Earth Syst Dynam 9 1159ndash1168 2018httpsdoiorg105194esd-9-1159-2018copy Author(s) 2018 This work is distributed underthe Creative Commons Attribution 40 License

Linking resilience and robustness and uncovering theirtrade-offs in coupled infrastructure systems

Mehran Homayounfar1 Rachata Muneepeerakul1 John M Anderies2 andChitsomanus P Muneepeerakul3

1Department of Agricultural and Biological Engineering University of Florida Gainesville Florida USA2School of Sustainability and School of Human Evolution and Social Change Arizona State University

Tempe Arizona USA3independent researcher Gainesville Florida USA

Correspondence Mehran Homayounfar (homayounfarufledu)and Rachata Muneepeerakul (rmuneepeufledu)

Received 28 December 2017 ndash Discussion started 15 January 2018Revised 26 July 2018 ndash Accepted 29 August 2018 ndash Published 5 October 2018

Abstract Robustness and resilience are concepts in systems thinking that have grown in importance and pop-ularity For many complex social-ecological systems however robustness and resilience are difficult to quantifyand the connections and trade-offs between them difficult to study Most studies have either focused on qualitativeapproaches to discuss their connections or considered only one of them under particular classes of disturbancesIn this study we present an analytical framework to address the linkage between robustness and resilience moresystematically Our analysis is based on a stylized dynamical model that operationalizes a widely used con-ceptual framework for social-ecological systems The model enables us to rigorously delineate the boundariesof conditions under which the coupled system can be sustained in a long run define robustness and resiliencerelated to these boundaries and consequently investigate their connections The results reveal the trade-offsbetween robustness and resilience They also show how the nature of such trade-offs varies with the choiceof certain policies (eg taxation and investment in public infrastructure) internal stresses and uncertainty insocial-ecological settings

1 Introduction

The concepts of ldquoresiliencerdquo and ldquorobustnessrdquo have grownconsiderably in popularity as desirable properties for a widerange of systems Terms like ldquoresilient communitiesrdquo andldquorobust citiesrdquo have been used more frequently in public dis-course (eg Chang and Shinozuka 2004 Longstaff et al2010 Chang et al 2014) The UKrsquos Water Act 2014 even in-cluded ldquoprimary duty to secure resiliencerdquo as one of the gen-eral duties of its Water Services Regulation Authority (Wa-ter Act 2014) Growing with that popularity is some confu-sion and potential misuse of the terms ldquorobustnessrdquo and ldquore-siliencerdquo due to imprecision vagueness and multiplicity oftheir definitions Such a lack of consistency and rigor hindersadvances in our understanding of the interplay between thesetwo important system properties

Relatively speaking robustness has been defined moreconsistently and rigorously ndash as it can be linked to a morefamiliar concept of sensitivity For example according toCarlson and Doyle (2002) robustness in engineering systemsrefers to the maintenance of system performance either whensubjected to external disturbances or internal uncertain pa-rameters In other words in robust systems performance isless sensitive to disturbances or uncertainty

Robustness may very well be a desirable property of asystem but it seems to come with a price Recent researchshows that tuning a system to be robust against certain dis-turbance regimes almost always reduces system performanceand likely increases its vulnerability to other disturbanceregimes (Ostrom et al 2007 Anderies et al 2007 Bode1945 Csete and Doyle 2002 Wolpert and Macready 1997)

Published by Copernicus Publications on behalf of the European Geosciences Union

1160 M Homayounfar et al Linking resilience and robustness and uncovering their trade-offs

Now if resilience is also a desirable property of the samesystem does it also come at the expense of performance androbustness Put it another way is there a trade-off amongperformance robustness and resilience Such a trade-off ifit exists is a crucial consideration for governing andor man-aging social-ecological systems (SESs)

But resilience as alluded to above is trickier to defineAccording to Holling (1973) resilience refers to the amountof change or disruption required to shift the maintenanceof a system along different sets of mutually reinforcingprocesses and structures In other words resilience can bethought of as how far the system is from certain thresholds orboundaries beyond which the system will undergo a regimeshift or a quantitative change in system structure or identityHolling (1996) categorized resilience into two types engi-neering resilience which refers to the ability of a system toreturn to a steady state following a perturbation and ecolog-ical resilience which refers to the capacity of system to re-main in a particular stability domain in the face of pertur-bations The latter category is used by many researchers todiscuss resilience of SESs or more generally coupled in-frastructure systems (CISs Carpenter et al 2001 Anderieset al 2006 Folke 2006 Biggs et al 2012 Barrett and Con-stas 2014 Redman 2014 Walker et al 2002 2004 Gun-derson and Holling 1995 Berkes and Folke 1998 Carpen-ter et al 1999a b Scheffer et al 2000 Berkes et al 2003Carpenter and Brock 2004 Janssen et al 2004 Folke etal 2002 2010 2016 Cote and Nightingale 2012 Mitra etal 2015 Cumming and Peterson 2017) The CISs term hasbeen introduced to generalize the notions of coupled naturalhuman systems (CNHSs) and SESs in this context infras-tructure is broadly defined to include human-made socialand natural infrastructure (eg see Anderies et al 2016)The problem is that these CISs are complex and thus identi-fying thresholds and potential regime shifts associated withthose thresholds is often difficult if not impossible In manycases major aspects of resilience in CISs may not be directlyobservable and must be actualized indirectly via surrogate at-tributes (Carpenter et al 2005 Kerner and Thomas 2014)Recent significant advances have been made toward identi-fying early-warning signals that indicate whether a criticalthreshold is being approached for a wide class of systems(Scheffer et al 2009 2012) Still there are gaps in our un-derstanding of how indicators of resilience and robustnesswill behave in more complex situations This lack of a rigor-ous metric for resilience makes the investigation into theirconnections interplay and trade-offs with robustness andperformance impossible

But these knowledge gaps need to be filled if one wishesto make advances in understanding the interplay between so-cial dynamics and planetary boundaries Given the magni-tude of impacts that human activities have on pushing Earthsystems toward their planetary boundaries we need clearerunderstanding of how social and biophysical factors come to-gether to define the nature of these boundaries This paper is

a step in that direction In particular we will build on recentwork that mathematically operationalizes the robustness of aSES framework (Anderies et al 2004) into a formal stylizeddynamical model (Muneepeerakul and Anderies 2017) Wewill exploit the relative simplicity of the model to rigorouslydefine robustness and resilience of the coupled system Themodeled system will be subject to uncertainty in social andecological factors which will affect the well-defined robust-ness and resilience thereby enabling us to investigate the in-terplay and trade-offs between these important properties aswell as investigate how the nature of the interplay and trade-offs are affected by policies implemented by social agents

2 Methods

Here we analyze a mathematical model developed byMuneepeerakul and Anderies (2017) by subjecting the cou-pled system to uncertainty in ecological and social factorsThe model captures the essential features of a system inwhich a group of agents shares infrastructure to producevalued flows Such a system is the archetype of most ifnot all of human sociality groups produce infrastructurethat they cannot produce individually (security defense ir-rigation canals roads markets financial systems coordina-tion mechanisms etc) that significantly increases productiv-ity The challenge is maintaining this shared infrastructure(eg decaying infrastructure is a major problem in the US atthe time of writing ASCE RCIA Advisory Council 2013)The model allows for mathematical definitions of the bound-aries of policy domain that result in a sustainable system inwhich both human-made and natural infrastructure can bemaintained over the long run Based on these boundariesand uncertainty in the exogenous factors we define met-rics of resilience and robustness associated with each policychoice and investigate the trade-off between them The basicmodel presented by Muneepeerakul and Anderies (2017) isdescribed in the Appendix

Here a policy is defined as a combination of taxation levelC and the proportion of tax revenue invested in infrastruc-ture maintenance y that the public infrastructure providers(PIPs) decide to implement in the system The infrastructure(eg canals) enable resource users (RUs) to produce valuedgoods from a natural resource The two uncertain factors arethe replenishment rate of the natural resource g and the wagew that RUs would earn from working outside the system ndasha combination of g and w defines a ldquosocial-ecological set-tingrdquo or simply ldquosettingrdquo There are two boundaries that oncecrossed will cause the system to collapse The first boundaryis called PIP participation constraint (PPC) when the PIPsmust invest too much in maintaining the public infrastruc-ture (exceeding the opportunity cost of wP) andor cannotretain enough revenue for themselves they will abandon thesystem for another The second boundary is the stability con-dition of the nontrivial equilibrium point (ie the ldquosocietyrdquo

Earth Syst Dynam 9 1159ndash1168 2018 wwwearth-syst-dynamnet911592018

M Homayounfar et al Linking resilience and robustness and uncovering their trade-offs 1161

in which both PIPs for example the state and RUs for exam-ple citizens participate in and public infrastructure is suffi-ciently maintained in a long run) Together these two bound-aries delineate a set of policies (Cndashy combinations) that cor-respond to sustainable outcomes The resilience metric tobe developed below can be thought of as a metric of howfar the system is from these boundaries As the two exoge-nous factors defining settings namely g and w change thetwo boundaries and thus the resilience metric also changewith them How sensitive the resilience metric is to thesesettings is used to define robustness Here it is worth not-ing that while it is possible to examine recovery-based re-silience (or the so-called ldquoengineering resiliencerdquo) this paperfocuses on regime shift-based resilience (traditionally calledldquoecological resiliencerdquo) and its robustness Quantification ofand trade-offs between resilience and robustness is a novelconcept that requires expositional clarity Presenting severalmetrics of resilience let alone studying their trade-offs withpotentially different metrics of robustness may confuse thematter and dilute the key messages we attempt to convey Assuch in what follows we will focus on developing a metricfor regime shift-based resilience With the scope of analysisclarified and suitably bounded we will now define resilienceand robustness more formally

21 Resilience metric

Direct measurement of the above-mentioned resilience as aspecified form of resilience (Walker et al 2004) in SESs isdifficult because boundaries and thresholds that separate do-mains of dynamics for SESs are difficult to identify (Carpen-ter et al 2005 Scheffer et al 2009 2012) In this stylizedmodel however such boundaries can be clearly identified bythe stability condition (SC) and the PPC Here we are inter-ested in the resilience of a systemrsquos ability to provide suffi-cient livelihoods for the PIPs and RUs The basin of attrac-tion for system resilience is defined by those system states(ie infrastructure state) in which this is possible and thesesystem states are directly mapped to the SC and PPC Wewill thus define resilience metrics based on the SC and PPCboundaries Here our goal is to develop resilience metricsthat can be meaningfully compared to one another As suchwe identify some desired properties that guide the defini-tions of these resilience metrics First they should be zero attheir respective boundaries Second positive values indicategreater resilience of the system in a desirable state Thesefirst two properties align with how resilience has been mea-sured ie the distance from the boundary of a basin of at-traction (eg Anderies et al 2002 Carpenter et al 1999a)Third to facilitate the consideration of relative risks associ-ated with different types of regime shifts that the system maybe facing the metrics should be comparable in magnitudeThese properties guide us toward the following definitions ofthe resilience metrics

We define the resilience of the system against abandon-ment by PIPs as follows

RPPC = (πPwP)minus 1 (1)

where πP is the net revenue that PIPs collect andwP is the op-portunity cost that they will earn if they choose to work withanother system Positive values of RPPC indicate that the sys-tem is resilient against being abandoned by PIPs while neg-ative values indicate that the system will eventually collapsedue to the PIPsrsquo abandonment It is important to note thatπP results from the coupled dynamics of the CIS this meansthat RPPC has already integrated the dynamics of infrastruc-ture resource and RUs (Eqs A1 A4 and A5) making it ametric of the system not of an individual component

Numerical analysis of the model indicates that the equilib-rium becomes unstable when the following RouthndashHurwitzcondition (eg May 2001 Kot 2001) is violated

Dminus T(J11J22+ J21J12+ J23J32+ J13J31

)gt 0 (2)

where D T and J are determinant trace and entries re-spectively of the Jacobian matrix of the dynamical system(Eqs A1 A4 and A5) evaluated at the nontrivial equilib-rium point (Eq A6) ndash when such an equilibrium point existsHere it is worth noting that we focus on the equilibrium pointrelated to the nontrivial sustainable long-term outcome An-alyzing other bifurcations related to other equilibria may bemathematically interesting but it could make the study lessaccessible and dilute its key message about the resiliencendashrobustness trade-off of different policies aiming at keepingthe system in the basin of attraction of the non-trivial sus-tainable long-term outcome

