## 5 Linear Hypothesis  
The hypothesis introduced by [23], which considers that in the case of no information, the recovery function may be represented with a linear function, are herein discussed. In particular, such assumptions may be particularly important for the definition of interdependencies. Figure 4 shows the case of linear hypotheses of Q\({}_{\text{N}}\), considering the functionality at the time of occurrence t\({}_{\text{IE}}\) 0 for all the quantities (as above specified). Figure 4: Linear hypothesis of Q\({}_{\text{N}}\)(t), (t\({}_{\text{IE}}\): time of occurrence of the earthquake E, RT: the repair time, Q\({}_{\text{i}}\)(t): recovery functions, t: time). Figure 3: Multidimensional definition of Q\({}_{\text{N}}\)(t), (t\({}_{\text{IE}}\): time of occurrence of the earthquake E, RT: the repair time, Q\({}_{\text{i}}\)(t): recovery functions, t: time). Such formulation may be described as:  
\[\text{Q}_{\text{N}};\text{t}=\sqrt{\sum_{\text{i}=1}^{\text{N}}\left(\frac{\text{ Q}_{\text{i}}}{\text{c}_{\text{i}}}\right)^{2}}+\text{t}_{\text{OE}} \tag{5}\]  
That represents a conical solid with different angles with the t-axis, depending on the coefficient c\({}_{\text{i}}\). In case of:  
\[\text{Q}_{\text{i}}\
eq 0\text{ and }\text{Q}_{\text{j}=\text{i}-1,\text{N}}=0 \tag{6}\]  
Then, Equation (5) simplifies as:  
\[\text{Q}_{\text{i}}=\text{c}_{\text{i}}(\text{t}-\text{t}_{\text{OE}}) \tag{7}\]  
That is the formulation represented in Figure 2 for the linear case. Now, it is worth noting that the coefficients c\({}_{\text{i}}\) represent the trends of reparation for the various dimensions since the rapidity of the recovery may differs among the various dimensions. Therefore, the angles of the surface of the cone with the t-axis (vertical axis) are described by c\({}_{\text{i}}\). In addition, such parameter may be calculated as:  
\[\text{c}_{\text{i}}=\frac{\text{Q}_{\text{F},\text{i}}}{\text{RT}_{\text{i}}} \tag{8}\]  
where  
\(\text{Q}_{\text{F},\text{i}}\) is the final value of the functionality that system i reaches at the end of the recovery process. This final functionality may vary from 1% to 100% respect with the originally functionality (that it in paper is considered 0). Sometimes recovery procedures allow an improvement relative to the original functionality, and this value can be higher than 100%, as described in [7]. Therefore, Equation (5) may be expanded in:  
\[\text{Q}_{\text{N}};\text{t}=\sqrt{\sum_{\text{i}=1}^{\text{N}}\left(\text{ RT}_{\text{i}}\frac{\text{Q}_{\text{i}}}{\text{Q}_{\text{F},\text{i}}}\right)^{2}}+ \text{t}_{\text{OE}} \tag{9}\]  
Such formulation accounts for the fact that the various systems may reach different levels of functionalities Q\({}_{\text{F},\text{i}}\) and that such level is reached at different repair times RT\({}_{\text{i}}\). In particular, applying Equation (9) instead of Equation (5) is fundamental in making decisions and defining recovery plans because the seismic resilience of the system depends on the RT of the various dimensions. Therefore, when seismic resilience is quantified, the value of RT needs to be calculated as:  
\[\text{RT}=\text{max}\{\text{RT}_{\text{i}}\} \tag{10}\]  
Consequently, (3) becomes:  
\[\text{SR}=\int_{\text{t}0\text{E}}^{\text{t}0\text{E}+\text{RT}}\frac{\sqrt{ \sum_{\text{i}=1}^{\text{N}}\left(\text{RT}_{\text{i}}\frac{\text{Q}_{\text{ i}}}{\text{Q}_{\text{F},\text{i}}}\right)^{2}}+\text{t}_{\text{OE}}}{ \text{RT}}\text{dt} \tag{11}\]  
It is worth considering that the choice to calculate RT as the maximum among the RT calculated for the various systems is a realistic assumption that we did. Another important assumption herein considered is that the functionality at the time of the event is 0 for all the cases, meaning that the solid has a vertex in V (t\({}_{\text{OE}}\), 0,... 0). This may be considered a limit of the present formulation that needs further development.