Communities and their supporting civil infrastructure systems can be viewed as an assembly of, often numerous, interacting components. Tools that can identify components relevant for community disaster resilience can help to e- ciently allocate limited resources to reach community resilience goals. We use Sobol' indices to measure the importance of vulnerability and recoverability of components for disaster resilience of communities with interdependent civil in- frastructure systems. The initial component importance analysis requires no prior knowledge regarding component's vulnerability and recoverability. We rst rank components based on their importance, using their Sobol' indices. Secondly, we illustrate how the results of the component importance analysis can be used to improve community disaster resilience. Finally, we use compo- nent importance to show how model complexity can be reduced by abstracting less important components. Keywords: interdependent civil infrastructure systems; Sobol' indices; disaster resilience; sensitivity analysis; component importance 1. Introduction Developing strategies to improve disaster resilience of communities presents a new frontier in our e orts to face and survive natural and man-made disasters Corresponding author Email address: blagojevic@ibk.baug.ethz.ch (Nikola Blagojevi c) 1Department of Civil, Environmental and Geomatic Engineering, ETH Zurich, Switzerland Preprint submitted to September 15, 2021 Draft(EERI (2016); OECD (2019); RRMC (2019); NIBS & BSSC (2020)). Identifying components relevant for disaster resilience of a community can aid our e orts 5 to reach community resilience goals in several ways: i) component importance can guide resilience-improvement measures; ii) high- delity simulations can be performed for components that signi cantly a ect community resilience, while simpli ed, computationally less expensive, models can be used for less impor- tant components; iii) data acquisition can be prioritized based on component 10 importance; and iv) communities can be designed so that component impor- tance is evenly distributed, and, hence, no component is too big to fail (Taleb (2012)). Barker et al. (Barker et al. (2013)) proposed some of the rst component importance measures (CIMs) for system disaster resilience. They quantify re- 15 silience as a time-dependent ratio of the recovered system performance at the considered time step over the initial loss of performance following a disruption, and, propose two resilience-based CIMs. The rst CIM measures the weighted contribution of a disrupted component to the time until full system performance is restored. The weighing factor is the ratio between the loss of performance 20 due to the disruption of the considered component over the maximum loss of performance among all components. The second CIM measures how the in- vulnerability (i.e., robustness) of a component a ects the system recovery time. Since both proposed CIMs are stochastic, the authors used a modi ed Copeland score, based on comparing quantiles of CIM distributions, to rank components. 25 Baroud et al. (Baroud et al. (2014)), then, used these CIMs to quantify com- ponent importance for the resilience of inland waterway networks. Fang et al. (Fang et al. (2016)) pointed out that the resilience-based CIMs proposed by Barker et al. (Barker et al. (2013)) do not capture how improving components' recoverability could a ect system resilience. Furthermore, the way 30 in which system resilience is quanti ed is \memoryless" since it only includes the system performance at the considered time step. Hence, the system be- haviour before the considered time step is not included in the proposed system resilience metric. Fang et al. (Fang et al. (2016)) quantify resilience using cumu- 2 Draftlative system functionality that considers the entire time period starting with 35 initial recovery actions and spanning until the time step at which the resilience is measured. Based on this resilience metric, they propose two CIMs: resilience reduction worth and optimal repair time . Resilience reduction worth quanti es the loss in system resilience if the recovery of a component is delayed. Opti- mal repair time recognizes components whose recovery should be prioritized to 40 maximize system resilience. Xu et al. (Xu et al. (2020)) assume that after a disruption there are mul- tiple states that a network can reach to satisfy the minimum ow demand of the network users, following di erent recovery paths. For each recovery path, authors quantify component importance in two ways: i) by assessing the contri- 45 bution of a recovered component to the ow capacity of the network and ii) by calculating the ratio of the recovery time of the link to the total recovery time of the network. These two CIMs are, then, uni ed over all possible recovery paths and all considered disruptions to form a single stochastic resilience based CIM. CIMs are, then, ranked using Copeland score, similarly to (Barker et al. 50 (2013)). Espinoza et al. (Espinoza et al. (2020)) performed a probabilistic assessment of seismic resilience of the Northern Chilean electric power supply system. They quanti ed component importance as the change in system performance, if a component is assumed to be invulnerable. The system performance measure is 55 the expected interruption cost de ned for a certain disaster return period. A community is a system of systems, where each system consists of numerous components that interact both within and between systems (e.g., inoperability of an electric power plant can lead to the inoperability of components belonging to other systems, such as communication system or water supply system). These 60 interdependencies additionally increase the vulnerability of civil infrastructure systems (CISs), when exposed to man-made or natural disasters (Rinaldi et al. (2001); Kr oger (2008)). Hence, CIMs should account for these e ects. Almoghathawi and Barker (Almoghathawi & Barker (2019)) de ned two CIMs that consider both the vulnerability and the recovery of physically inter- 65 3 Draftdependent