Following the guideline provided by the three desirableproperties above we rearrange terms in Eq (2) and define theresilience of the system against instability (increased proba-bility of collapse of infrastructure) as follows

Rstability =D∣∣T (J11J22+ J21J12+ J23J32+ J13J31

)∣∣ minus 1 (3)

This formulation is parallel to that of the first resiliencemetric (Eq 1) it possesses the following three propertiesRstability of zero indicates the boundary between stability andinstability positive Rstability means the system at the equilib-rium point is stable and the magnitudes of Rstability are com-parable to those of RPPC (see Fig 1) Note that Rstability toois determined from the coupled dynamics of the CIS thismeans that it has integrated the dynamics of infrastructureresource and RUs (Eqs A1 A4 and A5)

This allows us to meaningfully define the overall systemresilience as the minimum between the two resilience met-rics namely

Rsystem =

Min

[RPPCRStability

]RPPCRstability ge 0

0 otherwise (4)

wwwearth-syst-dynamnet911592018 Earth Syst Dynam 9 1159ndash1168 2018

1162 M Homayounfar et al Linking resilience and robustness and uncovering their trade-offs

0 02 04 06 08 1C

0

02

04

06

08

1

y

-05

0

05

1

15

2

0 02 04 06 08 1C

0

02

04

06

08

1

y

01

02

03

04

05

06

07

08

0 02 04 06 08 1C

0

02

04

06

08

1

y

0

02

04

06

08

(a) (b) (c)

Figure 1 Resilience metrics for a specific setting (a gndashw combination) inside the sustainable region in the policy space (ie Cndashy plane)(a) RPPC contours (b) Rstability contours and (c) Rsystem contours The black star in panels (a) (b) and (c) indicates the policy with thehighest RPPC Rstability and Rsystem respectively

Equation (4) implies that Rsystem is positive only when thenontrivial equilibrium point (Eq A6) exists and is stable oth-erwise the system is considered not resilient and denoted byRsystem = 0 Rsystem thus represents the tension between thePIPs being too greedy (high C and low y) whereby they getclose to the stability boundary and ldquonot greedy enoughrdquo (lowC and high y) whereby they get close to the PPC given a par-ticular choice for wP Note that the values of w and wP rep-resent the socioeconomic embedding of the CIS Thereforethe biophysical structure of the CIS along with the socioe-conomic context in which it is embedded co-determine themaximum resilience that can be achieved Given that the non-trivial equilibrium point exists and is stable the following ap-plies If the system is at a greater risk of being abandoned bythe PIPs (and eventually collapsing) Rsystem = RPPC if thesystem is at a greater risk of becoming unstable (and even-tually collapsing) Rsystem = Rstability Figure 1 illustrates therelationships between RPPC Rstability and Rsystem

22 Linking robustness and resilience

As discussed earlier robustness can be thought of as the op-posite of sensitivity A commonly used measure of sensitiv-ity is variance Thus variance of a given function under spe-cific disturbance or uncertainty regimes may be used to in-dicate robustness of that function against those disturbancesor uncertainty regimes (robustness of what to what) In thiscase the system function of interest is the system resilienceRsystem By choosing Rsystem as our function we can usefullylink these concepts If we use a ball and cup metaphor for re-silience the robustness of resilience refers to the degree atwhich the geometry of the cup changes as a result of externaldisturbances andor parameter changes However as we willargue below relating high variance in Rsystem to low robust-ness may be misleading and should not be used in evaluatinga given policy By definition the variance treats ldquogood de-viationsrdquo and ldquobad deviationsrdquo from the mean equally For

functions with preferred values such as resilience or profitvalues greater than the mean and those lower should not betreated in the same way Specifically contribution to a highvariance from a heavy tail in the good direction should notbe translated to less robustness This problem also arises inassessing financial risk what makes an asset risky is the val-ues on the ldquobad tailrdquo of the distribution (ie low or nega-tive profits) This has motivated more and more analyses toswitch to considering other measures of risk such as the con-ditional value at risk in evaluating their portfolios of invest-ment (Rockafellar and Uryasev 2000 Krokhmal et al 2002Sarykalin et al 2014 AitSahlia et al 2011 Zymler et al2013) Intuitively this means that the shape of the ldquocuprdquo canbe asymmetric and we need to take this into account

Following this logic we propose to use a ldquobelow-meanmeanrdquo as a new robustness metric the mean of all resiliencevalues lower than the mean This new definition of the robust-ness metric has several desirable features First it can now beappropriately thought of as a robustness metric in the sensethat the higher the value the more robust the system (unlikethe variance for which low variance means high robustness)Second by using the mean as the threshold value for bad de-viations we remove some arbitrariness associated with pre-scribing a certain quantile (eg 5th or 10th quantile) in cal-culating the conditional value at risk Third it still carriessome information about the sensitivity of the resilience met-ric to outside factors ndash the information that variance conveysthat is the higher the below-mean mean (ie the bad devia-tions from the mean are small and the below-mean mean isclose to the mean) the less sensitive ndash and thus more robustndash the resilience metric

In this study we subject the modeled system to uncertaintyin one natural factor and one social factor namely the natu-ral replenishment rate of the resource g and the payoff thata RU earns from working outside the system w Thus weare computing how the resilience of the system to shocks

Earth Syst Dynam 9 1159ndash1168 2018 wwwearth-syst-dynamnet911592018

M Homayounfar et al Linking resilience and robustness and uncovering their trade-offs 1163

Figure 2 Variation of Rsystem of a CIS with a fixed policy (Cy) over 10 000 settings associated with uncertainty characterized byg isin [75125] w isin [075125] (a)Rsystem surface and (b)Rsystem contours The values ofRsystem are used to calculate the meanmicroRsystem and the below-mean mean microRltmicro In this particular case the resilience does not change much when g is greater than about 100 but becomesmore sensitive to both g and w when g is lower than 100

andor variation in state variables changes as the parametersg and w change (ie we are uncertain about the underlyingsocial-ecological setting of the system) In particular we as-sume that g is uniformly distributed over the range [75 125]and w is uniformly distributed over the range [075 125]A social-ecological setting or setting is defined as a com-bination of g and w For a given policy (a Cndashy combina-tion) we calculate Rsystem for 10 000 settings (ie 10 000gndashw combinations see Fig 2) Then from these 10 000 val-ues of the resilience metric Rsystem we calculate the meanmicroRsystem = E

[Rsystem

] and use it as the resilience metric of

the coupled system with a given policy and the below-meanmean microRltmicro = E

[Rsystem|Rsystem lt microRsystem

] as the metric

for robustness of resilience This metric measures the robust-ness of the capacity of the system to cope with variation instate variables IHM R and U to fundamental uncertaintyabout the underlying setting of the system

3 Results

The surfaces and contours of the system resilience metricmicroRsystem and the associated different policies (Cndashy) over thepolicy space are shown in Fig 3a and b respectively Thepolicies with sustainable outcomes are located in the mid-dle of the policy space with microRsystem peaking in the centerand declining as policies become more extreme in either di-rection Our analysis also shows that microRsystem is more or lessproportional to the fraction of settings (gndashw combinations)under which the system with that particular policy (a Cndashycombination) results in a sustainable outcome (Rsystem gt 0)A similar concept has been used in the robust decision mak-ing literature (eg Groves and Lempert 2007 Bryant andLempert 2010)

The surfaces and contours of the robustness of microRltmicro as-sociated with different policies over the policy space are

shown in Fig 3c and d respectively The microRltmicro ldquolandscaperdquois more irregular having two local maxima with one beingmore dominant than the other The region with high robust-ness appears to be in the same general areas as the regionwith high resilience These features reflect the nonlinear in-terplay between the model parameters and model structureand may affect the nature of the trade-off between robustnessand resilience reported in Figs 4 and 5

We explore the interplay between microRsystem and microRltmicro inFig 4 Figure 4 shows that there are no perfect policies inthe sense that no policies yield both maximum resilience andmaximum robustness Recall that the robustness indicateshow sensitiveRsystem itself is to uncertainty in the underlyingsetting of the system (eg g and w) The best policies arethose along the Pareto frontier in the resiliencendashrobustnessspace among this set of Pareto-optimal policies an increasein resilience is necessarily accompanied by a decrease in ro-bustness clearly illustrating the trade-off between robustnessand resilience Fig 5 illustrates where the Pareto-optimalpolicies are located in the policy space

4 Discussion and conclusions

In this paper we exploit the simplicity of a stylized modelto quantitatively link resilience and robustness by computinghow the CIS resilience to shocks in state variables changeswith parameters In this way we compute the robustness ofCIS resilience to uncertainty in the underlying CIS settingThe resilience metric developed here is a measure of how farthe CIS is from the boundaries beyond which it will collapseThe model affords us with expressions of these boundarieswhich clearly show how social and biophysical factors inter-play to define these boundaries With a concrete definition ofresilience resilience itself can be considered as the ldquoof whatrdquoin the ldquorobustness of what to whatrdquo notion In particular we

wwwearth-syst-dynamnet911592018 Earth Syst Dynam 9 1159ndash1168 2018

1164 M Homayounfar et al Linking resilience and robustness and uncovering their trade-offs

0 02 04 06 08 1C

0

02

04

06

08

1

y

01

02

03

04

05

06

07

0 02 04 06 08 1C

0

02

04

06

08

1

y

01

02

03

04

05

(a) (b)

(d)(c)

Figure 3 The mean microRsystem and the below-mean mean of Rsystem microRltmicro over the entire decision space (a) Surface of the resiliencemetric microRsystem (b) Contours of microRsystem (c) Surface of the robustness the below-mean mean (microRltmicro) (d) Contours of microRltmicro

0 02 04 06 08

R-system

0

01

02

03

04

05

06

(R-s

yste

mlt

R-s

yste

m)

Figure 4 Resiliencendashrobustness trade-off Each point representsmicroRsystem and microRltmicro of the coupled system with a given policy Theblack dots represent a set of Pareto-optimal policies

use the below-mean mean of the quantitatively defined re-silience metric as the metric of robustness Consequently thisenables us to rigorously investigate the interplay between thetwo important but not always well-defined system proper-ties A key finding is the fundamental trade-off between re-silience and robustness there are no perfect policies in gov-erning a CIS only Pareto-optimal ones Specifically policiesdesigned to maximize the resilience of a CIS to shocks ontimescales at which the state variables play out may be verysensitive to being wrong about our understanding of the un-derlying dynamics of the CIS in question

Importantly we hope this work will stimulate further ad-vances in rigorous studies of CISs that address such sub-tle policy-relevant questions a few of which we briefly dis-cussed here More dimensions can be considered in definingPareto-optimality Figure 5 may give an impression that theset of Pareto-optimal policies is confined to a small region inthe policy space which would imply that PIPs do not havethat many choices ndash even in a simple CIS like the one stud-ied But that would be a wrong impression In addition toresilience and robustness (as defined here) a policy makeror a social planner may be interested in other types of ro-bustness with different ldquoof whatrdquo and ldquoto whatrdquo components

Earth Syst Dynam 9 1159ndash1168 2018 wwwearth-syst-dynamnet911592018

M Homayounfar et al Linking resilience and robustness and uncovering their trade-offs 1165

0

02

04

08

1

01

02

03

04

05

06

07

06y

160 800 20 40C

01

02

03

04

05

0

02

04

08

1

06y

160 800 20 40C

(a) (b)

Figure 5 Pareto-optimal policies represented by black dots in the policy space (Cndashy plane) superimposed with resilience (microRsystem ) (a)and robustness (microRltmicro) contours (b)

They may also be concerned about other system propertieseg productivity user participation etc As more dimensionsare considered the set of Pareto-optimal policies grow In thesame spirit as that of the work done here these other dimen-sions should be defined rigorously

This work also lends itself to more rigorous studies ofldquoadaptive governancerdquo In the present study the governancestructure represented by a policy (a combination of C andy) is fixed A natural next step is to explore if a policy isallowed to change how one may improve the resilience androbustness of a CIS andor alter the nature of their inher-ent trade-offs For example if C and y are to be functionsof other factors eg resource availability and outside incen-tives what functional forms should they take to improve thesystemrsquos resilience and robustness Indeed in the absence oftransparent metrics attempts to explore such adaptive poli-cies are severely limited

