## 1 Introduction  
Natural and man-made hazards can cause devastating damage to civil infrastructures, such as transportation, water, gas, and power networks. Complex lifelines have been constructed densely throughout entire cities; hence, disconnecting main components could cause not only massive direct damage (e.g., repair costs), but also indirect damage (e.g., disruption of commercial and residential activities). In particular, bridge transportation networks are extensively constructed to meet the needs of commercial, industrial, and residential activities by providing products or supplies from source to destination nodes through complex transportation systems. The frequency of recent natural disastersnecessitates seismic risk assessment for complex bridge transportation networks and a post-hazard recovery strategy. Therefore, it is essential to analyze the seismic performance of bridge transportation networks, including disconnection of major components, and assess post-hazard performance under disaster conditions [1; 2]. For this purpose, numerous researchers have evaluated the performance of various lifeline networks including bridge transportation networks under seismic conditions. In previous studies, lifeline structures were analyzed through network system analysis, and the introduced analyses of system performance evaluation can be classified into connectivity- and flow-based analyses. Connectivity-based analysis identifies the connectivity between source and sink nodes by dividing the damage states of a network's components into bi-states (i.e., damaged or intact). Because such a connectivity-based method is relatively simple and easy to handle, numerous prior studies have evaluated networks based on connectivity analysis. For example, [PERSON] et al. [3] analyzed the seismic reliability of a gas distribution network in Italy, considering key components such as metering/pressure reducing stations of gas plants. [PERSON] et al. [2] performed the seismic risk analysis of a water transmission network in South Korea. In addition, [PERSON] et al. [4] evaluated the seismic hazards of urban-level power, water, and road networks, and [PERSON] et al. [5] estimated system reliability considering the interdependencies between water distribution and power networks. [PERSON] et al. [6] introduced Markov Chain Monte Carlo (MCMC) simulations to evaluate the reliability of a deteriorated highway bridge network. They also proposed a recovery priority of bridge components and recovery strategies for an entire network based on the network topology. [PERSON] et al. [7] proposed a performance evaluation methodology for bridge transportation networks based on a non-sampling-based methodology. However, such a connectivity-based network analysis has limited accuracy in predicting network performance because it does not reflect physical conditions such as the serviceability and capacity of component facilities according to their various damage states. To overcome this limitation, flow-based network analysis methodologies have been developed. For example, [PERSON] et al. [8] conducted a flow analysis of a water transmission network in South Korea by introducing a network update strategy based on pressure-driven analysis. [PERSON] and [PERSON] [9] and [PERSON] and [PERSON] [10] developed the GIRAFFE software to evaluate the system performance of five water network districts in the Los Angeles region, USA, against seismic conditions. Moreover, [PERSON] et al. [11] proposed a flow equation that considers the damage states of major components to evaluate the seismic safety of power networks in Sicily, Italy. In the case of transportation networks, [PERSON] and [PERSON] [12] developed a multi-group non-dominated sorting genetic algorithm and conducted research to explore critical post-disaster scenarios by employing network impact measures such as network flow capacity in the Eastern Massachusetts highway network, USA, and a Jeju transportation network, South Korea. [PERSON] et al.