civil infrastructure systems. The authors simulate interdependency as a link between components from di erent networks, where the functionality of the dependent node depends on the functionality of the parent node. They de ne optimal recovery time CIM as the optimal time to recover a component to maximize the resilience of interdependent networks. The resilience reduc- 70 tion worth CIM considers the extent to which the inoperability of a component a ects the functionality of interdependent networks in a certain time step. Yu and Baroud (Yu & Baroud (2020)) consider network interdependencies to be uncertain and dynamic, where the probability of two nodes being interde- pendent is characterized using stochastic block models. Resilience is measured 75 using the same method as in (Barker et al. (2013)). Authors de ne compo- nent importance as the relative improvement in the resilience of interdependent networks at a certain time step, if the considered component is recovered. Com- ponents with higher importance are recovered rst. Component importance can be static, if component ranking is performed once before starting the recovery 80 process, or dynamic, if component importance is repeatedly calculated at each time step, thus, constantly updating the recovery strategy. The aforementioned system resilience related CIMs originate from reliability analysis or network theory. Formal global sensitivity analysis methods (Iooss & Lema^ tre (2015)) o er another way to assess the importance of components in 85 system. In this paper, we use Sobol' indices (Sobol (1993)), to quantify com- ponent importance for disaster resilience of communities with interdependent civil infrastructure systems. Sobol' indices belong to the class of variance-based methods { a subclass of global sensitivity analysis methods. Sobol' indices quan- tify input importance by decomposing the output variance into contributions 90 from each input and/or input group. Hence, the importance of an input or an input group is measured by how much its variance contributes to the output variance. Di erent component characteristics are often ranked using di erent CIMs: they are, therefore, not directly comparable. We quantify the importance of 95 both vulnerability and recoverability of components using the same tool { Sobol' 4 Draftindices. Therefore, the importance of di erent component characteristics is directly comparable. Since variance-based methods require inputs de ned as probability distri- butions, the sensitivity analysis presented here is implicitly probabilistic. As 100 opposed to quantifying component importance on a scenario-by-scenario basis, Sobol' indices quantify component importance over all possible scenarios con- sidered by the input probability distribution. We use the iRe-CoDeS extension (Blagojevi c et al. (2021b)) of the Re-CoDeS framework (Didier et al. (2018a)) to quantify disaster resilience of communities 105 with interdependent systems. The iRe-CoDeS resilience quanti cation model treats interdependencies among components in a dynamic manner: the ow of resources and services in the community determines how community components and systems interact at each instant of the post-disaster recovery process. 2. Disaster Resilience Quanti cation for Communities with Interde- 110 pendent Civil Infrastructure Systems Most disaster resilience quanti cation methods are based on the framework proposed by the Multidisciplinary Centre for Earthquake Engineering Research (MCEER), in which the disaster resilience of a system is quanti ed based on its post-disaster functionality (Bruneau et al. (2003)). Didier et al. (Didier 115 et al. (2018a,b)) extended the MCEER framework by presenting the Re-CoDeS framework that uses three measures of system functionality over time: the sup- ply, the demand and the consumption of a resource or a service a system is producing or making available. The Re-CoDeS framework accounts for the im- pact of the disaster on both the demand and the supply side of a system. Lack 120 of resilience (LoR) of a system regarding a resource or a service is de ned as the unmet demand of the system for the considered resource or service over the considered time frame. Disregarding the demand side of a system reduces the Re-CoDeS system LoR metric to the MCEER system functionality metric. 5 Draft2.1. Re-CoDeS framework for disaster resilience quanti cation 125 The Re-CoDeS framework consists of three main elements: the demand layer, the supply layer and the system resource and service distribution model. A system is de ned as a set of components that exchange resources and services, according to the system resource and service distribution model. A component of the system can belong to the supply layer (i.e., be a supplier), to the demand 130 layer (i.e., be a user), or to both the demand and supply layers (i.e., be a supplier and a user). A resource or a service produced and consumed in the system and considered in the resilience analysis is labelled in the following as R/S, for convenience. The system's demand for a R/S over time, Dsys;R=S (t), represents the sum 135 of the demands of all components i2f1;:::;Ig, where Iis the total number of components of the considered system and Di;R=S (t) is the demand of a compo- nent ifor a R/S at time t: Dsys;R=S (t) =X i2f1;:::;IgDi;R=S (t) (1) The supply capacity of a component ifor a R/S at time tis labelled as SC i;R=S(t). System supply capacity, SC sys;R=S(t), is similarly obtained by summing 140 the supply capacities of individual components: SC sys;R=S (t) =X i2f1;:::;IgSC i;R=S (t) (2) The system R/S distribution model describes the ow of R/Ss from sup- pliers to users. It considers the system's topology, the technical and physical laws governing the distribution of R/Ss (e.g., hydraulic ow laws), and possible dispatch/allocation strategies (e.g., prioritization of some users, restrictions for 145 others). The interaction between the demand and the supply layer, de ned by the system R/S distribution model, determines the R/S consumption of each component, Ci;R=S (t). It represents the amount of R/S that is consumed by the 6 Draftuser iat time t. The system's consumption, Csys;R=S (t), is then the sum of the R/S consumptions of all components: 150 Csys;R=S (t) =X i2f1;:::;IgCi;R=S (t) (3) The Re-CoDeS disaster resilience metric, Lack of Resilience of a system regarding a R/S, LoR sys;R=S , is amount of the unmet demand for the R/S the considered system is producing over the resilience assessment period, calculated as: LoR sys;R=S =Ztf t0(Dsys;R=S (t)Csys;R=S (t))dt=Ztf t0LoR sys;R=S (t)dt(4) wheret0andtfare the start and end times of system resilience assessment, 155 respectively. 