Additionally agents in a CIS may have the capacity tochange their behavior in response to changes in policy en-vironmental conditions technological changes and so on Inthis study strategic behavior and the decision-making pro-cess are assumed unchanged in the analysis Adaptation instrategic behavior of agents will subsequently alter the natureof resilience its robustness and their trade-offs Capturingsuch effects of adaptation requires structural changes to themodel eg in terms of specification of payoffs or even theformulation of the dynamical equations With such adaptivesocial agents how should one devise adaptive governance toenhance resilience and robustness of a CIS Addressing sucha question is a theoretically intriguing future research direc-tion with great practical implications

In keeping with the theme of ldquosocial dynamics and plane-tary boundaries in Earth system modellingrdquo our results shedlight on how social and biophysical factors may interplay todefine boundaries of a sustainable coupled infrastructure sys-tem While the modeled system here is admittedly simpleour methodology and results constitute a step toward quanti-tatively and meaningfully combining social and biophysicalfactors into indicators of boundaries of more complex sys-tems Just as in this work once those boundaries are clearlydefined calculation and discussion of resilience and robust-ness can become concrete

Data availability No data sets were used in this article

wwwearth-syst-dynamnet911592018 Earth Syst Dynam 9 1159ndash1168 2018

1166 M Homayounfar et al Linking resilience and robustness and uncovering their trade-offs

Appendix A

Basic model Here we briefly describe the basic model pre-sented by Muneepeerakul and Anderies (2017) The modelshows dynamic behavior of three principal variables namelythe state of the public infrastructure IHM resource level Rand the fraction of time a user makes use of infrastructureU through Eqs (A1 A4 and A5) The schematic diagramof this system of equations is shown in Fig A1

In this context IHM depends on PIPs in term of mainte-nance cost and has a positive relationship with the capacityof users to create resource flows Eq (A1) illustrates the dy-namics of IHM as follows

dIHMdt=M ( )minus δH (IHM ) (A1)

where δ is the infrastructurersquos depreciation rate andH (IHM )states functional relationship of public infrastructure and pro-ductivity of each RU According to Muneepeerakul and An-deries (2017) many shared infrastructures can be modeledby threshold functions Given that H (IHM ) shows thresh-old behavior they used a piecewise linear function to capturesuch behavior through Eq (A2)

H (IHM )=

0IHM lt I0

hIHM minus I0

Imminus I0I0 le IHM le Im

hIHM ge Im

(A2)

where h represents the maximum amount of harvest byeach user under no restriction and I0 and Im are the lowerbound and upper bound thresholds of IHM respectivelyAlso M ( ) is the maintenance function (Eq A3) and de-pends on the social structure of the system

M ( )= micro2yCpRUNH (IHM ) (A3)

In Eq (A3) given the number of users N RUNH (IHM ) isthe total harvest from the natural infrastructure The RUs selltotal harvest at price p to generate revenue Subsequentlythey assign a proportion C of revenue to PIPrsquos for their con-tribution Meanwhile the PIPrsquos spend proportion y of C onmaintaining public infrastructure through the maintenancefunction M ( ) Also micro2 is the maintenance effectivenessof PIPrsquos investment

The second variable is the resource levelR They assumedthe dynamics of resource to be

dRdt=G (R)minusRUNH (IHM ) (A4)

Natural infrastructure is assumed to invoke the conservationlaw comprising of regenerating capacity (G (R)= gminus dR)and total unit of harvest RUNH (IHM ) The definition pre-sented for G is the simplest model for natural infrastructurewhere g and d are the natural replenishment and the lossrates respectively

HM IHM

Resource users

N

Natural infrastructure

Public infrastructure

R G(R)wp

infrastructurePublic

providersy

w

M()

UNI

minusδ

HMUNRH( )

IHMH( ) IHMH( )

IHMH( )

IHMCpUNRH( )

I

Figure A1 Schematic diagram of the dynamical system modelTaken from Muneepeerakul and Anderies (2017)

The strategic behavior of the resource users (RUs) is cap-tured by employing a replicator equation Indeed replica-tor dynamics provide modelers with simple realistic so-cial mechanism where agents follow and replicate better-offstrategies The two possible strategies considered for RUsare staying inside the system with the associated payoff ofπU = (1minusC)pRH (IHM ) or leaving the system with thepayoff of πw = w According to the replicator equation

dUdt= rU (1minusU ) (πU minusw) (A5)

The replicator equation represents the fraction of time thatRUs assign to working inside system given C and y LikeRUs there is also two alternatives for PIPs working insidethe system or working for another CIS which leads to sys-tem failure Meanwhile C and y characterize the strategyor policy of PIPs The PIPs will participate in this coupledsystem only when πp = (1minus y)pCRUNH (IHM )ge wP Inother words the PIPs maintain the system when they arebetter-off than working outside This condition is termed thePIP Participation Constraint (PPC)

Based on the system of three differential equations(Eqs A1 A4 and A5) the sustainable equilibria ie long-term system outcomes that satisfy the stability condition andPPC can be expressed as follows

ilowastHM =yCUlowastNRlowast

gH(I lowastHM

)Rlowast =

g

d

(1minus

ilowastHM

yC

)

Ulowast =(1minusC)yC

φ1ilowast

HM (A6)

where ilowastHM =IlowastHMδ

micro2pg(indicates dimensionless) and φ1 =

pgwN

a dimensionless group representing the relative lucrativenessof the system namely the ratio of potential income ndash withthe entire resource flow turned into income ndash relative to out-side wage The results reported in this study are based on thefollowing parameter values h= 00005 δ = 01 I0 = 03Im = 3 g = 100 d = 002 N = 1000 r = 015 p = 10w = 1 wp = 100 and micro2 = 0001

Earth Syst Dynam 9 1159ndash1168 2018 wwwearth-syst-dynamnet911592018

M Homayounfar et al Linking resilience and robustness and uncovering their trade-offs 1167

Author contributions MH RM JMA and CPM designed thestudy MH carried out the analysis MH and RM prepared themanuscript with contributions from all co-authors

Competing interests The authors declare that they have no con-flict of interest

Special issue statement This article is part of the special issueldquoSocial dynamics and planetary boundaries in Earth system mod-ellingrdquo It is not associated with a conference

Acknowledgements John M Anderies and Rachata Muneepeer-akul acknowledge the support from the grant NSF GEO-1115054Rachata Muneepeerakul and Mehran Homayounfar acknowl-edge support from the Army Research OfficeArmy ResearchLaboratory The research reported in this grant was supported inwhole or in part by the Army Research OfficeArmy ResearchLaboratory under award no W911NF1810267 (Multi-UniversityResearch Initiative) The views and conclusions contained in thisdocument are those of the authors and should not be interpreted asrepresenting the official policies either expressed or implied of theArmy Research Office or the US Government The authors alsothank the editor and three anonymous referees for their constructiveand useful comments

Edited by Jonathan DongesReviewed by three anonymous referees

References

AitSahlia F Wang C J Cabrera V E Uryasev S and FraisseC W Optimal crop planting schedules and financial hedgingstrategies under ENSO-based climate forecasts Ann Oper Res190 201ndash220 2011

Anderies J M Janssen M A and Walker B H Graz-ing Management Resilience and the Dynamics of aFire-Driven Rangeland System Ecosystems 5 23ndash44httpsdoiorg101007s10021-001-0053-9 2002

Anderies J M Janssen M and Ostrom E A Frameworkto Analyze the Robustness of Social-Ecological Systems froman Institutional Perspective Ecol Soc 9 18 httpwwwecologyandsocietyorgvol9iss1art18inlinehtml 2004

Anderies J M Ryan P and Walker B H Loss of ResilienceCrisis and Institutional Change Lessons from an Intensive Agri-cultural System in Southeastern Australia Ecosystems 9 865ndash878 httpsdoiorg101007s10021-006-0017-1 2006

Anderies J M Rodriguez A A Janssen M A and CifdalozO Panaceas Uncertainty and the Robust Control Frameworkin Sustainability Science P Natl Acad Sci USA 104 15194ndash15199 httpsdoiorg101073pnas0702655104 2007

Anderies J M Janssen M and Schlager E Institutions andthe performance of coupled infrastructure systems Int J Com-mons 10 495ndash516 httpsdoiorg1018352ijc651 2016

ASCE RCIA Advisory Council Report card for Americarsquos Infras-trcutre Tech Rep American Society of Civil Engineers 2013

Barrett C B and Constas M A Toward a Theoryof Resilience for International Development Applica-tions P Natl Acad Sci USA 111 14625ndash14630httpsdoiorg101073pnas1320880111 2014

Berkes F and Folke C Linking Social and Ecological Systemsfor Resilience and Sustainability Linking Social and Ecologi-cal Systems Management Practices and Social Mechanisms forBuilding Resilience 1 13ndash20 1998

Berkes F Colding J and Folke C Navigating Social-EcologicalSystems Building Resilience for Complexity and change Build-ing p 393 httpsdoiorg101016jbiocon200401010 2003

Biggs R Schluumlter M Biggs D Bohensky E L BurnSil-ver S Cundill G Dakos V Daw T M Evans L SKotschy K Leitch A M Meek C Quinlan A Raudsepp-Hearne C Robards M D Schoon M L Schultz L andWest P C Toward Principles for Enhancing the Resilienceof Ecosystem Services Annu Rev Env Resour 37 421ndash448httpsdoiorg101146annurev-environ-051211-123836 2012

Bode H W Network Analysis and Feedback Amplifier DesignBell Telephone Laboratories Series p 551 1945

Bryant B P and Lempert R J Thinking inside the BoxA Participatory Computer-Assisted Approach to Sce-nario Discovery Technol Forecast Soc 77 34ndash49httpsdoiorg101016jtechfore200908002 2010

Carlson J M and Doyle J Complexity and Ro-bustness P Natl Acad Sci USA 99 2538ndash2545httpsdoiorg101073pnas012582499 2002

Carpenter S Walker B Anderies J M and Abel N FromMetaphor to Measurement Resilience of What to WhatEcosystems 4 765ndash781 httpsdoiorg101007s10021-001-0045-9 2001

Carpenter S R and Brock W A Spatial Complexity Resilienceand Policy Diversity Fishing on Lake-Rich Landscapes EcolSoc 9 8 2004

Carpenter S R Ludwig D and Brock W A Management of Eu-trophication for Lakes Subject to Potentially Irreversible ChangeEcol Appl 9 751ndash771 httpsdoiorg10230726413271999a

Carpenter S Brock W and Hanson P Ecological and SocialDynamics in Simple Models of Ecosystem Management Con-serv Ecol 3 1ndash31 httpsdoiorg105751ES-00122-0302041999b

Carpenter S R Westley F and Turner M G Surrogates for Re-silience of Social-Ecological Systems Ecosystems 8 941ndash944httpsdoiorg101007s10021-005-0170-y 2005

Chang S E and Shinozuka M Measuring Improvements in theDisaster Resilience of Communities Earthq Spectra 20 739ndash755 httpsdoiorg10119311775796 2004

Chang S E McDaniels T Fox J Dhariwal R and LongstaffH Toward Disaster-Resilient Cities Characterizing Resilienceof Infrastructure Systems with Expert Judgments Risk Anal 34416ndash434 httpsdoiorg101111risa12133 2014

Cote M and Nightingale A J Resilience ThinkingMeets Social Theory Prog Hum Geog 36 475ndash489httpsdoiorg1011770309132511425708 2012

Csete M E and Doyle J C Reverse Engineeringof Biological Complexity Science 295 1664ndash1669httpsdoiorg101126science1069981 2002

wwwearth-syst-dynamnet911592018 Earth Syst Dynam 9 1159ndash1168 2018

1160 M Homayounfar et al Linking resilience and robustness and uncovering their trade-offs

Now if resilience is also a desirable property of the samesystem does it also come at the expense of performance androbustness Put it another way is there a trade-off amongperformance robustness and resilience Such a trade-off ifit exists is a crucial consideration for governing andor man-aging social-ecological systems (SESs)

