## 3 Materials and Methods
### Security Metrics for Satellite Networks
#### 3.5.2 Vulnerability Score  
The vulnerability score, which is widely used as a metric for assessing network security, provides a quantitative evaluation based on known vulnerabilities and related metrics (e.g., CVSS and CVE) [39]. In this study, the vulnerability scoring method used in TV-HARM was adjusted to adapt to the unique characteristics of the satellite network environment. The attack success probability and risk were calculated based on host and protocol vulnerabilities in the satellite network. Herein, an individual vulnerability, \(v_{i}\), represents a specific security vulnerability identified in a network node, where \(p_{v_{i}}\) denotes the probability of a successful attack, and \(r_{v_{i}}\) indicates the risk resulting from the vulnerability. \[p_{v_{l}}=\frac{\text{CVSS BS}_{v_{l}}}{10},\quad p_{pv_{l}}=\frac{\text{CVSS BS}_{pv_{l}}}{10} \tag{1}\]  
\[r_{v_{l}}=Impact_{v_{l}}\times p_{v_{l}},\quad r_{p_{v_{l}}}=Impact_{p_{v_{l}}} \times p_{p_{v_{l}}} \tag{2}\]  
Equation (1) defines the attack success probability (\(p_{v_{l}}\)), normalizing it based on the CVSS base score within a range from 0 to 1. Herein, CVSSCVS BS\({}_{v_{l}}\) represents the CVSS base score of a host vulnerability (\(v_{l}\)), while CVSS BS\({}_{pv_{l}}\) represents the CVSS base score of a protocol vulnerability (\(pv_{l}\)). The CVSS base score quantifies exploitability on a scale from 0 to 10, and normalization ensures its usability as an attack success probability within a range from 0 to 1. The risk (\(r_{v_{l}}\)) is calculated as the product of the attack success probability and the impact, as shown in Equation (2). Herein, \(Impact_{v_{l}}\) denotes the impact score from the CVSS metric, quantifying a successful exploit's potential consequences. Applying Equations (1) and (2), \(p_{v_{l}}\) and \(r_{v_{l}}\) numerically express the likelihood of exploitation and the risk for each specific vulnerability. The next step involves computing security metrics for hosts. By analyzing the vulnerabilities associated with a host, the model determines its attack success probability (\(p_{h}\)) and risk (\(r_{h}\)). Equations (3) and (4) formally define these metrics. \[p_{h}= \begin{cases}1-\prod_{v\in A_{h}}(1-p_{v}),&\text{if only host vulnerability exists}\\ 1-\prod_{pv\in A_{h}}(1-p_{pv}),&\text{if only protocol vulnerability exists}\\ 1-\prod_{v\in A_{h}}(1-p_{v})\cdot\prod_{pv\in P_{h}}(1-p_{pv}),&\text{if both exist}\\ \end{cases}  \tag{3}\]  
\[r_{h}= \begin{cases}\sum_{v\in A_{h}}p_{v}\cdot r_{v_{l}},&\text{if only host vulnerability exists}\\ \sum_{pv\in P_{h}}p_{pv}\cdot r_{v_{l}},&\text{if only protocol vulnerability exists}\\ \sum_{v\in A_{h}}p_{v}\cdot r_{v_{l}}+\sum_{pv\in P_{h}}p_{pv}\cdot r_{p_{v_{l}}},&\text{if both exist}\\ \end{cases}  \tag{4}\]  
Equation (3) defines the attack success probability (\(p_{h}\)), which is categorized into three cases based on host and protocol vulnerabilities: Host vulnerabilities: The attack success probability (\(p_{h}\)) for host-based vulnerabilities (\(v\in A_{h}\)) in a node (\(h\)) is calculated by determining the combined failure probability (\((1-p_{v})\)) for all the vulnerabilities and subtracting it from 1. This calculation assumes independent effects, representing the probability that at least one host vulnerability is successfully exploited. Protocol vulnerabilities: When a node (\(h\)) has only protocol vulnerabilities, the attack success probability is determined similarly. The failure probabilities (\((1-p_{pv})\)) of all the protocol vulnerabilities are multiplied, and the result is subtracted from 1, indicating the probability that at least one protocol vulnerability is exploited. Independent effects of vulnerabilities are assumed. Host and protocol vulnerabilities: When both host and protocol vulnerabilities are present, the probabilities of failure for each set are computed independently, multiplied together, and subtracted from 1. This yields the overall attack success probability for the node.