2.2. Extending the Re-CoDeS framework to consider system interdependencies Blagojevi c et al. (Blagojevi c et al. (2021b)) proposed an extension of the Re-CoDeS framework to enable disaster resilience quanti cation for systems-of- interdependent-systems, the iRe-CoDeS framework. As in Re-CoDeS, a com- 160 ponent is the lowest level of system discretization considered in the iRe-CoDeS framework. Each component is characterized by instantaneous R/S supply and demand values. The supply values represent the supply capacities of a compo- nent at a given time in the resilience assessment interval. The demand values represent the amounts of R/Ss the component needs to operate and, conse- 165 quently, supply the system with its supply capacities. The iRe-CoDeS frame- work simulates interdependency e ects among components, by decreasing the supply capacity of those components whose demand cannot be met. At every time step of the resilience assessment interval any supplier of an R/S can meet the demand of any user for that R/S, and, hence, a component is dependent on 170 a R/S, not on a pre-determined component that produces that R/S. Therefore, interdependency relations between components are dynamic, not static. 7 DraftComponent properties that a ect the instantaneous R/S demand Di;R=S (t), supply capacity SC i;R=S(t) and consumption Ci;R=S (t), as well as the system LoR metric (Eq. 4) are damage level ,repair rate and functionality level . In 175 addition to these component properties, component locations, system topology and system operation also a ect the LoR metric. In the iRe-CoDeS framework and the subsequent sensitivity analysis, these component and system properties are considered to be independent of each other. 2.2.1. Damage, recovery and functionality of components 180 A continuous variable, labelled damage level represents the state of compo- nents' physical damage at each time step in the resilience analysis. Immediately after a disaster, an initial damage level is assigned to each component using that component's disaster vulnerability function, a conditional probability that a component will be at a certain (or larger) damage level if it experiences a 185 disaster of a given intensity (Blagojevi c et al. (2020a,b)). In iRe-CoDeS, the damage level of a component is a real number that varies from 0 { no damage, to 1 - complete damage. The recovery of a component from disaster-induced damage is simulated in iRe-CoDeS by decreasing its damage level at each time step (e.g., one day) of 190 the resilience analysis using component's repair rate , which characterizes the recoverability of the component. The repair rate of a component can span from 0 { the component never recovers, to 1 { component recovers in one time step, regardless of the initial damage level. The component is considered fully recovered once its damage level reaches 0. In iRe-CoDeS resilience analysis used 195 herein, the repair rate of a component is assumed to be constant over the entire recovery period and its damage level is decreased by the repair rate at each time step until the component is fully recovered. These assumptions are sucient to illustrate the Sobol' index component sensitivity analysis, but can be replaced by more realistic ones if such information is available (Blagojevi c et al. (2021a)). 200 The functionality level of a component describes how much of component's function has been lost due to damage and recovered due to repairs after a disas- 8 Draftter. iRe-CoDeS accommodates di erent relations between component damage and functionality levels. In this study, binary and linear damage-functionality component functions (Blagojevi c et al. (2021b)) are used, together with an as- 205 sumption that the supply capacities of a component are reduced in proportion to its current post-disaster functionality level. Therefore, as opposed to CIMs that assume components functionality is binary (e.g., the component is either operational or not), the iRe-CoDeS framework allows for partial component functionality. 210 2.2.2. Localities and links The spatial distribution of components is de ned using geographically lo- calized units, localities . An example of a locality is a district, if the considered system is a city, or a city, if the considered system is composed of several cities. Components can either be in a locality (e.g., a facility) or between localities. 215 Such components are named links (e.g., roads or pipes) that connect localities, enabling the transfer of R/Ss from one locality to the other. The transfer of R/S within a locality (e.g., between two facilities at the same locality) is assumed to be direct, without any links, and unconstrained. The functionality of a link can be pre-conditioned on the functionality of 220 another component, the link carrier (e.g., a bridge) carrying the link. The functional dependency between a link and a link carrier is also modelled using the supply/demand approach. The links that need to be carried have a demand forcarrier R/S, which the link carrier provides. 