## 3 System Model
### Construction of Maze Model  
Inspired by the attack graph model, we model the intranet node topology as a maze \(M\), shown in Figure 3. We also limit the number of neighbours one node can have to reduce the probability of attack surface exploitation. The maze is modeled as a \(5\times 5\) grid graph, assigning index values to each grid, and we classify the grids into five classes, where the red grid is the entry node \(E\), which is the initial node of lateral movement in the attacker's inner network. For the real network structure, an entry node can be a web server, IOT devices, and other nodes that can be directly accessed from the external network. We set the attacker breaking through the entry node as the start of the attack. The blue grid is the defense node \(D\), which is the initial node of the defender's intranet scanning and positioning. Generally, it is the management node in the intranet, and the management node has the ability to detect the scanning and can find the location of the Intranet attacker. The green grid is the target node \(T\), that is, the high-value node in the internal network, generally a database, file server, mail server, etc. The gray grid is the offline node \(O\). We assume that there are offline nodes in the internal network, and these nodes cannot be accessed through the internal network link due to downtime, maintenance, or other reasons and cannot be recovered in a short time, just like the walls in the maze. The white grid is the intermediate node \(I\), the value of this kind of node is low and the attacker can normally enter and use this kind of node as a springboard to find the target node in the internal network. The maze model can be expressed as \(M=(E,D,T,I,O)\), Among them \(E=\{0\}\), \(D=\{4\}\), \(T=\{24\}\), \(O=\{3,17\}\), \(I=\{1,2,5,6,7,8,9,10,11,12,13,14,15,16,18,19,20,21,22,23\}\). We have a set of valid actions for each node \(i\), expressed as \(VA_{i},0\leq i\leq 24\). We use it to describe the connectivity of node \(i\), and although the valid action set can be unlimited, we still limit the valid actions of each node for the sake of security, because the more neighbors one node has, the higher the probability of attack surface exploitation happening. In order to fit the maze model we proposed, we set the original valid action set as \(VA_{i}=\{Up,Down,Left,Right\}\), but it will be adjusted according to the situation of the model. For example, at node \(0\), due to its boundary position, its valid actions are only down and right, expressed as \(VA_{0}=\{Down,Right\}\). For the presence of offline nodes in the adjacent nodes, such as node \(7\), its valid action set is \(VA_{7}=\{Up,Down,Right\}\). Similarly, the valid action sets of the remaining nodes can be obtained. In the process of lateral movement within the intranet, attackers incur certain costs. These costs arise predominantly from activities such as information gathering and deploying attack tools. Nonetheless, these expenses can be somewhat mitigated by exploiting the  
Figure 3: Maze model. value of intermediary nodes. Similarly, defenders also bear costs when investigating the intranet. The cost is mainly generated by system vulnerability scanning, malicious program detection, and other tools. The attacker and defender will only get a positive payoff if they reach their goals, but because of the overhead, both players expect to be able to reach their goals in the shortest number of steps. Since the attacker does not know the topology of the entire intranet, he needs to try to find the shortest path from the entry node to the destination node through continuous probing and movement. In the model shown in Figure 3, there are obviously multiple paths from the entry node to the target node, and also there exist multiple interception paths for the defender. The path is represented in the maze as shown in Figure 4.  
The analysis shows that there are Nash equilibrium paths for both attackers and defenders in the maze model.