But resilience as alluded to above is trickier to defineAccording to Holling (1973) resilience refers to the amountof change or disruption required to shift the maintenanceof a system along different sets of mutually reinforcingprocesses and structures In other words resilience can bethought of as how far the system is from certain thresholds orboundaries beyond which the system will undergo a regimeshift or a quantitative change in system structure or identityHolling (1996) categorized resilience into two types engi-neering resilience which refers to the ability of a system toreturn to a steady state following a perturbation and ecolog-ical resilience which refers to the capacity of system to re-main in a particular stability domain in the face of pertur-bations The latter category is used by many researchers todiscuss resilience of SESs or more generally coupled in-frastructure systems (CISs Carpenter et al 2001 Anderieset al 2006 Folke 2006 Biggs et al 2012 Barrett and Con-stas 2014 Redman 2014 Walker et al 2002 2004 Gun-derson and Holling 1995 Berkes and Folke 1998 Carpen-ter et al 1999a b Scheffer et al 2000 Berkes et al 2003Carpenter and Brock 2004 Janssen et al 2004 Folke etal 2002 2010 2016 Cote and Nightingale 2012 Mitra etal 2015 Cumming and Peterson 2017) The CISs term hasbeen introduced to generalize the notions of coupled naturalhuman systems (CNHSs) and SESs in this context infras-tructure is broadly defined to include human-made socialand natural infrastructure (eg see Anderies et al 2016)The problem is that these CISs are complex and thus identi-fying thresholds and potential regime shifts associated withthose thresholds is often difficult if not impossible In manycases major aspects of resilience in CISs may not be directlyobservable and must be actualized indirectly via surrogate at-tributes (Carpenter et al 2005 Kerner and Thomas 2014)Recent significant advances have been made toward identi-fying early-warning signals that indicate whether a criticalthreshold is being approached for a wide class of systems(Scheffer et al 2009 2012) Still there are gaps in our un-derstanding of how indicators of resilience and robustnesswill behave in more complex situations This lack of a rigor-ous metric for resilience makes the investigation into theirconnections interplay and trade-offs with robustness andperformance impossible

But these knowledge gaps need to be filled if one wishesto make advances in understanding the interplay between so-cial dynamics and planetary boundaries Given the magni-tude of impacts that human activities have on pushing Earthsystems toward their planetary boundaries we need clearerunderstanding of how social and biophysical factors come to-gether to define the nature of these boundaries This paper is

a step in that direction In particular we will build on recentwork that mathematically operationalizes the robustness of aSES framework (Anderies et al 2004) into a formal stylizeddynamical model (Muneepeerakul and Anderies 2017) Wewill exploit the relative simplicity of the model to rigorouslydefine robustness and resilience of the coupled system Themodeled system will be subject to uncertainty in social andecological factors which will affect the well-defined robust-ness and resilience thereby enabling us to investigate the in-terplay and trade-offs between these important properties aswell as investigate how the nature of the interplay and trade-offs are affected by policies implemented by social agents

2 Methods

Here we analyze a mathematical model developed byMuneepeerakul and Anderies (2017) by subjecting the cou-pled system to uncertainty in ecological and social factorsThe model captures the essential features of a system inwhich a group of agents shares infrastructure to producevalued flows Such a system is the archetype of most ifnot all of human sociality groups produce infrastructurethat they cannot produce individually (security defense ir-rigation canals roads markets financial systems coordina-tion mechanisms etc) that significantly increases productiv-ity The challenge is maintaining this shared infrastructure(eg decaying infrastructure is a major problem in the US atthe time of writing ASCE RCIA Advisory Council 2013)The model allows for mathematical definitions of the bound-aries of policy domain that result in a sustainable system inwhich both human-made and natural infrastructure can bemaintained over the long run Based on these boundariesand uncertainty in the exogenous factors we define met-rics of resilience and robustness associated with each policychoice and investigate the trade-off between them The basicmodel presented by Muneepeerakul and Anderies (2017) isdescribed in the Appendix

Here a policy is defined as a combination of taxation levelC and the proportion of tax revenue invested in infrastruc-ture maintenance y that the public infrastructure providers(PIPs) decide to implement in the system The infrastructure(eg canals) enable resource users (RUs) to produce valuedgoods from a natural resource The two uncertain factors arethe replenishment rate of the natural resource g and the wagew that RUs would earn from working outside the system ndasha combination of g and w defines a ldquosocial-ecological set-tingrdquo or simply ldquosettingrdquo There are two boundaries that oncecrossed will cause the system to collapse The first boundaryis called PIP participation constraint (PPC) when the PIPsmust invest too much in maintaining the public infrastruc-ture (exceeding the opportunity cost of wP) andor cannotretain enough revenue for themselves they will abandon thesystem for another The second boundary is the stability con-dition of the nontrivial equilibrium point (ie the ldquosocietyrdquo

Earth Syst Dynam 9 1159ndash1168 2018 wwwearth-syst-dynamnet911592018

M Homayounfar et al Linking resilience and robustness and uncovering their trade-offs 1161

in which both PIPs for example the state and RUs for exam-ple citizens participate in and public infrastructure is suffi-ciently maintained in a long run) Together these two bound-aries delineate a set of policies (Cndashy combinations) that cor-respond to sustainable outcomes The resilience metric tobe developed below can be thought of as a metric of howfar the system is from these boundaries As the two exoge-nous factors defining settings namely g and w change thetwo boundaries and thus the resilience metric also changewith them How sensitive the resilience metric is to thesesettings is used to define robustness Here it is worth not-ing that while it is possible to examine recovery-based re-silience (or the so-called ldquoengineering resiliencerdquo) this paperfocuses on regime shift-based resilience (traditionally calledldquoecological resiliencerdquo) and its robustness Quantification ofand trade-offs between resilience and robustness is a novelconcept that requires expositional clarity Presenting severalmetrics of resilience let alone studying their trade-offs withpotentially different metrics of robustness may confuse thematter and dilute the key messages we attempt to convey Assuch in what follows we will focus on developing a metricfor regime shift-based resilience With the scope of analysisclarified and suitably bounded we will now define resilienceand robustness more formally

21 Resilience metric

Direct measurement of the above-mentioned resilience as aspecified form of resilience (Walker et al 2004) in SESs isdifficult because boundaries and thresholds that separate do-mains of dynamics for SESs are difficult to identify (Carpen-ter et al 2005 Scheffer et al 2009 2012) In this stylizedmodel however such boundaries can be clearly identified bythe stability condition (SC) and the PPC Here we are inter-ested in the resilience of a systemrsquos ability to provide suffi-cient livelihoods for the PIPs and RUs The basin of attrac-tion for system resilience is defined by those system states(ie infrastructure state) in which this is possible and thesesystem states are directly mapped to the SC and PPC Wewill thus define resilience metrics based on the SC and PPCboundaries Here our goal is to develop resilience metricsthat can be meaningfully compared to one another As suchwe identify some desired properties that guide the defini-tions of these resilience metrics First they should be zero attheir respective boundaries Second positive values indicategreater resilience of the system in a desirable state Thesefirst two properties align with how resilience has been mea-sured ie the distance from the boundary of a basin of at-traction (eg Anderies et al 2002 Carpenter et al 1999a)Third to facilitate the consideration of relative risks associ-ated with different types of regime shifts that the system maybe facing the metrics should be comparable in magnitudeThese properties guide us toward the following definitions ofthe resilience metrics

We define the resilience of the system against abandon-ment by PIPs as follows

RPPC = (πPwP)minus 1 (1)

where πP is the net revenue that PIPs collect andwP is the op-portunity cost that they will earn if they choose to work withanother system Positive values of RPPC indicate that the sys-tem is resilient against being abandoned by PIPs while neg-ative values indicate that the system will eventually collapsedue to the PIPsrsquo abandonment It is important to note thatπP results from the coupled dynamics of the CIS this meansthat RPPC has already integrated the dynamics of infrastruc-ture resource and RUs (Eqs A1 A4 and A5) making it ametric of the system not of an individual component

Numerical analysis of the model indicates that the equilib-rium becomes unstable when the following RouthndashHurwitzcondition (eg May 2001 Kot 2001) is violated

Dminus T(J11J22+ J21J12+ J23J32+ J13J31

)gt 0 (2)

where D T and J are determinant trace and entries re-spectively of the Jacobian matrix of the dynamical system(Eqs A1 A4 and A5) evaluated at the nontrivial equilib-rium point (Eq A6) ndash when such an equilibrium point existsHere it is worth noting that we focus on the equilibrium pointrelated to the nontrivial sustainable long-term outcome An-alyzing other bifurcations related to other equilibria may bemathematically interesting but it could make the study lessaccessible and dilute its key message about the resiliencendashrobustness trade-off of different policies aiming at keepingthe system in the basin of attraction of the non-trivial sus-tainable long-term outcome

Following the guideline provided by the three desirableproperties above we rearrange terms in Eq (2) and define theresilience of the system against instability (increased proba-bility of collapse of infrastructure) as follows

Rstability =D∣∣T (J11J22+ J21J12+ J23J32+ J13J31

)∣∣ minus 1 (3)

This formulation is parallel to that of the first resiliencemetric (Eq 1) it possesses the following three propertiesRstability of zero indicates the boundary between stability andinstability positive Rstability means the system at the equilib-rium point is stable and the magnitudes of Rstability are com-parable to those of RPPC (see Fig 1) Note that Rstability toois determined from the coupled dynamics of the CIS thismeans that it has integrated the dynamics of infrastructureresource and RUs (Eqs A1 A4 and A5)

This allows us to meaningfully define the overall systemresilience as the minimum between the two resilience met-rics namely

Rsystem =

Min

[RPPCRStability

]RPPCRstability ge 0

0 otherwise (4)

wwwearth-syst-dynamnet911592018 Earth Syst Dynam 9 1159ndash1168 2018

1162 M Homayounfar et al Linking resilience and robustness and uncovering their trade-offs

0 02 04 06 08 1C

0

02

04

06

08

1

y

-05

0

05

1

15

2

0 02 04 06 08 1C

0

02

04

06

08

1

y

01

02

03

04

05

06

07

08

0 02 04 06 08 1C

0

02

04

06

08

1

y

0

02

04

06

08

(a) (b) (c)

Figure 1 Resilience metrics for a specific setting (a gndashw combination) inside the sustainable region in the policy space (ie Cndashy plane)(a) RPPC contours (b) Rstability contours and (c) Rsystem contours The black star in panels (a) (b) and (c) indicates the policy with thehighest RPPC Rstability and Rsystem respectively

Equation (4) implies that Rsystem is positive only when thenontrivial equilibrium point (Eq A6) exists and is stable oth-erwise the system is considered not resilient and denoted byRsystem = 0 Rsystem thus represents the tension between thePIPs being too greedy (high C and low y) whereby they getclose to the stability boundary and ldquonot greedy enoughrdquo (lowC and high y) whereby they get close to the PPC given a par-ticular choice for wP Note that the values of w and wP rep-resent the socioeconomic embedding of the CIS Thereforethe biophysical structure of the CIS along with the socioe-conomic context in which it is embedded co-determine themaximum resilience that can be achieved Given that the non-trivial equilibrium point exists and is stable the following ap-plies If the system is at a greater risk of being abandoned bythe PIPs (and eventually collapsing) Rsystem = RPPC if thesystem is at a greater risk of becoming unstable (and even-tually collapsing) Rsystem = Rstability Figure 1 illustrates therelationships between RPPC Rstability and Rsystem

22 Linking robustness and resilience

As discussed earlier robustness can be thought of as the op-posite of sensitivity A commonly used measure of sensitiv-ity is variance Thus variance of a given function under spe-cific disturbance or uncertainty regimes may be used to in-dicate robustness of that function against those disturbancesor uncertainty regimes (robustness of what to what) In thiscase the system function of interest is the system resilienceRsystem By choosing Rsystem as our function we can usefullylink these concepts If we use a ball and cup metaphor for re-silience the robustness of resilience refers to the degree atwhich the geometry of the cup changes as a result of externaldisturbances andor parameter changes However as we willargue below relating high variance in Rsystem to low robust-ness may be misleading and should not be used in evaluatinga given policy By definition the variance treats ldquogood de-viationsrdquo and ldquobad deviationsrdquo from the mean equally For

functions with preferred values such as resilience or profitvalues greater than the mean and those lower should not betreated in the same way Specifically contribution to a highvariance from a heavy tail in the good direction should notbe translated to less robustness This problem also arises inassessing financial risk what makes an asset risky is the val-ues on the ldquobad tailrdquo of the distribution (ie low or nega-tive profits) This has motivated more and more analyses toswitch to considering other measures of risk such as the con-ditional value at risk in evaluating their portfolios of invest-ment (Rockafellar and Uryasev 2000 Krokhmal et al 2002Sarykalin et al 2014 AitSahlia et al 2011 Zymler et al2013) Intuitively this means that the shape of the ldquocuprdquo canbe asymmetric and we need to take this into account