2.2.3. R/S distribution and interdependency modelling 225 Two groups of R/Ss are considered in iRe-CoDeS: utilities (e.g., electric power supplied by the electric power plants); and transfer services (e.g., elec- tric power transfer service supplied by electric power transmission lines). To obtain the utility R/S consumption of community components, the utility R/Ss are distributed among components at each time step following the utility R/S 230 distribution model. To account for the state of the distribution networks, the 9 Drafttransfer service supply capacity of links is used to form optimal paths among localities, as explained in (Blagojevi c et al. (2021b)). Before a utility R/S is transferred between components from di erent localities, the transfer service supply capacity of the optimal path connecting considered localities is compared 235 to the transfer service demand of such a utility transfer. If the supply is higher than the demand, the utility R/S is transferred, otherwise, the supply available to the component is set to zero and the component ceases to operate, simu- lating interdependency e ects. Additionally, the same interdependency e ect occurs if the amount of utility R/S transferred to the component is not enough 240 to meet its demand in full. The R/S distributions are repeated several times in a time step to simulate the interdependency feedback loops and thus-induced cascading component shutdowns. A component cascading shutdown sequence is assumed to unfold completely within one resilience analysis time step. Hence, the interdependency e ects do not propagate to the next time step. 245 Once the recovery target is met (e.g., all components are fully recovered), an iRe-CoDeS resilience analysis is ended. Then, the LoR values for each of the considered R/Ss in the community are computed following equations 1-4 to quantify the disaster resilience of the community and its interdependent systems. 3. Quanti cation of Community Component Importance for Disaster 250 Resilience Component importance for community disaster resilience is quanti ed using a Sobol' index variance-based global sensitivity analysis of community disaster resilience, evaluated using the iRe-CoDeS framework and the LoR metric, with respect to component vulnerability and recoverability properties. Recall that 255 component vulnerability de nes its initial disaster-induced damage level, while the component recoverability de nes the rate of component repair and recovery of its function in the community. These component properties directly a ect the LoR metric (Eq. 4), since they directly a ect component demand, supply and consumption. Additionally, intervening to change component vulnerabil- 260 10 Draftity and recoverability is localized (component-scale) and involves conventional planning, means and resources typically readily available to community sys- tem operators. Therefore, the majority of studies on component importance for system resilience focus on components' vulnerability and/or recoverability (e.g. Almoghathawi & Barker (2019); Espinoza et al. (2020)). However, other 265 component properties, such as their locations, and community properties, such as the topology and the operation of community systems, are important as well. Interventions to change these system-level properties are often global (community-scale) and involve extraordinary planning, means and resources. Based on the previous studies done on this topic and the aforementioned as- 270 sumptions regarding the ability of system operators to intervene in systems, the sensitivity analysis performed in this study involves only component vulnerabil- ity and recoverability properties. Variance-based sensitivity analysis requires that the component parameters to which the system sensitivity is investigated are de ned as probability dis- 275 tributions (Iooss & Lema^ tre (2015)). Probability distributions characterizing the vulnerability and recoverability of components depend on numerous factors, such as the considered hazard, the pre-disaster preparedness of the community, the type, age and level of maintenance of components, etc.. A variety of un- bounded or bounded, discrete or continuous probability distributions could be 280 assumed based on the available data about the components and the commu- nity, as well as considering the investigated disaster scenarios. This sensitivity study is constructed assuming a minimal amount of data regarding component's vulnerability or recoverability is available at the outset: these are the bounds of component property values. In the iRe-CoDeS framework, both the initial 285 damage level and the repair rate are bounded between 0 and 1. When only the bounds of a component property probability distribution are known, the appropriate decision for a sensitivity study is to assign a uniform probability distribution to both component properties to maximize the entropy with respect to the available information (Kapur (1989)). Therefore, it is assumed that the 290 initial damage level and repair rate of all components are uniformly distributed. 11 DraftThe importance of community components with respect to its disaster re- silience is quanti ed using Sobol' indices (Sobol (1993)), also described brie y in the following section. Sobol' indices quantify component property (input) im- portance by decomposing the variance of the community LoR metric (output) 295 into contributions from each input and input group. Hence, the importance of a component is measured by how much the variance of its properties contributes to the variance of the community LoR metric. Such variance contributions are comparable to each other, enabling the ranking of community components in terms of their importance to community disaster resilience, from those with 300 highest variance contributions (most important) to those with lowest variance contributions (least important) Sobol suggested several ways in which Sobol' indices can be used (Sobol (2001)). Sobol' index community component importance ranking is in itself an important decision-making basis. One way to use component importance 305 ranking is to focus on the most important components and improve their vul- nerability and recoverability properties. For example, if the sensitivity analysis shows that the initial damage level and the repair rate of a bridge are important for disaster resilience of a community, engineering resources can be invested to more precisely determine the current state of the bridge and obtain a more accu- 310 rate bridge vulnerability