## 2 Methodology
### Uncertainty Propagation  
Significant uncertainties are associated with community-level resilience assessment and quantification, propagating from hazard intensity to damage estimation. The distribution of community-level functionality (\(F(\cdot)\)), conditioned on a predefined flooding scenario (\(S_{f}\)) characterized by the specific intensity flood depth, can be formulated according to [PERSON]'s work [16] as shown in Equation (3).  
\[P_{(F\leq F|S_{F})}=\iiint F_{F|DV}(f|i)f_{DV|DS}(i|j)f_{DS|IM}(j|h)f_{IM|S_{F} }(h|S_{F})di\;di\;dh \tag{3}\]  
where \(f_{IM|S_{F}}(h|S_{F})\), \(f_{DS|IM}(j|h)\), and \(f_{DV|DS}(i|j)\) are the probability density functions (PDFs) of flood depth conditioned for the flood scenario event, DS conditioned on flooding intensity, and damage value (DV) conditioned on DS, respectively. Additionally, \(\iiint F_{F|DV}(f|i)\) is the cumulative distribution function (CDF) of community-level functionality conditioned on DV. This formulation directly relates the community-level functionality to the flooding hazard scenario. In this convolution, an MCS is used to capture the uncertainties propagating from hazard intensity to damage estimation. This analysis was conducted in MATLAB, where the MCS framework simulated 10,000 iterations to convolve the probability density functions (PDFs) of flood depth, damage states, and damage values, as defined in Equation (3). The tool's matrix computation capabilities facilitated the integration of these distributions, yielding the cumulative distribution function (CDF) of community-level functionality. This process ensured a robust quantification of uncertainties across the virtual community's building sectors.  
To prevent the possible occurrence of significant population dislocation, the vital community functions must be in regular service even when flooding hazards occur. Population dislocation is the result of several factors. For example, if we consider four building sectors and the functional interdependencies among them, then we respectively use \(P_{d,i}\), \(P_{d,ij}\), \(P_{d,ijk}\), and \(P_{d,ijkl}\) to denote the population dislocation based on the loss of a single sector \(i\), two sectors (\(i\) and \(j\)), three sectors (\(i,j\), and \(k\)), and all four sectors (\(i,j\), \(k\), and \(l\)) simultaneously. Here, \(j\), \(k\), and \(l\) represent the remaining three sectors. Letting \(P_{i}\) represent the probability of functionality loss in sector \(i\), the expected population dislocation ratio, \(P_{pd}\), is calculated according to the total probability theorem and can be simplified by neglecting the last two higher order terms in the whole probability equation as follows:  
\[P_{pd}=\sum_{i=1}^{4}P_{d,i}P_{i}\prod_{j=1;j\
eq i}^{4}\left(1-P_{j}\right)+ \sum_{i,j=1;j\
eq j}^{4}P_{d,ij}P_{i}P_{j}\prod_{k=1;k\
eq i,j}^{4}(1-P_{k}) \tag{4}\]  
It can also be expressed in matrix form, as shown in Equation (5):  
\[P_{pd}=I_{1}\times 4[A]\{P\}_{1\times 4} \tag{5}\]  
It should be noted that Equation (4) has been simplified to this format based on the assumption that the probability of functionality loss for three or four essential building sectors' functions is typically small [8]. In Equation (5), [A] is a damage augmented adjacency matrix, which is explained in detail in the following section.