# A Vulnerability Assessment Approach for Transportation Networks Subjected to Cyber-Physical Attacks
## 3 Vulnerability Assessment Approach
### Cyber-Physical Attack Scenarios and Vulnerability States  
The vulnerability assessment approach commences with Activity 1 and 2 (refer to Figure 2). It includes the selection of a sensing area cyber-physical attack scenario (i.e., Activity 1) and its division into vulnerability states in physical and cyber space (i.e., Activity 2). The selection of cyber-physical attack scenarios necessitates the role of stakeholders who should have a bespoke knowledge of their system deficiencies [65], held on account of either their existing knowledge from previous cyberattacks [66] or by availing themselves of the use of public databases. Specifically, existing public catalogues, databases and studies that detail attack scenarios are commonly used and can be applied in this approach. For example, public catalogues (e.g., CAPEC catalogue [60]) and databases (e.g., Common Vulnerabilities and Exposures database [67]) detail the exploitation of vulnerabilities based on real-world events. In addition to these databases and catalogues, both theoretical [68] and experimental studies [57] detail attack scenarios against IoT devices.  
The division of the cyber-physical attack scenarios into vulnerability states in the physical and cyber spaces (i.e., Activity 2) relies on a BN attack graph. The BN attack graph models the vulnerability states that the attacker should exploit to succeed in the selected cyber-physical attack scenario (i.e., Activity 1) in a probabilistic manner. The BN attack graph is represented by a set of _vertices_ that describe the vulnerability states, _arcs_ that describe the conditional relationships (i.e., OR, AND) between the _vertices_ and _conditional probability tables_ that includes the probability score of every vertex. The use of BN attack graph enables us to model the probability of a vulnerability state based on vulnerability states that have occurred and thus update the posterior probabilities. The application of the chain rule of probability theory allows to factorize joint probabilities for a set of _v vertexes_, ranging from \(X_{1}\) to \(X_{\mathrm{v}}\), by using Equation (1),  
\[\mathrm{P}(X_{1},X_{2},\ldots,X_{\mathrm{v}})=\prod_{i=1}^{n}\mathrm{P}(X_{i} |\;\mathrm{Pa}(X_{i}))X_{i}\;X_{\mathrm{v}} \tag{1}\]  
where \(\mathrm{Pa}(X_{i})\) is the collection of all parent vertexes of the vertex, \(X_{i}\). A numerical example of a BN attack graph is shown in Figure 3 to demonstrate the application of a BN attack graph. In this abstract example, in order the attacker to succeed in the goal of the attack, the attacker should exploit Vulnerability C (i.e., \(V_{C}\)) given the prior exploitation of either Vulnerability A (i.e., \(V_{A}\)) OR Vulnerability B (i.e., \(V_{B}\)). For the purposes of this example, abstract numerical values are used. To simplify the generation of CPTs, an assumption of probability scores is made, namely, that \(V_{A}\) is equal to 0.3, \(V_{B}\) is equal to 0.4 and \(V_{C}\) is equal to 0.5. The True (T) and False (F) statements are used to indicate the condition of vulnerability state.  
Figure 2: Process diagram of vulnerability assessment approach.  
Based on Equation (1), the calculation of the successful exploitation of \(\rm V_{C}\) given that \(\rm V_{A}\) OR \(\rm V_{B}\) has been exploited is shown below:  
\[P(\rm Vc=T) = \sum\limits_{\{A,B\in T,\Phi\}}P(\rm V_{C}=T,\rm V_{A},\rm V_{B}\ )\] \[= P(\rm V_{C}=T,\rm V_{A}=\it F,\rm V_{B}=\it F)+P(\rm V_{C}=\it T, \rm V_{A}=\it F,\rm V_{B}=\it T)\] \[+P(\rm V_{C}=\it T,\rm V_{A}=\it T,\rm V_{B}=\it F)+P(\rm V_{C}= \it T,\rm V_{A}=\it T,\rm V_{B}=\it T)\] \[= 0+0.5\times 0.7\times 0.4+0.5\times 0.3\times 0.6+0.5\times 0.3 \times 0.4\] \[= 0.29\]  
To highlight the significance of a BN attack graph, the possibility of exploitation of \(\rm V_{C}\) given the initial exploitation of \(\rm V_{A}\) OR \(\rm V_{B}\) is lower (i.e., 0.29) than the initial belief of individual exploitation of \(\rm V_{C}\) (i.e., 0.50).