function. Alternatively, better recovery plans for that bridge can be developed prior to the disaster, thus shortening the time needed for bridge repair and return to function. Due to such actions, the probability distributions representing the bridge vulnerability and recoverability properties would be changed from a uniform to one that has a smaller variance, thus re- 315 ducing the contribution of the bridge to the community disaster LoR metric variance and reducing its importance for community disaster resilience. Another way to use component importance ranking is to focus on the least important components. If the variance of these component properties does not signi cantly a ect the variance of community disaster resilience, the value of 320 information about these components is low. It follows that community model complexity can be reduced by assuming the properties of these less important 12 Draftcomponents are deterministic (and have pre-assigned constant values). It follows further that resilience-improvement measures aimed at these component would have little or no e ect on the community disaster resilience. 325 Sobol' indices are usually obtained using Monte Carlo sampling (Sobol (2001)). However, the dimensionality and complexity of community disaster resilience models can grow quickly with the size of the community and detail of the anal- ysis. Thus, an important limitation of the Sobol' sensitivity analysis, the high computational cost, must be kept in mind. This issue is usually solved us- 330 ing meta-modelling techniques (Sudret (2008); Abbiati et al. (2021)). However, thanks to the streamlined implementation of the iRe-CoDeS framework on high- performance computers and the relatively low computational time of the case study, the computational cost of computing Sobol' indices using Monte Carlo sampling was relatively low (i.e., a few seconds per one iRe-CoDeS resilience 335 quanti cation). Thus, meta-models were not used in the case study. 4. Sobol' indices The Sobol' index sensitivity analysis can be applied to any deterministic model with multiple independent inputs and a nite-variance output. The main idea behind Sobol' indices is to de ne an expansion of the model output compris- 340 ing summands, which are functions of the input variables of increasing dimen- sions (Sobol (1993)). Then, the contribution of the variance of each summand to the total variance of the model output is quanti ed. Given an integrable function (i.e., a computational model) y=f(x), where yis a scalar output and x=x1;:::;x nis an input vector of ndimensions, f(x) 345 can be decomposed into: f(x) =f0+nX i=1fi(xi) +nX i=1nX j>ifij(xixj) +:::+f1;2;:::;n(x1;x2;:::;x n) (5) 13 DraftOr using a shortened notation: f(x) =f0+nX s=1nX i1<:::<i sfi1;:::;is(xi1;:::;x is) (6) where total number of summands is 2n1. The expansion is unique if the integrals of the summands with respect to their own variables are equal to zero: Z fi1;:::;is(x1;:::;x s)dxk= 0for 1ks (7) Therefore, summands can be calculated by integrating f(x): 350 Z f(x)dx=f0 (8) Z f(x)dxi=f0+fi(xi) (9) Z f(x)dxi;j=f0+fi(xi) +fj(xj) +fij(xixj) (10) where xistands for vector xthat excludes its i-th element. The higher order summands are calculated in the same manner. Iff(x) is square-integrable, it follows that all the summands are also square- integrable. Squaring the expansion and integrating over the input domain we get: 355 Z f2(x)dxf2 0=nX s=1nX i1<:::<i sZ f2 i1;:::;is(xi1;:::;x is)dx1;:::dx s (11) The total variance of the function is given by the equation: D=Z f2(x)dxf2 0 (12) It is, then, decomposed into contributions from each group of inputs: D=nX s=1nX i1<:::<i sZ f2 i1;:::;is(xi1;:::;x is)dx1;:::dx s=nX s=1nX i1<:::<i sDi1;:::;is(13) 14 DraftIn the expanded form: D=nX i=1Di+nX i=1nX j>iDij+:::+D1;2;:::;n (14) Now, Sobol' indices can be de ned as variances of inputs and/or input groups normalized by the total variance: 360 Si1;:::;is=Di1;:::;is D(15) All indices are non-negative and sum up to 1. The integer sde nes the order of the Sobol' index. For example, S1is the rst order Sobol' index that shows how much the variance of input 1 alone contributes to the variance of the output. Setting an input with a high rst-order index to a constant value will signi cantly reduce output variance. Hence, this input is considered important. 365 However, a low rst-order index does not necessarily mean that the input is not important: it may be important when interacting with other inputs. This is captured in higher order Sobol' indices containing the considered input. Total Sobol' indices (Homma & Saltelli (1996)) are used to account for the total e ect of an input, including the rst-order e ects and the interactions with other 370 inputs. The total Sobol' index of an input irepresents the sum of Sobol' indices of all orders that contain input i: ST i=X i2[i1;:::;is]Si1;:::;is (16) Saltelli et al. (Saltelli et al. (2010)) presented ecient methods to calculate Sobol' indices using Monte-Carlo (MC) sampling. In this study, the Python library SALib is used to calculate the MC-based Sobol' indices (Herman & 375 Usher (2017)). The total number of samples that needs to be generated to calculate rst-order and total Sobol' indices is: NTot=NInputDim(n+ 2) (17) 15 DraftwhereNTotis the total number of samples, NInputDim is the chosen number of samples per input and nis the number of model inputs. 