# Research on Power Cyber-Physical Cross-Domain Attack Paths Based on Graph Knowledge
## 3 Method for Constructing Attack Graph of Electric Power Cyber-Physical Fusion System
### Attack Graph Building Process
If it can have an impact, the value of \(R_{\mathrm{k}}\) is 1, otherwise it is 0. It is defined here that the \(\mathrm{v_{i}}\) value of each vertex in the collection \(\mathrm{Y}\) is 1. 4. \(\beta_{\mathrm{i}}\) represents the probability of the attack event represented by vertex \(\mathrm{v_{i}}\) occurring. \(\beta_{\mathrm{i}}\) calculation method can refer to the Bayesian network or Markov model method in [21] for calculation. Define \(\beta_{\mathrm{i}}\) of each vertex in the source set \(\mathrm{X}\) to be 1. Step 4: Calculate the geodesic distance from the leaf node to the target root node. To calculate the geodesic distance between two vertices, you first need to define the edge weights in the attack directed graph. Due to the hierarchical algorithm of the attack graph, in the \(\mathrm{L_{l}}\) layer, the weights of the directed edges connecting leaf nodes representing various cyber-physical cross-domain attack types are consistent with the occurrence probabilities of various attack types. For the calculation method, please refer to step 3. The directed edge weights between vertices such as control master station, measurement and control equipment failure, transient stable node disturbance, etc., can be determined in the following way:  
1. First, give the definition of edge weights. Let \(\Delta=\Delta_{\mathrm{s}}\cup\Delta_{\mathrm{a}}\) represent the weight of the directed edge. Among them, \(\Delta_{\mathrm{s}}\) corresponds to the directed edge \(\mathrm{E_{s}}\), which refers to the one-step transition probability from the previous node to the current node, indicating the probability of the current node occurring when the previous state satisfies the conditions, which is called the previous transition probability. \(\Delta_{\mathrm{a}}\) corresponds to the directed edge \(\mathrm{E_{a}}\), which represents the probability that the current node reaches the subsequent node state, which is called the posterior transition probability here. 2. Therefore, from the definition of edge weights in (1), it can be seen that in the attack graph of the power cyber-physical fusion system, the pre-transition probability of the leaf nodes representing various cyber-physical cross-domain attacks is consistent with the probability of occurrence of various attacks. The weight of the edge connected to the root node representing the attack target is consistent with the post-transition probability of the preceding node. The weight of the edge connected to the intermediate vertex is determined by the forward and backward transition probabilities of the vertex. 3. Let the element \(\Delta\mathrm{s_{ij}}\) be the weight of the directed edge \(\mathrm{E_{s}}=\left(\mathrm{a_{i}},\mathrm{a_{j}}\right)\), which represents the probability of node \(\mathrm{a_{j}}\) occurring if node \(\mathrm{a_{i}}\) is satisfied, \(\Delta\mathrm{s_{ij}}=P\left(a_{j}\middle|a_{i}\right).\) The pre-transition probability and the post-transition probability of the edges connected to intermediate nodes such as control master station, measurement and control equipment failure, transient stable node disturbance, etc., will be affected by factors such as the node's own functional attribute GN, the scene attribute CJ, and the edge JO between cyber-physical networks. The influence can be used as the basis for determining the weight value \(\Delta\mathrm{s}\) of the directed edge.The directed edge weight can be defined as Formula (6). \[\Delta\mathrm{s}=2 GN+CJ+JO\] (6)  
In actual business scenarios, state transfer between nodes is often caused by network congestion, jitter, system failure, and other factors. For example, the operating status of the physical system due to network congestion cannot be monitored in real time, causing the control master station to issue incorrect operation instructions. The state transition between nodes caused by such factors can be determined based on statistical analysis of historical fault types. The state transition probability between nodes determined in this article is as shown in Table 1 below.