Following this logic we propose to use a ldquobelow-meanmeanrdquo as a new robustness metric the mean of all resiliencevalues lower than the mean This new definition of the robust-ness metric has several desirable features First it can now beappropriately thought of as a robustness metric in the sensethat the higher the value the more robust the system (unlikethe variance for which low variance means high robustness)Second by using the mean as the threshold value for bad de-viations we remove some arbitrariness associated with pre-scribing a certain quantile (eg 5th or 10th quantile) in cal-culating the conditional value at risk Third it still carriessome information about the sensitivity of the resilience met-ric to outside factors ndash the information that variance conveysthat is the higher the below-mean mean (ie the bad devia-tions from the mean are small and the below-mean mean isclose to the mean) the less sensitive ndash and thus more robustndash the resilience metric

In this study we subject the modeled system to uncertaintyin one natural factor and one social factor namely the natu-ral replenishment rate of the resource g and the payoff thata RU earns from working outside the system w Thus weare computing how the resilience of the system to shocks

Earth Syst Dynam 9 1159ndash1168 2018 wwwearth-syst-dynamnet911592018

M Homayounfar et al Linking resilience and robustness and uncovering their trade-offs 1163

Figure 2 Variation of Rsystem of a CIS with a fixed policy (Cy) over 10 000 settings associated with uncertainty characterized byg isin [75125] w isin [075125] (a)Rsystem surface and (b)Rsystem contours The values ofRsystem are used to calculate the meanmicroRsystem and the below-mean mean microRltmicro In this particular case the resilience does not change much when g is greater than about 100 but becomesmore sensitive to both g and w when g is lower than 100

andor variation in state variables changes as the parametersg and w change (ie we are uncertain about the underlyingsocial-ecological setting of the system) In particular we as-sume that g is uniformly distributed over the range [75 125]and w is uniformly distributed over the range [075 125]A social-ecological setting or setting is defined as a com-bination of g and w For a given policy (a Cndashy combina-tion) we calculate Rsystem for 10 000 settings (ie 10 000gndashw combinations see Fig 2) Then from these 10 000 val-ues of the resilience metric Rsystem we calculate the meanmicroRsystem = E

[Rsystem

] and use it as the resilience metric of

the coupled system with a given policy and the below-meanmean microRltmicro = E

[Rsystem|Rsystem lt microRsystem

] as the metric

for robustness of resilience This metric measures the robust-ness of the capacity of the system to cope with variation instate variables IHM R and U to fundamental uncertaintyabout the underlying setting of the system

3 Results

The surfaces and contours of the system resilience metricmicroRsystem and the associated different policies (Cndashy) over thepolicy space are shown in Fig 3a and b respectively Thepolicies with sustainable outcomes are located in the mid-dle of the policy space with microRsystem peaking in the centerand declining as policies become more extreme in either di-rection Our analysis also shows that microRsystem is more or lessproportional to the fraction of settings (gndashw combinations)under which the system with that particular policy (a Cndashycombination) results in a sustainable outcome (Rsystem gt 0)A similar concept has been used in the robust decision mak-ing literature (eg Groves and Lempert 2007 Bryant andLempert 2010)

The surfaces and contours of the robustness of microRltmicro as-sociated with different policies over the policy space are

shown in Fig 3c and d respectively The microRltmicro ldquolandscaperdquois more irregular having two local maxima with one beingmore dominant than the other The region with high robust-ness appears to be in the same general areas as the regionwith high resilience These features reflect the nonlinear in-terplay between the model parameters and model structureand may affect the nature of the trade-off between robustnessand resilience reported in Figs 4 and 5

We explore the interplay between microRsystem and microRltmicro inFig 4 Figure 4 shows that there are no perfect policies inthe sense that no policies yield both maximum resilience andmaximum robustness Recall that the robustness indicateshow sensitiveRsystem itself is to uncertainty in the underlyingsetting of the system (eg g and w) The best policies arethose along the Pareto frontier in the resiliencendashrobustnessspace among this set of Pareto-optimal policies an increasein resilience is necessarily accompanied by a decrease in ro-bustness clearly illustrating the trade-off between robustnessand resilience Fig 5 illustrates where the Pareto-optimalpolicies are located in the policy space

4 Discussion and conclusions

In this paper we exploit the simplicity of a stylized modelto quantitatively link resilience and robustness by computinghow the CIS resilience to shocks in state variables changeswith parameters In this way we compute the robustness ofCIS resilience to uncertainty in the underlying CIS settingThe resilience metric developed here is a measure of how farthe CIS is from the boundaries beyond which it will collapseThe model affords us with expressions of these boundarieswhich clearly show how social and biophysical factors inter-play to define these boundaries With a concrete definition ofresilience resilience itself can be considered as the ldquoof whatrdquoin the ldquorobustness of what to whatrdquo notion In particular we

wwwearth-syst-dynamnet911592018 Earth Syst Dynam 9 1159ndash1168 2018

1164 M Homayounfar et al Linking resilience and robustness and uncovering their trade-offs

0 02 04 06 08 1C

0

02

04

06

08

1

y

01

02

03

04

05

06

07

0 02 04 06 08 1C

0

02

04

06

08

1

y

01

02

03

04

05

(a) (b)

(d)(c)

Figure 3 The mean microRsystem and the below-mean mean of Rsystem microRltmicro over the entire decision space (a) Surface of the resiliencemetric microRsystem (b) Contours of microRsystem (c) Surface of the robustness the below-mean mean (microRltmicro) (d) Contours of microRltmicro

0 02 04 06 08

R-system

0

01

02

03

04

05

06

(R-s

yste

mlt

R-s

yste

m)

Figure 4 Resiliencendashrobustness trade-off Each point representsmicroRsystem and microRltmicro of the coupled system with a given policy Theblack dots represent a set of Pareto-optimal policies

use the below-mean mean of the quantitatively defined re-silience metric as the metric of robustness Consequently thisenables us to rigorously investigate the interplay between thetwo important but not always well-defined system proper-ties A key finding is the fundamental trade-off between re-silience and robustness there are no perfect policies in gov-erning a CIS only Pareto-optimal ones Specifically policiesdesigned to maximize the resilience of a CIS to shocks ontimescales at which the state variables play out may be verysensitive to being wrong about our understanding of the un-derlying dynamics of the CIS in question

Importantly we hope this work will stimulate further ad-vances in rigorous studies of CISs that address such sub-tle policy-relevant questions a few of which we briefly dis-cussed here More dimensions can be considered in definingPareto-optimality Figure 5 may give an impression that theset of Pareto-optimal policies is confined to a small region inthe policy space which would imply that PIPs do not havethat many choices ndash even in a simple CIS like the one stud-ied But that would be a wrong impression In addition toresilience and robustness (as defined here) a policy makeror a social planner may be interested in other types of ro-bustness with different ldquoof whatrdquo and ldquoto whatrdquo components

Earth Syst Dynam 9 1159ndash1168 2018 wwwearth-syst-dynamnet911592018

M Homayounfar et al Linking resilience and robustness and uncovering their trade-offs 1165

0

02

04

08

1

01

02

03

04

05

06

07

06y

160 800 20 40C

01

02

03

04

05

0

02

04

08

1

06y

160 800 20 40C

(a) (b)

Figure 5 Pareto-optimal policies represented by black dots in the policy space (Cndashy plane) superimposed with resilience (microRsystem ) (a)and robustness (microRltmicro) contours (b)

They may also be concerned about other system propertieseg productivity user participation etc As more dimensionsare considered the set of Pareto-optimal policies grow In thesame spirit as that of the work done here these other dimen-sions should be defined rigorously

This work also lends itself to more rigorous studies ofldquoadaptive governancerdquo In the present study the governancestructure represented by a policy (a combination of C andy) is fixed A natural next step is to explore if a policy isallowed to change how one may improve the resilience androbustness of a CIS andor alter the nature of their inher-ent trade-offs For example if C and y are to be functionsof other factors eg resource availability and outside incen-tives what functional forms should they take to improve thesystemrsquos resilience and robustness Indeed in the absence oftransparent metrics attempts to explore such adaptive poli-cies are severely limited

Additionally agents in a CIS may have the capacity tochange their behavior in response to changes in policy en-vironmental conditions technological changes and so on Inthis study strategic behavior and the decision-making pro-cess are assumed unchanged in the analysis Adaptation instrategic behavior of agents will subsequently alter the natureof resilience its robustness and their trade-offs Capturingsuch effects of adaptation requires structural changes to themodel eg in terms of specification of payoffs or even theformulation of the dynamical equations With such adaptivesocial agents how should one devise adaptive governance toenhance resilience and robustness of a CIS Addressing sucha question is a theoretically intriguing future research direc-tion with great practical implications

In keeping with the theme of ldquosocial dynamics and plane-tary boundaries in Earth system modellingrdquo our results shedlight on how social and biophysical factors may interplay todefine boundaries of a sustainable coupled infrastructure sys-tem While the modeled system here is admittedly simpleour methodology and results constitute a step toward quanti-tatively and meaningfully combining social and biophysicalfactors into indicators of boundaries of more complex sys-tems Just as in this work once those boundaries are clearlydefined calculation and discussion of resilience and robust-ness can become concrete

Data availability No data sets were used in this article

wwwearth-syst-dynamnet911592018 Earth Syst Dynam 9 1159ndash1168 2018

1166 M Homayounfar et al Linking resilience and robustness and uncovering their trade-offs

Appendix A

Basic model Here we briefly describe the basic model pre-sented by Muneepeerakul and Anderies (2017) The modelshows dynamic behavior of three principal variables namelythe state of the public infrastructure IHM resource level Rand the fraction of time a user makes use of infrastructureU through Eqs (A1 A4 and A5) The schematic diagramof this system of equations is shown in Fig A1

In this context IHM depends on PIPs in term of mainte-nance cost and has a positive relationship with the capacityof users to create resource flows Eq (A1) illustrates the dy-namics of IHM as follows

dIHMdt=M ( )minus δH (IHM ) (A1)

where δ is the infrastructurersquos depreciation rate andH (IHM )states functional relationship of public infrastructure and pro-ductivity of each RU According to Muneepeerakul and An-deries (2017) many shared infrastructures can be modeledby threshold functions Given that H (IHM ) shows thresh-old behavior they used a piecewise linear function to capturesuch behavior through Eq (A2)

H (IHM )=

0IHM lt I0

hIHM minus I0

Imminus I0I0 le IHM le Im

hIHM ge Im

(A2)

where h represents the maximum amount of harvest byeach user under no restriction and I0 and Im are the lowerbound and upper bound thresholds of IHM respectivelyAlso M ( ) is the maintenance function (Eq A3) and de-pends on the social structure of the system

M ( )= micro2yCpRUNH (IHM ) (A3)

In Eq (A3) given the number of users N RUNH (IHM ) isthe total harvest from the natural infrastructure The RUs selltotal harvest at price p to generate revenue Subsequentlythey assign a proportion C of revenue to PIPrsquos for their con-tribution Meanwhile the PIPrsquos spend proportion y of C onmaintaining public infrastructure through the maintenancefunction M ( ) Also micro2 is the maintenance effectivenessof PIPrsquos investment

The second variable is the resource levelR They assumedthe dynamics of resource to be

dRdt=G (R)minusRUNH (IHM ) (A4)

Natural infrastructure is assumed to invoke the conservationlaw comprising of regenerating capacity (G (R)= gminus dR)and total unit of harvest RUNH (IHM ) The definition pre-sented for G is the simplest model for natural infrastructurewhere g and d are the natural replenishment and the lossrates respectively