5. Case Study 380 The subject of the case study is a virtual community (Figure 1) with 3600 inhabitants, located in 9 Building Stock Units (BSUs), supported by three in- terdependent CISs. The considered CISs are: the Electric Power Supply System (EPSS), the Water Supply System (WSS), and the Cellular Communication Sys- tem (CCS). Modelling how these CISs function is simpli ed for this case study, 385 as shown in Figure 2. The EPSS consists of Electric Power Plants (EPPs) producing the electric power (EP) R/S and Electric Power Transmission Lines (EPTL), transferring the EP to the users. The CCS consists of Base Station Controllers (BSCs) that manage the cellular communication trac between Base Transceiver Stations (BTSs), through the High Level Communication (HLC) 390 R/S. BTSs then supply the community inhabitants with communication ser- vices, labelled as Low Level Communication (LLC) R/S. Both communication R/Ss are measured in Erlang [E] (Erlang (1925)) and are transferred wirelessly. The WSS provides Potable Water (PW) and Cooling Water (CW) R/Ss to the virtual community. Potable Water Pipes (PWPs) transfer the PW from Potable 395 Water Facilities (PWFs) to the BSUs and community inhabitants who occupy them. Cooling Water Pipes (CWPs) are used to transfer CW from the Cooling Water Facilities (CWFs) to the end users, which are components in other CISs. PWPs, CWPs and EPTLs share the same links shown in Figure 1, where the community localities are represented by circles and the links between localities 400 are shown using black lines. Further details on this virtual community can be found in (Blagojevi c et al. (2021b)). The virtual community experiences a disaster and recovers from it. The ef- fects of a disaster are represented by an initial damage level of each community component. The recovery of each component is assumed to proceed indepen- 405 dently and simultaneously, at a component repair rate. Since the damage level 16 Draft Figure 1: Virtual Community set-up. Components in localities are Electric Power Plants (EPPs), Building Stock Units (BSUs), Base Station Controllers (BSCs), Potable Water Facil- ities (PWFs), Base Transceiver Stations (BTSs) and Cooling Water Facilities (CWFs) of a component is reduced by its repair rate at each time step following the dis- aster, the recovery time of a component depends on its initial damage level and its repair rate. In this study, impeding factors are not considered, and recovery of all components starts immediately after the disaster. The assumption that 410 recovery is linear and uninterrupted is a simpli cation of the actual recovery process occurring in a community after a disaster. More sophisticated recovery modelling methods can also be implemented (Blagojevi c et al. (2021a)). From a sensitivity study standpoint, each virtual community component is characterized by two inputs, its initial damage level and its repair rate. These 415 two inputs describe the vulnerability and the recoverability of that component, respectively. The virtual community is composed of 32 facilities, 2 bridges, 27 PWPs, CWPs and EPTLs, summing up to in total 115 components. Each dis- aster scenario is then characterized by a total of 230 inputs representing the speci c initial damage and repair rate of each community component in that 420 disaster scenario. The iRe-CoDeS framework is used to quantify the disaster re- 17 Draft Figure 2: Simpli ed CIS functionality, a) electric power supply system, b) high level and low level communication systems, c) potable water supply system, d) cooling water supply system silience of the virtual community in each scenario by computing ve LoR values (Eq. 4), one for each of the ve considered utility R/Ss. The virtual community disaster resilience sensitivity is assessed separately for each utility R/S using Sobol' indices, setting up ve sensitivity analyses each with a 230-dimensional 425 input and one-dimensional output space. The MC-based Sobol' indices are computed using 10,000 samples per input, creating 2,320,000 (Eq. 17) distinct disaster scenarios, each assessed in a separate iRe-CoDeS run. To prevent long recovery times, and to decrease computational time, the lower bound of the uniform distribution for the repair rate was set to 0.01 instead of 0, leading to 430 a maximal recovery time of 100 time steps. The required computations were performed on the Euler HPC cluster, part of the ETH parallel computing facil- ities. The computation involved running 5000 jobs in parallel, where each job ran 464 disaster scenarios. One job was performed using 4 processing threads, required on average 6 (mostly from 4 to 10) minutes of computational time and 435 around 7.5GB of memory. 6. Results and Discussion First, we use the computed total Sobol' indices to assess the disaster re- silience of the virtual community regarding each considered R/S and discuss the ndings to identify and quantify CIS interdependencies. Then, we examine 440 two ways to use the computed Sobol' indices to improve community disaster re- silience. On one hand, we focus on the most important community components 18 Draft(those that have high total Sobol' index values) with respect to community disaster resilience and examine how changing the parameters of the probabil- ity distributions that describe the relevant characteristics of these components 445 changes the disaster resilience of the entire community. On the other hand, we focus on the least important community components and show how to use this information to reduce the complexity of the community model without sacri c- ing the quality of its disaster resilience assessment. 6.1. Evaluation of component importance using total Sobol' indices 450 Figure 3 to Figure 7 present mean values and 95% con dence intervals of the community component total Sobol' index values obtained from disaster re- silience sensitivity analyses for the ve considered R/Ss (i.e., EP, LLC, HLC, PW and CW). For clarity, only 30 (out of 115) components with the highest total Sobol' index values are shown in each gure. Components are labelled on 455 the x-axis. For example, EPP 301 refers to the electric power plant at locality 301 and Bridge (201, 301) refers to the bridge connecting localities 201 to 301 in Figure 1. 