal state and the failed state. [PERSON] et al. [32] proposed a cascading failure model of complex networks considering overloaded nodes. When the load exceeds the capacity of a node, the node will fail with a certain probability in the certain bearing range \(\delta\). This state is called the critical state. When the load exceeds the above range, the node fails directly. Motivated by this, the specific rules for judging the failure situation after the node obtains the reallocated load are as follows:  
(1) If \(L_{i}(t)\leq C_{i}\), the node \(v_{i}\) does not fail.  
(2) If \(C_{i}<L_{i}(t)\leq(1+\delta)C_{i}\), the node \(v_{i}\) is in the critical failure state. Although the load exceeds the capacity of a node, it is still within the affordable range. The more the node is overloaded, the greater the probability of node failure and the failure probability is  
\[p_{i}(t)=\frac{L_{i}(t)-C_{i}}{\delta C_{i}}. \tag{14}\]  
When \(p_{i}(t)\) is greater than a random number, the node \(v_{i}\) fails.  
(3) if \(L_{i}(t)>(1+\delta)C_{i}\), the node \(v_{i}\) fails immediately.  
In summary, taking the cascading failure of the communication net as an example, the failure probability of  
nodes to make them invalid, while the intended attack is to make nodes invalid according to a certain order of node importance. Since DGCN is a double-layer network, there are different attack modes for different networks. For example, in different combat phases, the enemy may select specific functional net nodes to attack or destroy the nodes of the communication net randomly. According to the attack object of the combat network and the attack intention, the attack mode can be divided into the following six types:  
(1) Random single communication network attack (RSCA): Randomly select nodes with a ratio \(f\) from the communication net to attack and let these nodes fail.  
(2) Intended single communication network attack (ISCA): Select nodes with a ratio \(f\) from the communication net according to the descending node degree, and let them fail by attacking.  
(3) Random single functional network attack (RSFA): Randomly select nodes with a ratio \(f\) from the communication net to attack and let these nodes fail.  
(4) Intended single functional network attack (ISFA): Select nodes with a ratio \(f\) from the functional net according to the descending node degree, and let them fail by attacking.  
(5) Random double networks attack (RDA): Randomly select nodes with a ratio \(0.5f\), respectively, from the communication and functional net to attack, and let these nodes fail.  
(6) Intended double networks attack (IDA): Select nodes with a ratio \(0.5f\), respectively, from the communication net and functional net according to the descending node degree, and let them fail by attacking.  
Different attack modes may have different effects on robustness. Combining the asymmetric dependent failure model and overload cascading failure model, the failure process of the DGCN of CSOS can be represented as Figure 6, in which the dashed boxes of three different colors represent different attacked objects. These nodes in a double-layer network will suffer random attacks or intended attacks. As can be seen from the figure, the failure process of the communication net always leads to the failure of the functional net, so the functional net is more fragile. Any disturbance in the network may cause serious consequences of \"failure, disconnection and paralysis\".