HM IHM

Resource users

N

Natural infrastructure

Public infrastructure

R G(R)wp

infrastructurePublic

providersy

w

M()

UNI

minusδ

HMUNRH( )

IHMH( ) IHMH( )

IHMH( )

IHMCpUNRH( )

I

Figure A1 Schematic diagram of the dynamical system modelTaken from Muneepeerakul and Anderies (2017)

The strategic behavior of the resource users (RUs) is cap-tured by employing a replicator equation Indeed replica-tor dynamics provide modelers with simple realistic so-cial mechanism where agents follow and replicate better-offstrategies The two possible strategies considered for RUsare staying inside the system with the associated payoff ofπU = (1minusC)pRH (IHM ) or leaving the system with thepayoff of πw = w According to the replicator equation

dUdt= rU (1minusU ) (πU minusw) (A5)

The replicator equation represents the fraction of time thatRUs assign to working inside system given C and y LikeRUs there is also two alternatives for PIPs working insidethe system or working for another CIS which leads to sys-tem failure Meanwhile C and y characterize the strategyor policy of PIPs The PIPs will participate in this coupledsystem only when πp = (1minus y)pCRUNH (IHM )ge wP Inother words the PIPs maintain the system when they arebetter-off than working outside This condition is termed thePIP Participation Constraint (PPC)

Based on the system of three differential equations(Eqs A1 A4 and A5) the sustainable equilibria ie long-term system outcomes that satisfy the stability condition andPPC can be expressed as follows

ilowastHM =yCUlowastNRlowast

gH(I lowastHM

)Rlowast =

g

d

(1minus

ilowastHM

yC

)

Ulowast =(1minusC)yC

φ1ilowast

HM (A6)

where ilowastHM =IlowastHMδ

micro2pg(indicates dimensionless) and φ1 =

pgwN

a dimensionless group representing the relative lucrativenessof the system namely the ratio of potential income ndash withthe entire resource flow turned into income ndash relative to out-side wage The results reported in this study are based on thefollowing parameter values h= 00005 δ = 01 I0 = 03Im = 3 g = 100 d = 002 N = 1000 r = 015 p = 10w = 1 wp = 100 and micro2 = 0001

Earth Syst Dynam 9 1159ndash1168 2018 wwwearth-syst-dynamnet911592018

M Homayounfar et al Linking resilience and robustness and uncovering their trade-offs 1167

Author contributions MH RM JMA and CPM designed thestudy MH carried out the analysis MH and RM prepared themanuscript with contributions from all co-authors

Competing interests The authors declare that they have no con-flict of interest

Special issue statement This article is part of the special issueldquoSocial dynamics and planetary boundaries in Earth system mod-ellingrdquo It is not associated with a conference

Acknowledgements John M Anderies and Rachata Muneepeer-akul acknowledge the support from the grant NSF GEO-1115054Rachata Muneepeerakul and Mehran Homayounfar acknowl-edge support from the Army Research OfficeArmy ResearchLaboratory The research reported in this grant was supported inwhole or in part by the Army Research OfficeArmy ResearchLaboratory under award no W911NF1810267 (Multi-UniversityResearch Initiative) The views and conclusions contained in thisdocument are those of the authors and should not be interpreted asrepresenting the official policies either expressed or implied of theArmy Research Office or the US Government The authors alsothank the editor and three anonymous referees for their constructiveand useful comments

Edited by Jonathan DongesReviewed by three anonymous referees

References

AitSahlia F Wang C J Cabrera V E Uryasev S and FraisseC W Optimal crop planting schedules and financial hedgingstrategies under ENSO-based climate forecasts Ann Oper Res190 201ndash220 2011

Anderies J M Janssen M A and Walker B H Graz-ing Management Resilience and the Dynamics of aFire-Driven Rangeland System Ecosystems 5 23ndash44httpsdoiorg101007s10021-001-0053-9 2002

Anderies J M Janssen M and Ostrom E A Frameworkto Analyze the Robustness of Social-Ecological Systems froman Institutional Perspective Ecol Soc 9 18 httpwwwecologyandsocietyorgvol9iss1art18inlinehtml 2004

Anderies J M Ryan P and Walker B H Loss of ResilienceCrisis and Institutional Change Lessons from an Intensive Agri-cultural System in Southeastern Australia Ecosystems 9 865ndash878 httpsdoiorg101007s10021-006-0017-1 2006

Anderies J M Rodriguez A A Janssen M A and CifdalozO Panaceas Uncertainty and the Robust Control Frameworkin Sustainability Science P Natl Acad Sci USA 104 15194ndash15199 httpsdoiorg101073pnas0702655104 2007

Anderies J M Janssen M and Schlager E Institutions andthe performance of coupled infrastructure systems Int J Com-mons 10 495ndash516 httpsdoiorg1018352ijc651 2016

ASCE RCIA Advisory Council Report card for Americarsquos Infras-trcutre Tech Rep American Society of Civil Engineers 2013

Barrett C B and Constas M A Toward a Theoryof Resilience for International Development Applica-tions P Natl Acad Sci USA 111 14625ndash14630httpsdoiorg101073pnas1320880111 2014

Berkes F and Folke C Linking Social and Ecological Systemsfor Resilience and Sustainability Linking Social and Ecologi-cal Systems Management Practices and Social Mechanisms forBuilding Resilience 1 13ndash20 1998

Berkes F Colding J and Folke C Navigating Social-EcologicalSystems Building Resilience for Complexity and change Build-ing p 393 httpsdoiorg101016jbiocon200401010 2003

Biggs R Schluumlter M Biggs D Bohensky E L BurnSil-ver S Cundill G Dakos V Daw T M Evans L SKotschy K Leitch A M Meek C Quinlan A Raudsepp-Hearne C Robards M D Schoon M L Schultz L andWest P C Toward Principles for Enhancing the Resilienceof Ecosystem Services Annu Rev Env Resour 37 421ndash448httpsdoiorg101146annurev-environ-051211-123836 2012

Bode H W Network Analysis and Feedback Amplifier DesignBell Telephone Laboratories Series p 551 1945

Bryant B P and Lempert R J Thinking inside the BoxA Participatory Computer-Assisted Approach to Sce-nario Discovery Technol Forecast Soc 77 34ndash49httpsdoiorg101016jtechfore200908002 2010

Carlson J M and Doyle J Complexity and Ro-bustness P Natl Acad Sci USA 99 2538ndash2545httpsdoiorg101073pnas012582499 2002

Carpenter S Walker B Anderies J M and Abel N FromMetaphor to Measurement Resilience of What to WhatEcosystems 4 765ndash781 httpsdoiorg101007s10021-001-0045-9 2001

Carpenter S R and Brock W A Spatial Complexity Resilienceand Policy Diversity Fishing on Lake-Rich Landscapes EcolSoc 9 8 2004

Carpenter S R Ludwig D and Brock W A Management of Eu-trophication for Lakes Subject to Potentially Irreversible ChangeEcol Appl 9 751ndash771 httpsdoiorg10230726413271999a

Carpenter S Brock W and Hanson P Ecological and SocialDynamics in Simple Models of Ecosystem Management Con-serv Ecol 3 1ndash31 httpsdoiorg105751ES-00122-0302041999b

Carpenter S R Westley F and Turner M G Surrogates for Re-silience of Social-Ecological Systems Ecosystems 8 941ndash944httpsdoiorg101007s10021-005-0170-y 2005

Chang S E and Shinozuka M Measuring Improvements in theDisaster Resilience of Communities Earthq Spectra 20 739ndash755 httpsdoiorg10119311775796 2004

Chang S E McDaniels T Fox J Dhariwal R and LongstaffH Toward Disaster-Resilient Cities Characterizing Resilienceof Infrastructure Systems with Expert Judgments Risk Anal 34416ndash434 httpsdoiorg101111risa12133 2014

Cote M and Nightingale A J Resilience ThinkingMeets Social Theory Prog Hum Geog 36 475ndash489httpsdoiorg1011770309132511425708 2012

Csete M E and Doyle J C Reverse Engineeringof Biological Complexity Science 295 1664ndash1669httpsdoiorg101126science1069981 2002

wwwearth-syst-dynamnet911592018 Earth Syst Dynam 9 1159ndash1168 2018

M Homayounfar et al Linking resilience and robustness and uncovering their trade-offs 1161

in which both PIPs for example the state and RUs for exam-ple citizens participate in and public infrastructure is suffi-ciently maintained in a long run) Together these two bound-aries delineate a set of policies (Cndashy combinations) that cor-respond to sustainable outcomes The resilience metric tobe developed below can be thought of as a metric of howfar the system is from these boundaries As the two exoge-nous factors defining settings namely g and w change thetwo boundaries and thus the resilience metric also changewith them How sensitive the resilience metric is to thesesettings is used to define robustness Here it is worth not-ing that while it is possible to examine recovery-based re-silience (or the so-called ldquoengineering resiliencerdquo) this paperfocuses on regime shift-based resilience (traditionally calledldquoecological resiliencerdquo) and its robustness Quantification ofand trade-offs between resilience and robustness is a novelconcept that requires expositional clarity Presenting severalmetrics of resilience let alone studying their trade-offs withpotentially different metrics of robustness may confuse thematter and dilute the key messages we attempt to convey Assuch in what follows we will focus on developing a metricfor regime shift-based resilience With the scope of analysisclarified and suitably bounded we will now define resilienceand robustness more formally

21 Resilience metric

Direct measurement of the above-mentioned resilience as aspecified form of resilience (Walker et al 2004) in SESs isdifficult because boundaries and thresholds that separate do-mains of dynamics for SESs are difficult to identify (Carpen-ter et al 2005 Scheffer et al 2009 2012) In this stylizedmodel however such boundaries can be clearly identified bythe stability condition (SC) and the PPC Here we are inter-ested in the resilience of a systemrsquos ability to provide suffi-cient livelihoods for the PIPs and RUs The basin of attrac-tion for system resilience is defined by those system states(ie infrastructure state) in which this is possible and thesesystem states are directly mapped to the SC and PPC Wewill thus define resilience metrics based on the SC and PPCboundaries Here our goal is to develop resilience metricsthat can be meaningfully compared to one another As suchwe identify some desired properties that guide the defini-tions of these resilience metrics First they should be zero attheir respective boundaries Second positive values indicategreater resilience of the system in a desirable state Thesefirst two properties align with how resilience has been mea-sured ie the distance from the boundary of a basin of at-traction (eg Anderies et al 2002 Carpenter et al 1999a)Third to facilitate the consideration of relative risks associ-ated with different types of regime shifts that the system maybe facing the metrics should be comparable in magnitudeThese properties guide us toward the following definitions ofthe resilience metrics

We define the resilience of the system against abandon-ment by PIPs as follows

RPPC = (πPwP)minus 1 (1)

where πP is the net revenue that PIPs collect andwP is the op-portunity cost that they will earn if they choose to work withanother system Positive values of RPPC indicate that the sys-tem is resilient against being abandoned by PIPs while neg-ative values indicate that the system will eventually collapsedue to the PIPsrsquo abandonment It is important to note thatπP results from the coupled dynamics of the CIS this meansthat RPPC has already integrated the dynamics of infrastruc-ture resource and RUs (Eqs A1 A4 and A5) making it ametric of the system not of an individual component

Numerical analysis of the model indicates that the equilib-rium becomes unstable when the following RouthndashHurwitzcondition (eg May 2001 Kot 2001) is violated

Dminus T(J11J22+ J21J12+ J23J32+ J13J31

)gt 0 (2)

where D T and J are determinant trace and entries re-spectively of the Jacobian matrix of the dynamical system(Eqs A1 A4 and A5) evaluated at the nontrivial equilib-rium point (Eq A6) ndash when such an equilibrium point existsHere it is worth noting that we focus on the equilibrium pointrelated to the nontrivial sustainable long-term outcome An-alyzing other bifurcations related to other equilibria may bemathematically interesting but it could make the study lessaccessible and dilute its key message about the resiliencendashrobustness trade-off of different policies aiming at keepingthe system in the basin of attraction of the non-trivial sus-tainable long-term outcome

Following the guideline provided by the three desirableproperties above we rearrange terms in Eq (2) and define theresilience of the system against instability (increased proba-bility of collapse of infrastructure) as follows