6.1.1. EP LoR Figure 3 presents the sensitivity analysis outcome for electric power disaster 460 resilience of the virtual community: it depends primarily on the suppliers of EP R/S, the two EPPs in localities 301 and 305. The total Sobol' indices show that the repair rate is more important than the initial damage level of these suppliers. Bridges connecting localities 301 and 302, and 201 and 301 are also recognized as important. This is a consequence of the virtual community layout, namely, 465 locality 301 is connected to the rest of the network only by the two bridges that carry EPTLs and CWPs to the EPP (Figure 1). Further, several EPTLs are important, out of which the EPTL connecting localities 105 and 205 stands out. This is also a consequence of the topology of the virtual community. Several paths between di erent localities include the link connecting localities 105 to 470 205. Hence, when optimal EP paths are calculated in iRe-CoDeS, this link is 19 Draftoftentimes included, and hence a ects the EPTL transfer supply capacity. Figure 3: Total Sobol' indices for EP LoR 6.1.2. HLC LoR Figure 4 presents the component total Sobol' indices for high-level commu- nication community disaster resilience. The sensitivity analysis indicates that 475 the suppliers of HLC, the BSCs, are most important. As in the case of EP disaster resilience sensitivity, the recoverability had higher importance than the vulnerability of the components. Once again, several EPTLs emerged as im- portant for HLC disaster resilience, recognizing that supply of EP is crucial for the BSCs. Additionally, CWFs have notable Sobol' index values. This is due 480 to the assumed BSC demand for CW (Blagojevi c et al. (2021b)). Importantly, these ndings show that the virtual community CISs are interdependent, and that the iRe-CoDeS framework is capable of identifying and quantifying such interdependencies. 20 Draft Figure 4: Total Sobol' indices for HLC LoR 6.1.3. LLC LoR 485 The component importance ranking for community disaster resilience re- garding the low-level communication R/S is presented in Figure 5. The inputs with the highest importance are the initial damage level and the repair rate of several EPTLs, BSCs and CWFs. These EPTLs have a high Sobol' index because they transfer the EP needed to operate the BTSs (for LLC), as well as 490 BSCs and CWFs. Note that BTSs need HLC supplied by the BSCs to operate, while the BSCs, in turn, need CW supplied by CWFs to operate. Therefore, the iRe-CoDeS framework was able to represent, and the Sobol' index sensi- tivity analysis was able to recognize such multi-level interdependencies among components of di erent CISs. The larger number of LLC suppliers, the BTSs, 495 compared to the number of suppliers of CW and HLC in the virtual commu- nity can explain why the vulnerability and the recoverability of the BTSs is less important than the vulnerability and recoverability of the components that the BTSs depend on. In fact, there are only two BSCs and 4 CWFs, while there are 11 BTSs. Therefore, there is more redundancy in the production of LLC due to 500 21 Drafta larger number of suppliers. Figure 5: Total Sobol' indices for LLC LoR 6.1.4. PW LoR Figure 6 presents the component importance for community disaster re- silience regarding the potable water R/S. Evidently, it depends almost exclu- sively on the vulnerability and recoverability of PW suppliers, the PWFs. This 505 means that the supply of PW is independent of other community CISs. No- tably, PWPs that connect the PWFs to the BSUs (and the inhabitants) are not identi ed as important. This is a consequence of the highly redundant water pipe network of the virtual community (Figure 1). 22 Draft Figure 6: Total Sobol' indices for the PW LoR 6.1.5. CW LoR 510 Component importance for community disaster resilience regarding the cool- ing water R/S is shown in Figure 7. The total Sobol' indices are high for the repair rate and the initial damage levels of CWFs, the supplier of CW. Notably, several EPTLs and the two bridges also have high Sobol' index values. These EPTLs connect the CWFs with the EPPs and the bridges carry the EPTLs, 515 enabling the ow of EP from EPPs to the CWFs. This indicates the depen- dence of the community WSS on its EPSS. The CWPs, connecting CWFs and the components of the EPSS and CCS that need CW to operate, are not as important because the redundancy of the virtual community cooling water pipe network diminishes the reverse dependence of community EPSS and CCS on 520 the WSS's distribution network (Figures 3, 4). 23 Draft Figure 7: Total Sobol' indices for CW LoR 6.2. Making decision to increase community disaster resilience using total Sobol' indices The information about community component importance for its disaster resilience can be used to select the components whose improvement may improve 525 community disaster resilience the most. In this case study, let us assume that a decision maker wants to increase the virtual community disaster resilience regarding the EP R/S. Figure 3 shows that the recoverability characteristics of the four components (the two EPPs, the EPTLs from 105 to 205 and the Bridge from 201 to 301) have the highest importance. Let us imagine that 530 the uninformed priors of these four input variables { the uniform distributions, are now updated to a Beta distribution with parameters a= 9 andb= 3, an informed posterior. This may be based on component inspections, newly developed recovery plans, and/or better vulnerability and recoverability models of these four important components. The posterior distribution of the inputs has 535 a higher mean and a lower variance than the uninformed prior (Figure 8), and, hence, the mean and the variance of the output, (i.e., the EP LoR), as well as the 24 Draftvalue of the total Sobol' indices, also change. To illustrate this, the sensitivity analysis is run, rst, with all inputs being uniformly distributed. Then, the simulations are run with the repair rate of two EPPs, the EPTL from 105 to 540 205 and the Bridge from 201 to 301, sampled according to the Beta distribution. The di erence in the LoR EPand the total Sobol' indices are shown in Figure 8 and Figure 9, respectively. From Figure 8, it can be seen that the mean LoR is reduced by 24%, from LoREP= 174:34MWh toLoREP= 132:53MWh and its standard deviation is reduced by 34%, from 131.66MWh to 87MWh. 545 Furthermore, the 95% quantile of the LoR EPdistribution is reduced by 30%, from 387MWh to 273.4MWh. This is a result of changing the distribution of only 4 component properties, out of a total of 230 considered component properties. Additionally, due to the change in variance of the 4 component properties probability distributions, their total Sobol' indices, and, hence, their 550 importance for community disaster resilience, is now reduced (Figure 9). The EPTLs from localities 205 to 305, 201 to 301, 301 to 302 and 302 to 303, are now the 4 most important components. Compared to the sensitivity analysis results presented in section 6.1.1, their importance increased from being 6th to 10th most important components to 1st to 4th ranking. Interestingly, the bridge 555 from locality 301 to 302 remained the 5th most important component in both sensitivity analyses. 