## 4 Current Approaches to Measure Resilience in Smart ICI
### Dynamic System Approach  
System dynamic simulation approaches have emerged as a solution for scientists to study infrastructure services, the consequences of disturbances, and the associated downstream results. Dynamic simulations also possess the capability to assess the impacts of policies and regulations on infrastructure processes. Numerous studies have developed specific dynamic simulations for various interdependent infrastructures, including energy infrastructures (electricity, oil, and natural gas), telecommunications systems, transportation systems (waterways, highways, and rail), emergency services, banking and finance infrastructure, agricultural sector, water systems, shipping systems, and market sectors [20]. In [48], the authors present a method for evaluating the resilience of smart ICIs based on designing operative dependencies and examining the operational responses to disruptions. They represent the infrastructure as a network model, where nodes and arcs between them determine dependencies. This type of relationship is described as a network topology, represented by a dependency graph. A clear example of a dependency graph is shown in Figure 9.  
In this method, each infrastructure is defined based on the functionality loss rate \(\lambda\), recovery rate \(\mu\), and service threshold \(\sigma\). The authors conduct a simple examination of two interconnected infrastructures, as illustrated in Figure 10. If \(S1\) fails (i.e., its state threshold is exceeded), it ceases providing service to \(S2\), which subsequently begins to fail. Two primary scenarios exist: (1) \(S1\) recovers, meaning the state of \(S1\) returns within its threshold \(\sigma_{1}\), before \(S2\) fails; (2) \(S2\) fails before \(S1\) recovers. In the latter case, both infrastructures become inoperative and incapable of regaining functionality, resulting in a deadlock wherein each system awaits the other's functional restoration. The components of the interconnected infrastructures are combined into a hypercube \([0,1]^{n}\). This collection can be divided into four disjoint subsets: the operation region \(O\), the resilience region \(R\), the non-resilience region \(\bar{R}\), and the out-of-operation region \(\bar{O}\). The following ratio provides an evaluation of the network resilience capacity:  
\[c_{r}=\frac{M(R)}{M(R\cup\bar{R})}=\frac{M(R)}{M(R)+M(\bar{R}))}\in[0,1] \tag{10}\]  
Studying the resilience of interconnected infrastructures is an original objective linked to other challenging goals, such as designing resilient controllers. In situations involving large-scale events, understanding response dynamics in the face of an incident is crucial. In complex, realistic obstacle environments, the recovery to standard functionality must be executed within prescribed time constraints, e.g., the maximum functional blackout a given infrastructure can tolerate.  
_Limitations:_ This model faces limitations in its capacity to address all distinct stages and to incorporate every resilience capacity within each integrated model, as well as in overlapping with other concepts such as robustness, vulnerability, and fragility. Furthermore, scalability and accuracy also present significant challenges in this method.