## 4 Current Approaches to Measure Resilience in Smart ICI
### Vulnerability Analysis Approach  
The vulnerability analysis-based approach is a primary method for investigating and assessing the resilience of interdependent critical infrastructures (ICIs). In their study [57], the authors conducted a vulnerability analysis of ICIs, demonstrating the impact of dependencies on connected infrastructures. These effects can lead to significant economic disruptions and losses in various aspects. Their work examines two types of vulnerability: structural and functional vulnerability. A high-level overview of the vulnerability analysis process for ICIs is illustrated in Figure 15.  
The topology is represented as a graph \(G=(V,E)\), where \(V=v_{i}\) is the set of vertices and \(E\) is the set of edges. The shortest path lengths connecting two nodes in the topology are denoted by \(d(v_{i},v_{j})\). The structural efficiency \(X(G)\) of infrastructure can then be defined as:  
\[X(G)=\frac{1}{N_{r}N_{l}}\sum_{i\in G_{r},j\in G_{l}}\frac{1}{d(v_{i},v_{j})} \tag{14}\]  
Here, \(N_{r}\) represents the total number of support nodes, and \(N_{l}\) denotes the total number of load nodes. When two nodes are entirely unconnected or become disconnected due to attacks, their shortest path length \(d(v_{i},v_{j})\) becomes infinite, and \(1/d(v_{i},v_{j})\) equals zero. A high value for \(X(G)\) indicates that the infrastructure is well-connected and has excellent performance. In the case of ICIs, the authors introduce a new parameter called  
Figure 14: Average Restoration curves, where (**a**) shows the average power performance over disaster time, and (**b**) shows the average gas performance over disaster time in Harris County, Texas, USA [31].  
Figure 15: The vulnerability analysis process of interdependent infrastructures [57].  
the interdependent effect. This parameter is defined as the absolute difference between the independent and interdependent efficiency, normalized by the maximum independent efficiency achieved at any extraction fraction:  
\[\text{Independent}_{effect}=\frac{|\text{Interdependent Efficiency}-\text{ Independent Efficiency}|}{max(\text{Independent Efficiency})} \tag{15}\]  
For the independent scenario, the performance can be calculated for any given removal fraction. However, in the interdependent case, a fixed fraction of nodes will be removed from both infrastructures, and subsequently, the interdependent performance is evaluated for each network.  
_Limitations:_ The primary challenges of the vulnerability analysis-based approach are related to data availability and scalability, which can be further elaborated as follows:  
**Data Availability:** Obtaining accurate and up-to-date data for interdependent critical infrastructures is often difficult due to security and privacy concerns. Additionally, proprietary information or sensitive data might not be publicly accessible, resulting in incomplete or outdated information for the analysis. This limitation can hinder the accurate assessment of the ICIs' resilience and vulnerability, leading to suboptimal decision making and risk management.  
**Scalability:** The vulnerability analysis-based approach can become computationally expensive when applied to large-scale and complex ICIs, as it involves calculating the shortest path lengths and interdependent effects for numerous nodes and edges. As the size and complexity of the infrastructure networks increase, so does the required computational power and time. This challenge can make it difficult to efficiently analyze large-scale interdependent infrastructures and provide timely results for effective decision making and response planning.  
These limitations highlight the need for alternative methods or improvements in data collection and computational efficiency to overcome the challenges associated with data availability and scalability in the vulnerability analysis-based approach for assessing the resilience of interdependent critical infrastructures.