25 Draft Figure 8: Comparison between the LoREPwith uninformed priors and with informed poste- rior distributions for important components. Figure 9: Total Sobol' indices after modifying the probability distributions of important components 26 Draft6.3. Reducing model complexity using total Sobol' indices The information about community component importance for its disaster resilience can also be used to signi cantly reduce the complexity of the virtual 560 community model. Low-ranked component properties have small total Sobol' index values, indicating that the variance of their properties (vulnerability and recoverability) has a very small e ect on the variance of the community LoR metric. Recall that Figures 3-7 show only 30 components with the highest- ranked total Sobol indices. For EP LoR (Figure 3), 62 of the 115 iRe-CoDeS 565 virtual community components have a total Sobol index value for both their damage level and repair rate smaller than 0.001. These components may be considered unimportant for the community EP LoR metric. Further, the com- plexity of the virtual community model can be reduced by not considering the damage and recovery of these components for EP resilience assessment. To 570 demonstrate this, two sets of 10,000 simulations were run: rst, by setting the damage level and repair rate of the 62 unimportant components to 0 (no dam- age, no repairs needed) across all scenarios, while the damage levels and repair rates of the remaining 53 components followed a uniform distribution (i.e., the simpli ed model); and second, sampling the damage level and repair rate of all 575 component properties from a uniform distribution (i.e., the original full model). The histograms of LoR EPfor the full and the simpli ed model are compared in Figure 10. The curves tted to the histograms vary only in the peak LoR EP values, with the simpli ed model undershooting the full model by less than 2%. The shapes of the tted distributions are quite similar. 580 27 Draft Figure 10: Comparison between the LoREPdistribution obtained using the full virtual com- munity model and the simpli ed model, that does not consider the damage and recovery of 62 unimportant community components. 7. Conclusion Intervening to increase disaster resilience of communities with complex and interdependent systems and numerous hard-to-inspect components is dicult. Decision-makers are likely to undertake the least costly and most bene cial inter- ventions, preferably localized at a small number of components whose resilience- 585 related characteristics can be improved using already available means and re- sources. In this paper, we presented a sensitivity analysis approach aimed at identifying the community components whose importance for community dis- aster resilience is the highest, and whose resilience improvement is likely to improve the resilience of the entire community the most. The presented ap- 590 proach is based on a compositional, bottom up, approach to quantifying the disaster resilience of communities with interdependent systems, facilitated by the iRe-CoDeS framework (Blagojevi c et al. (2021b)), and a Sobol' index based 28 Draftcomponent sensitivity quanti cation and ranking. This combination makes it possible to investigate the sensitivity of the community disaster resilience with 595 respect to disaster-relevant properties of each of the community components. An important feature of the presented approach is that it works with mini- mal initial knowledge about the resilience-relevant component vulnerability and recoverability properties. The presented approach to quantify component importance for community 600 disaster resilience was illustrated using a case study on a moderately com- plex virtual community. The case study con rmed that the iRe-CoDeS frame- work and the Sobol' indices are capable of modelling the interdependencies among community components and systems, and re ecting such interdependen- cies when quantifying component importance for community disaster resilience. 605 Additionally, the case study demonstrated two important bene ts of using the Sobol' index based sensitivity analysis to rank community components by their importance. One is the ability to select a few most important (highest-ranked) components for resilience-improving interventions and to demonstrate the e ect of these interventions in a before-after comparison. The other is the ability 610 to signi cantly reduce the complexity of the community resilience model by treating the (often numerous) less important (low-rank) components determin- istically. The principal drawback of the proposed approach is the complexity and dimensionality of the community resilience model and the computational e ort 615 needed to compute Sobol' indices. However, use of meta-models for Sobol' index computation o ers a computationally ecient way to handle signi cantly more complex problems than the one analysed in the presented case study. Even though the presented approach can be used to identify components rel- evant for community disaster resilience, investing into improving the resilience- 620 related properties of high-ranked component may not be the option with the best cost-bene t ratio. It is possible that improving mid-ranked components, for which resilience-improvement measures are more a ordable than those for high-ranked components, represents a more rational investment strategy for in- 29 Draftcreasing community disaster resilience. This issue needs further investigation. 625