## 2 Seismic Risk Assessment and Retrofit Strategy  
### Seismic Risk Assessment of Transportation Networks  
A typical transportation network is composed of nodes, links, and critical components. Nodes represent the population centers and road intersections, and links represent the road segments connecting the nodes. Critical components usually include bridges, tunnels, and vulnerable road sections. The seismic risk of a transportation network is determined by the uncertainties in both seismic scenarios and component damage states. The uncertainty of seismic scenarios is modelled by identifying possible earthquake events [46; 47] and predicting the ground motion intensities at specific sites [48; 49], and the damage uncertainty of a component is generally described by its structural fragility curve [50; 51; 52]. In this paper, to make the problem simple and clear, the following sections will be specific to the bridge retrofit decision problem, because the impact of bridge damage on post-earthquake network functionality is the most concerned in existing studies [53; 54; 55]. However, it is noted that the proposed method is a general framework for transportation networks that can be easily expand to the retrofit decision, including also other vulnerable network components. Considering a transportation network consists of \(n\) bridges, let \(\mathbf{\lambda}=\{\lambda_{1},\lambda_{2},\ldots,\lambda_{n}\}\) denote the damage-state vector of all the bridges in the network after an earthquake event and \(\lambda_{i}\) denotes the damage state of the \(i\)th bridge. Then the joint probability density function (PDF) for \(\mathbf{\lambda}\) is computed as:  
\[p(\mathbf{\lambda})=\int p(\mathbf{\lambda}|\mathbf{IM})\,p(\mathbf{IM}|EQ)\,p(EQ)\,dEQ \tag{1}\]  
where \(p(\mathbf{\lambda}|\mathbf{IM})\) is the PDF for \(\mathbf{\lambda}\) associated with a given vector of intensity measures \(\mathbf{IM}=\{IM_{1},IM_{2},\ldots,IM_{n}\}\), where \(IM_{i}\) denotes the intensity measure at the location of the \(i\)th bridge. \(p(\mathbf{IM}|EQ)\) is the PDF for \(\mathbf{IM}\), which can be estimated by the ground motion prediction equation under a given earthquake event \(EQ\), and \(p(EQ)\) is the occurrence probability of the earthquake event. In practical, the above probability transfer process can be approximated by MCS. Considering the number of damage states of a bridge under seismic hazard is \(n_{s}\), the conditional probability of the \(i\)th bridge being in damage state \(DS_{j}\) can be written as \(p\big{(}\lambda_{i}=DS_{j}\big{|}IM_{i}\big{)}\), for \(j=1,2,\ldots,n_{s}\). In general, the damage states of transportation network components can be divided into five levels [56] (i.e., no damage, slight damage, moderate damage, extensive damage and complete damage). Additionally, it should be noted that the seismic uncertainty model formulated above may not be applicable to complex components, which need to be modelled and analyzed separately, such as suspension bridges [57], cable-stayed bridges [58] and long-span bridges [59]. In the seismic analysis of transportation network, the damage states of bridges can be used to estimate their post-earthquake serviceability [60; 61], and the corresponding network functionality is calculated accordingly. Depending on different research concerns, existing post-earthquake functionality assessment methods of the transportation network include connectivity-based [62; 63], traffic delay-based [64; 65], throughput-based [25; 66], etc., and the analysis result of each method will ultimately be reflected in the functionality indicator. For illustration, Section 4 introduces a quantitative method to measure the post-earthquake network functionality in terms of emergency response connectivity, and the method is utilized in the example of Section 5. However, it should be noted that the decision-making framework proposed in this paper is applicable to all kinds of network functionality analysis methods.