Rstability =D∣∣T (J11J22+ J21J12+ J23J32+ J13J31

)∣∣ minus 1 (3)

This formulation is parallel to that of the first resiliencemetric (Eq 1) it possesses the following three propertiesRstability of zero indicates the boundary between stability andinstability positive Rstability means the system at the equilib-rium point is stable and the magnitudes of Rstability are com-parable to those of RPPC (see Fig 1) Note that Rstability toois determined from the coupled dynamics of the CIS thismeans that it has integrated the dynamics of infrastructureresource and RUs (Eqs A1 A4 and A5)

This allows us to meaningfully define the overall systemresilience as the minimum between the two resilience met-rics namely

Rsystem =

Min

[RPPCRStability

]RPPCRstability ge 0

0 otherwise (4)

wwwearth-syst-dynamnet911592018 Earth Syst Dynam 9 1159ndash1168 2018

1162 M Homayounfar et al Linking resilience and robustness and uncovering their trade-offs

0 02 04 06 08 1C

0

02

04

06

08

1

y

-05

0

05

1

15

2

0 02 04 06 08 1C

0

02

04

06

08

1

y

01

02

03

04

05

06

07

08

0 02 04 06 08 1C

0

02

04

06

08

1

y

0

02

04

06

08

(a) (b) (c)

Figure 1 Resilience metrics for a specific setting (a gndashw combination) inside the sustainable region in the policy space (ie Cndashy plane)(a) RPPC contours (b) Rstability contours and (c) Rsystem contours The black star in panels (a) (b) and (c) indicates the policy with thehighest RPPC Rstability and Rsystem respectively

Equation (4) implies that Rsystem is positive only when thenontrivial equilibrium point (Eq A6) exists and is stable oth-erwise the system is considered not resilient and denoted byRsystem = 0 Rsystem thus represents the tension between thePIPs being too greedy (high C and low y) whereby they getclose to the stability boundary and ldquonot greedy enoughrdquo (lowC and high y) whereby they get close to the PPC given a par-ticular choice for wP Note that the values of w and wP rep-resent the socioeconomic embedding of the CIS Thereforethe biophysical structure of the CIS along with the socioe-conomic context in which it is embedded co-determine themaximum resilience that can be achieved Given that the non-trivial equilibrium point exists and is stable the following ap-plies If the system is at a greater risk of being abandoned bythe PIPs (and eventually collapsing) Rsystem = RPPC if thesystem is at a greater risk of becoming unstable (and even-tually collapsing) Rsystem = Rstability Figure 1 illustrates therelationships between RPPC Rstability and Rsystem

22 Linking robustness and resilience

As discussed earlier robustness can be thought of as the op-posite of sensitivity A commonly used measure of sensitiv-ity is variance Thus variance of a given function under spe-cific disturbance or uncertainty regimes may be used to in-dicate robustness of that function against those disturbancesor uncertainty regimes (robustness of what to what) In thiscase the system function of interest is the system resilienceRsystem By choosing Rsystem as our function we can usefullylink these concepts If we use a ball and cup metaphor for re-silience the robustness of resilience refers to the degree atwhich the geometry of the cup changes as a result of externaldisturbances andor parameter changes However as we willargue below relating high variance in Rsystem to low robust-ness may be misleading and should not be used in evaluatinga given policy By definition the variance treats ldquogood de-viationsrdquo and ldquobad deviationsrdquo from the mean equally For

functions with preferred values such as resilience or profitvalues greater than the mean and those lower should not betreated in the same way Specifically contribution to a highvariance from a heavy tail in the good direction should notbe translated to less robustness This problem also arises inassessing financial risk what makes an asset risky is the val-ues on the ldquobad tailrdquo of the distribution (ie low or nega-tive profits) This has motivated more and more analyses toswitch to considering other measures of risk such as the con-ditional value at risk in evaluating their portfolios of invest-ment (Rockafellar and Uryasev 2000 Krokhmal et al 2002Sarykalin et al 2014 AitSahlia et al 2011 Zymler et al2013) Intuitively this means that the shape of the ldquocuprdquo canbe asymmetric and we need to take this into account

Following this logic we propose to use a ldquobelow-meanmeanrdquo as a new robustness metric the mean of all resiliencevalues lower than the mean This new definition of the robust-ness metric has several desirable features First it can now beappropriately thought of as a robustness metric in the sensethat the higher the value the more robust the system (unlikethe variance for which low variance means high robustness)Second by using the mean as the threshold value for bad de-viations we remove some arbitrariness associated with pre-scribing a certain quantile (eg 5th or 10th quantile) in cal-culating the conditional value at risk Third it still carriessome information about the sensitivity of the resilience met-ric to outside factors ndash the information that variance conveysthat is the higher the below-mean mean (ie the bad devia-tions from the mean are small and the below-mean mean isclose to the mean) the less sensitive ndash and thus more robustndash the resilience metric

In this study we subject the modeled system to uncertaintyin one natural factor and one social factor namely the natu-ral replenishment rate of the resource g and the payoff thata RU earns from working outside the system w Thus weare computing how the resilience of the system to shocks

Earth Syst Dynam 9 1159ndash1168 2018 wwwearth-syst-dynamnet911592018

M Homayounfar et al Linking resilience and robustness and uncovering their trade-offs 1163

Figure 2 Variation of Rsystem of a CIS with a fixed policy (Cy) over 10 000 settings associated with uncertainty characterized byg isin [75125] w isin [075125] (a)Rsystem surface and (b)Rsystem contours The values ofRsystem are used to calculate the meanmicroRsystem and the below-mean mean microRltmicro In this particular case the resilience does not change much when g is greater than about 100 but becomesmore sensitive to both g and w when g is lower than 100

andor variation in state variables changes as the parametersg and w change (ie we are uncertain about the underlyingsocial-ecological setting of the system) In particular we as-sume that g is uniformly distributed over the range [75 125]and w is uniformly distributed over the range [075 125]A social-ecological setting or setting is defined as a com-bination of g and w For a given policy (a Cndashy combina-tion) we calculate Rsystem for 10 000 settings (ie 10 000gndashw combinations see Fig 2) Then from these 10 000 val-ues of the resilience metric Rsystem we calculate the meanmicroRsystem = E

[Rsystem

] and use it as the resilience metric of

the coupled system with a given policy and the below-meanmean microRltmicro = E

[Rsystem|Rsystem lt microRsystem

] as the metric

for robustness of resilience This metric measures the robust-ness of the capacity of the system to cope with variation instate variables IHM R and U to fundamental uncertaintyabout the underlying setting of the system

3 Results

The surfaces and contours of the system resilience metricmicroRsystem and the associated different policies (Cndashy) over thepolicy space are shown in Fig 3a and b respectively Thepolicies with sustainable outcomes are located in the mid-dle of the policy space with microRsystem peaking in the centerand declining as policies become more extreme in either di-rection Our analysis also shows that microRsystem is more or lessproportional to the fraction of settings (gndashw combinations)under which the system with that particular policy (a Cndashycombination) results in a sustainable outcome (Rsystem gt 0)A similar concept has been used in the robust decision mak-ing literature (eg Groves and Lempert 2007 Bryant andLempert 2010)

The surfaces and contours of the robustness of microRltmicro as-sociated with different policies over the policy space are

shown in Fig 3c and d respectively The microRltmicro ldquolandscaperdquois more irregular having two local maxima with one beingmore dominant than the other The region with high robust-ness appears to be in the same general areas as the regionwith high resilience These features reflect the nonlinear in-terplay between the model parameters and model structureand may affect the nature of the trade-off between robustnessand resilience reported in Figs 4 and 5

We explore the interplay between microRsystem and microRltmicro inFig 4 Figure 4 shows that there are no perfect policies inthe sense that no policies yield both maximum resilience andmaximum robustness Recall that the robustness indicateshow sensitiveRsystem itself is to uncertainty in the underlyingsetting of the system (eg g and w) The best policies arethose along the Pareto frontier in the resiliencendashrobustnessspace among this set of Pareto-optimal policies an increasein resilience is necessarily accompanied by a decrease in ro-bustness clearly illustrating the trade-off between robustnessand resilience Fig 5 illustrates where the Pareto-optimalpolicies are located in the policy space

4 Discussion and conclusions

In this paper we exploit the simplicity of a stylized modelto quantitatively link resilience and robustness by computinghow the CIS resilience to shocks in state variables changeswith parameters In this way we compute the robustness ofCIS resilience to uncertainty in the underlying CIS settingThe resilience metric developed here is a measure of how farthe CIS is from the boundaries beyond which it will collapseThe model affords us with expressions of these boundarieswhich clearly show how social and biophysical factors inter-play to define these boundaries With a concrete definition ofresilience resilience itself can be considered as the ldquoof whatrdquoin the ldquorobustness of what to whatrdquo notion In particular we

wwwearth-syst-dynamnet911592018 Earth Syst Dynam 9 1159ndash1168 2018

1164 M Homayounfar et al Linking resilience and robustness and uncovering their trade-offs

0 02 04 06 08 1C

0

02

04

06

08

1

y

01

02

03

04

05

06

07

0 02 04 06 08 1C

0

02

04

06

08

1

y

01

02

03

04

05

(a) (b)

(d)(c)

Figure 3 The mean microRsystem and the below-mean mean of Rsystem microRltmicro over the entire decision space (a) Surface of the resiliencemetric microRsystem (b) Contours of microRsystem (c) Surface of the robustness the below-mean mean (microRltmicro) (d) Contours of microRltmicro

0 02 04 06 08

R-system

0

01

02

03

04

05

06

(R-s

yste

mlt

R-s

yste

m)

Figure 4 Resiliencendashrobustness trade-off Each point representsmicroRsystem and microRltmicro of the coupled system with a given policy Theblack dots represent a set of Pareto-optimal policies

use the below-mean mean of the quantitatively defined re-silience metric as the metric of robustness Consequently thisenables us to rigorously investigate the interplay between thetwo important but not always well-defined system proper-ties A key finding is the fundamental trade-off between re-silience and robustness there are no perfect policies in gov-erning a CIS only Pareto-optimal ones Specifically policiesdesigned to maximize the resilience of a CIS to shocks ontimescales at which the state variables play out may be verysensitive to being wrong about our understanding of the un-derlying dynamics of the CIS in question

Importantly we hope this work will stimulate further ad-vances in rigorous studies of CISs that address such sub-tle policy-relevant questions a few of which we briefly dis-cussed here More dimensions can be considered in definingPareto-optimality Figure 5 may give an impression that theset of Pareto-optimal policies is confined to a small region inthe policy space which would imply that PIPs do not havethat many choices ndash even in a simple CIS like the one stud-ied But that would be a wrong impression In addition toresilience and robustness (as defined here) a policy makeror a social planner may be interested in other types of ro-bustness with different ldquoof whatrdquo and ldquoto whatrdquo components

Earth Syst Dynam 9 1159ndash1168 2018 wwwearth-syst-dynamnet911592018

M Homayounfar et al Linking resilience and robustness and uncovering their trade-offs 1165

0

02

04

08

1

01

02

03

04

05

06

07

06y

160 800 20 40C

01

02

03

04

05

0

02

04

08

1

06y

160 800 20 40C

(a) (b)

Figure 5 Pareto-optimal policies represented by black dots in the policy space (Cndashy plane) superimposed with resilience (microRsystem ) (a)and robustness (microRltmicro) contours (b)

They may also be concerned about other system propertieseg productivity user participation etc As more dimensionsare considered the set of Pareto-optimal policies grow In thesame spirit as that of the work done here these other dimen-sions should be defined rigorously

This work also lends itself to more rigorous studies ofldquoadaptive governancerdquo In the present study the governancestructure represented by a policy (a combination of C andy) is fixed A natural next step is to explore if a policy isallowed to change how one may improve the resilience androbustness of a CIS andor alter the nature of their inher-ent trade-offs For example if C and y are to be functionsof other factors eg resource availability and outside incen-tives what functional forms should they take to improve thesystemrsquos resilience and robustness Indeed in the absence oftransparent metrics attempts to explore such adaptive poli-cies are severely limited

