## 4 Current Approaches to Measure Resilience in Smart ICI
### Network-Based Approach  
The network-based approach seeks to understand data on dependencies by representing various components of the infrastructure as nodes within a network, where the presence of a connection between two nodes is depicted by a link connecting them. In paper [31], the authors present a resilience metric based on the infrastructure performance method during a period from 0 to \(T\), which may involve multiple disruptive events, as illustrated in Figure 13. For a disruption event, time is divided into three phases: (1) a disaster prevention phase (\(t_{0}\leq t\leq t_{1}\)), (2) a loss propagation phase (\(t_{1}\leq t\leq t_{2}\)), and (3) an evaluation and recovery phase (\(t_{2}\leq t\leq t_{3}\)). These three phases can sequentially demonstrate the resistant, absorptive, and restorative capabilities of the infrastructure experiencing the disaster, and these capacities, shown from 0 to \(T\), collectively define infrastructure resilience during that period. The overall resilience is then calculated based on the target performance curve \(P_{T}(t)\) and the original performance curve \(P_{R}(t)\):  
\[R(T)=\int_{0}^{T}P_{R}(t)dt/\int_{0}^{T}P_{T}(t)dt \tag{12}\]  
Note that various times \(T\) allow for different types of resilience: past resilience, present potential resilience, and anticipated potential resilience. This model primarily examines  
Figure 12: Static economic resilience quantification [54]. the present potential resilience, where infrastructure components are constant through 0 to \(T\) and similar to those currently in place. For the case in which the present potential resilience \(PT(t)\) is a fixed value 1.0 and when a risk has its existence governed by a Poisson process [56], the expected resilience \(E[R(T)]\) is:  
\[E[R(T)]=E\left[\frac{\int_{0}^{T}P_{R}(t)dt}{T}\right]=1-E\left[\frac{1}{T}\sum _{n=1}^{N(T)}IA_{n}(t_{n})\right]=1-\lambda E[AI] \tag{13}\]  
where \(E[*]\) is the expected value; \(n\) is the event experience number; \(N(T)\) is the total number of event experiences during \(T\); \(t_{n}\) is the occurrence time of the \(n\)th event. Additionally, the random variable \(IA_{n}(t_{n})\) is defined as the space between the original performance curve and the target performance curve, called the impact area for the \(n\)th incident occurring at time \(t_{n}\); \(E(IA)\) is the expected impact area following the accidents considering all possible forces; and \(\lambda\) is the occurrence rate of the accidents per year. In order to demonstrate a network-based framework for interdependent systems resilience assessment, the authors of [31] investigate the gas system in Harris County, Texas, USA. The gas compressors, gas storage facilities, gas delivery facilities, gas receipt facilities, and gas pipeline junctions are modeled as nodes, while the gas pipeline segments are described as links. The authors also investigate the effects resulting from the interdependencies among the systems by applying five different restoration strategies: random restoration, independent restoration, power first and gas second restoration, gas-aimed restoration, and power and gas compromised restoration. Figure 14 displays the average restoration curves of the power and gas systems in Harris County, Texas, USA, under a hurricane scenario. This network-based approach is useful for understanding the complexities of interdependent infrastructure systems, helping decision makers plan and implement effective resilience strategies based on the unique characteristics and relationships between different infrastructure components. By considering various restoration strategies and analyzing their impacts on the performance of interdependent systems, this approach provides valuable insights into the most effective ways to enhance resilience in the face of disruptive events. _Limitations:_ While the network-based approach offers valuable insights for understanding interdependent infrastructure systems, it may not be suitable for all types of ICIs due to the specific topology of certain systems.