## 4 Numerical Examples and Their Analysis
Then, in Step 3, among all temporal labels of the OR nodes, node 6 with the smallest temporal label is chosen and granted a permanent time label \(t_{6}\), as shown in row (3) in Table 1. This node is added to set \(\mathbb{F}\). Since the permanent time label for the target node 3, has not been found yet, the algorithm goes again to Step 1 (row (1) in the last line of Table 1). During the execution of Step 1, the permanent label for the AND node 3 can be defined because all its predecessors, i.e., nodes 2 and 6, are already in the set \(\mathbb{F}\). The algorithm terminates when the minimal time for the target node 3 is found. As can be seen in Table 1, the overall minimum attack time to compromise the target node 3 is 6 min. The corresponding attack path is shown in Figure 2b with a thick dashed line. This attack path is optimal, in the sense that it has the minimum length among the attack paths in the attack graph under consideration. A configuration of the resulting attack path, the subgraph in Figure 2b, is neither a chain nor a tree but a \"flower graph\". Recall that in this study, we assume that the attacker (which could be a group of people or a set of programs) can perform multiple malicious actions simultaneously. In the considered Example 1 in Figure 2b, this means that after the attacker has gained access to node 2, they have enough resources to simultaneously perform his/her transition actions on edges from 2 to 3 and from 2 to 6. Next, we will consider more general (and more realistic) formulations of the problem, in which the durations of the attacker's compromising activities on nodes are added to the model, which can significantly affect the duration and configuration of the attack path. In Example 2, we consider the path extraction problem on the attack graph with both node and edge times. **Example 2**. : _Consider the attack graph shown in Figure 3a. It is an acyclic AND/OR graph with weighted edges and nodes: nodes are equipped with known times (durations) marked in red, and all edge times are marked in blue._  
Here, the problem is to find the attack path of the smallest duration from the Start to node 3. All steps of the algorithm in solving the problem are shown in Table 2, and the extracted minimum-length attack path is shown in Figure 3b with a thick dashed line. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Step**Node** & \(Start\) & 2 & 3 & 4 & 5 & 6 \\ \hline Initial & \(t_{S}=0\) & & & & \(u_{5}=\infty\) & \(u_{6}=\infty\) \\ (1) & & \(t_{2}=12\) & & & \(u_{5}=4.5\) & \(u_{6}=14\) \\ (2) & & & & \(t_{4}=6\) & & \(u_{6}=9\) \\Table 2 shows that the minimum attack time to compromise target node 3 is 15 min. The extracted attack path presented in Figure 3b, shown with a thick dashed line, notably differs from the one found in Figure 2b, when the attack times on nodes were neglected. Although the attack path in Figure 3b includes more nodes to be compromised before the target node is reached compared to the one presented in Figure 2b, it is shorter in terms of the overall attack time when the times on nodes are considered: the alternative attack path AP = {start \(\to 2\to 6\to 3\)} will result in an overall attack time of 18 min. **Example 3**. : _Consider the attack graph shown in Figure 4a. It is a cycled graph with weighted edges and nodes. Nodes are equipped with known times marked in red, and all edge times are marked in blue. The intruder's goal is to access node 7._  
The proposed algorithm provides the solution presented in Table 3. The attack path is shown in Figure 4b. Figure 4: (**a**) Attack graph with cycles. (**b**) Extracted minimum-length attack path. Figure 3: (**a**) Acyclic attack graph equipped with the node times. (**b**) Extracted minimum-length attack path for Example 2. Unlike the attack graphs presented in Figures 1(a) and 2(b), the attack graph in Figure 3(a) contains a cycle, which makes the path-finding problem more difficult.