## 3 Probabilistic Joint Importance-Based Retrofit Strategy
### Computational Efficiency  
For the retrofit problem of the transportation network discussed in this paper, the majority of the computation cost comes from repeated calculation of the network functionality under various damage scenarios, since other numerical calculations are nearly negligible for a computer. In a specific transportation network, the computing time of network functionality under each damage scenario is similar. Therefore, the execution times of the functionality calculation procedure can be used to measure the computational cost of an algorithm, also known as time frequency \(T(n)\).  
In order to obtain the exact solution of Equation (5) in Section 2, the network functionality of each enumerated damage scenario must be calculated, so the time frequency is \(T(n)=(n_{s})^{n}\), considering the \(n_{s}\) possible damage states of each bridge. Since the number of bridges is at the exponential position, the computational cost increases significantly with the rise of problem dimension. Then, if MCS is used to estimate the network risk under each retrofit strategy and the sample size is set to be \(N_{s}\), the time frequency becomes \(T(n)=C_{n}^{m}\)\(N_{s}\). This is a typical NP-hard problem, so heuristic algorithms are usually needed to find the approximate optimal solutions. Taking the genetic algorithm [67] as an example, the time frequency can be reduced to \(T(n)=N_{Gen}\cdot N_{Chrom}\cdot N_{s}\), where \(N_{Gen}\) is the number of genetic generations and \(N_{Chrom}\) is the population number of each generation. However, as the dimension of problem rises, the required values of \(N_{Gen}\), \(N_{Chrom}\), and \(N_{s}\) for computation also increase, which makes the analysis of large-scale transportation networks still unacceptably time-consuming.  
In contrast, the method proposed in this paper aims at capturing the internal relationship between bridge damage uncertainties and the network seismic risk, rather than directly comparing the failure probabilities of various retrofit options. Based on the above illustration, the computation of \(JMI\) needs to establish \(P(\mathbf{\lambda}_{m})\), \(P(\varphi)\) and \(P(\mathbf{\lambda}_{m},\varphi)\), directly calculating which requires a time-consuming enumeration process. Considering the problem of feature subset selection is typically based on a limited set of data samples, we can also evaluate the importance of bridges based on a representative sample set. Given a set of stochastic samples that takes into account various uncertainties, the above probabilities can be estimated by counting the number of scenarios that match specific conditions in the sample set. This is a similar idea as using MCS to estimate the post-earthquake network functionality level instead of enumerating all damage scenarios. However, the difference is that the proposed joint importance-based method only relies on one set of stochastic samples, rather than re-sampling for each candidate retrofit strategy. So the computing time frequency of the proposed method is \(T(n)=N_{s}\), which only depends on the size of the sample. This high computing efficiency makes it easier to scale up to the analysis of large networks. In the following example, we also use the most common MCS for sampling to illustrate the applicability of the proposed method.