Additionally agents in a CIS may have the capacity tochange their behavior in response to changes in policy en-vironmental conditions technological changes and so on Inthis study strategic behavior and the decision-making pro-cess are assumed unchanged in the analysis Adaptation instrategic behavior of agents will subsequently alter the natureof resilience its robustness and their trade-offs Capturingsuch effects of adaptation requires structural changes to themodel eg in terms of specification of payoffs or even theformulation of the dynamical equations With such adaptivesocial agents how should one devise adaptive governance toenhance resilience and robustness of a CIS Addressing sucha question is a theoretically intriguing future research direc-tion with great practical implications

In keeping with the theme of ldquosocial dynamics and plane-tary boundaries in Earth system modellingrdquo our results shedlight on how social and biophysical factors may interplay todefine boundaries of a sustainable coupled infrastructure sys-tem While the modeled system here is admittedly simpleour methodology and results constitute a step toward quanti-tatively and meaningfully combining social and biophysicalfactors into indicators of boundaries of more complex sys-tems Just as in this work once those boundaries are clearlydefined calculation and discussion of resilience and robust-ness can become concrete

Data availability No data sets were used in this article

wwwearth-syst-dynamnet911592018 Earth Syst Dynam 9 1159ndash1168 2018

1166 M Homayounfar et al Linking resilience and robustness and uncovering their trade-offs

Appendix A

Basic model Here we briefly describe the basic model pre-sented by Muneepeerakul and Anderies (2017) The modelshows dynamic behavior of three principal variables namelythe state of the public infrastructure IHM resource level Rand the fraction of time a user makes use of infrastructureU through Eqs (A1 A4 and A5) The schematic diagramof this system of equations is shown in Fig A1

In this context IHM depends on PIPs in term of mainte-nance cost and has a positive relationship with the capacityof users to create resource flows Eq (A1) illustrates the dy-namics of IHM as follows

dIHMdt=M ( )minus δH (IHM ) (A1)

where δ is the infrastructurersquos depreciation rate andH (IHM )states functional relationship of public infrastructure and pro-ductivity of each RU According to Muneepeerakul and An-deries (2017) many shared infrastructures can be modeledby threshold functions Given that H (IHM ) shows thresh-old behavior they used a piecewise linear function to capturesuch behavior through Eq (A2)

H (IHM )=

0IHM lt I0

hIHM minus I0

Imminus I0I0 le IHM le Im

hIHM ge Im

(A2)

where h represents the maximum amount of harvest byeach user under no restriction and I0 and Im are the lowerbound and upper bound thresholds of IHM respectivelyAlso M ( ) is the maintenance function (Eq A3) and de-pends on the social structure of the system

M ( )= micro2yCpRUNH (IHM ) (A3)

In Eq (A3) given the number of users N RUNH (IHM ) isthe total harvest from the natural infrastructure The RUs selltotal harvest at price p to generate revenue Subsequentlythey assign a proportion C of revenue to PIPrsquos for their con-tribution Meanwhile the PIPrsquos spend proportion y of C onmaintaining public infrastructure through the maintenancefunction M ( ) Also micro2 is the maintenance effectivenessof PIPrsquos investment

The second variable is the resource levelR They assumedthe dynamics of resource to be

dRdt=G (R)minusRUNH (IHM ) (A4)

Natural infrastructure is assumed to invoke the conservationlaw comprising of regenerating capacity (G (R)= gminus dR)and total unit of harvest RUNH (IHM ) The definition pre-sented for G is the simplest model for natural infrastructurewhere g and d are the natural replenishment and the lossrates respectively

HM IHM

Resource users

N

Natural infrastructure

Public infrastructure

R G(R)wp

infrastructurePublic

providersy

w

M()

UNI

minusδ

HMUNRH( )

IHMH( ) IHMH( )

IHMH( )

IHMCpUNRH( )

I

Figure A1 Schematic diagram of the dynamical system modelTaken from Muneepeerakul and Anderies (2017)

The strategic behavior of the resource users (RUs) is cap-tured by employing a replicator equation Indeed replica-tor dynamics provide modelers with simple realistic so-cial mechanism where agents follow and replicate better-offstrategies The two possible strategies considered for RUsare staying inside the system with the associated payoff ofπU = (1minusC)pRH (IHM ) or leaving the system with thepayoff of πw = w According to the replicator equation

dUdt= rU (1minusU ) (πU minusw) (A5)

The replicator equation represents the fraction of time thatRUs assign to working inside system given C and y LikeRUs there is also two alternatives for PIPs working insidethe system or working for another CIS which leads to sys-tem failure Meanwhile C and y characterize the strategyor policy of PIPs The PIPs will participate in this coupledsystem only when πp = (1minus y)pCRUNH (IHM )ge wP Inother words the PIPs maintain the system when they arebetter-off than working outside This condition is termed thePIP Participation Constraint (PPC)

Based on the system of three differential equations(Eqs A1 A4 and A5) the sustainable equilibria ie long-term system outcomes that satisfy the stability condition andPPC can be expressed as follows

ilowastHM =yCUlowastNRlowast

gH(I lowastHM

)Rlowast =

g

d

(1minus

ilowastHM

yC

)

Ulowast =(1minusC)yC

φ1ilowast

HM (A6)

where ilowastHM =IlowastHMδ

micro2pg(indicates dimensionless) and φ1 =

pgwN

a dimensionless group representing the relative lucrativenessof the system namely the ratio of potential income ndash withthe entire resource flow turned into income ndash relative to out-side wage The results reported in this study are based on thefollowing parameter values h= 00005 δ = 01 I0 = 03Im = 3 g = 100 d = 002 N = 1000 r = 015 p = 10w = 1 wp = 100 and micro2 = 0001

Earth Syst Dynam 9 1159ndash1168 2018 wwwearth-syst-dynamnet911592018

M Homayounfar et al Linking resilience and robustness and uncovering their trade-offs 1167

Author contributions MH RM JMA and CPM designed thestudy MH carried out the analysis MH and RM prepared themanuscript with contributions from all co-authors

Competing interests The authors declare that they have no con-flict of interest

Special issue statement This article is part of the special issueldquoSocial dynamics and planetary boundaries in Earth system mod-ellingrdquo It is not associated with a conference

Acknowledgements John M Anderies and Rachata Muneepeer-akul acknowledge the support from the grant NSF GEO-1115054Rachata Muneepeerakul and Mehran Homayounfar acknowl-edge support from the Army Research OfficeArmy ResearchLaboratory The research reported in this grant was supported inwhole or in part by the Army Research OfficeArmy ResearchLaboratory under award no W911NF1810267 (Multi-UniversityResearch Initiative) The views and conclusions contained in thisdocument are those of the authors and should not be interpreted asrepresenting the official policies either expressed or implied of theArmy Research Office or the US Government The authors alsothank the editor and three anonymous referees for their constructiveand useful comments

Edited by Jonathan DongesReviewed by three anonymous referees

References

AitSahlia F Wang C J Cabrera V E Uryasev S and FraisseC W Optimal crop planting schedules and financial hedgingstrategies under ENSO-based climate forecasts Ann Oper Res190 201ndash220 2011

Anderies J M Janssen M A and Walker B H Graz-ing Management Resilience and the Dynamics of aFire-Driven Rangeland System Ecosystems 5 23ndash44httpsdoiorg101007s10021-001-0053-9 2002

Anderies J M Janssen M and Ostrom E A Frameworkto Analyze the Robustness of Social-Ecological Systems froman Institutional Perspective Ecol Soc 9 18 httpwwwecologyandsocietyorgvol9iss1art18inlinehtml 2004

Anderies J M Ryan P and Walker B H Loss of ResilienceCrisis and Institutional Change Lessons from an Intensive Agri-cultural System in Southeastern Australia Ecosystems 9 865ndash878 httpsdoiorg101007s10021-006-0017-1 2006

Anderies J M Rodriguez A A Janssen M A and CifdalozO Panaceas Uncertainty and the Robust Control Frameworkin Sustainability Science P Natl Acad Sci USA 104 15194ndash15199 httpsdoiorg101073pnas0702655104 2007

Anderies J M Janssen M and Schlager E Institutions andthe performance of coupled infrastructure systems Int J Com-mons 10 495ndash516 httpsdoiorg1018352ijc651 2016

ASCE RCIA Advisory Council Report card for Americarsquos Infras-trcutre Tech Rep American Society of Civil Engineers 2013

Barrett C B and Constas M A Toward a Theoryof Resilience for International Development Applica-tions P Natl Acad Sci USA 111 14625ndash14630httpsdoiorg101073pnas1320880111 2014

Berkes F and Folke C Linking Social and Ecological Systemsfor Resilience and Sustainability Linking Social and Ecologi-cal Systems Management Practices and Social Mechanisms forBuilding Resilience 1 13ndash20 1998

Berkes F Colding J and Folke C Navigating Social-EcologicalSystems Building Resilience for Complexity and change Build-ing p 393 httpsdoiorg101016jbiocon200401010 2003

Biggs R Schluumlter M Biggs D Bohensky E L BurnSil-ver S Cundill G Dakos V Daw T M Evans L SKotschy K Leitch A M Meek C Quinlan A Raudsepp-Hearne C Robards M D Schoon M L Schultz L andWest P C Toward Principles for Enhancing the Resilienceof Ecosystem Services Annu Rev Env Resour 37 421ndash448httpsdoiorg101146annurev-environ-051211-123836 2012

Bode H W Network Analysis and Feedback Amplifier DesignBell Telephone Laboratories Series p 551 1945

Bryant B P and Lempert R J Thinking inside the BoxA Participatory Computer-Assisted Approach to Sce-nario Discovery Technol Forecast Soc 77 34ndash49httpsdoiorg101016jtechfore200908002 2010

Carlson J M and Doyle J Complexity and Ro-bustness P Natl Acad Sci USA 99 2538ndash2545httpsdoiorg101073pnas012582499 2002

Carpenter S Walker B Anderies J M and Abel N FromMetaphor to Measurement Resilience of What to WhatEcosystems 4 765ndash781 httpsdoiorg101007s10021-001-0045-9 2001

Carpenter S R and Brock W A Spatial Complexity Resilienceand Policy Diversity Fishing on Lake-Rich Landscapes EcolSoc 9 8 2004

Carpenter S R Ludwig D and Brock W A Management of Eu-trophication for Lakes Subject to Potentially Irreversible ChangeEcol Appl 9 751ndash771 httpsdoiorg10230726413271999a

Carpenter S Brock W and Hanson P Ecological and SocialDynamics in Simple Models of Ecosystem Management Con-serv Ecol 3 1ndash31 httpsdoiorg105751ES-00122-0302041999b

Carpenter S R Westley F and Turner M G Surrogates for Re-silience of Social-Ecological Systems Ecosystems 8 941ndash944httpsdoiorg101007s10021-005-0170-y 2005

Chang S E and Shinozuka M Measuring Improvements in theDisaster Resilience of Communities Earthq Spectra 20 739ndash755 httpsdoiorg10119311775796 2004

Chang S E McDaniels T Fox J Dhariwal R and LongstaffH Toward Disaster-Resilient Cities Characterizing Resilienceof Infrastructure Systems with Expert Judgments Risk Anal 34416ndash434 httpsdoiorg101111risa12133 2014

Cote M and Nightingale A J Resilience ThinkingMeets Social Theory Prog Hum Geog 36 475ndash489httpsdoiorg1011770309132511425708 2012

Csete M E and Doyle J C Reverse Engineeringof Biological Complexity Science 295 1664ndash1669httpsdoiorg101126science1069981 2002

wwwearth-syst-dynamnet911592018 Earth Syst Dynam 9 1159ndash1168 2018

1162 M Homayounfar et al Linking resilience and robustness and uncovering their trade-offs

0 02 04 06 08 1C

0

02

04

06

08

1

y

-05

0

05

1

15

2

0 02 04 06 08 1C

0

02

04

06

08

1

y

01

02

03

04

05

06

07

08

0 02 04 06 08 1C

0

02

04

06

08

1

y

0

02

04

06

08

(a) (b) (c)

Figure 1 Resilience metrics for a specific setting (a gndashw combination) inside the sustainable region in the policy space (ie Cndashy plane)(a) RPPC contours (b) Rstability contours and (c) Rsystem contours The black star in panels (a) (b) and (c) indicates the policy with thehighest RPPC Rstability and Rsystem respectively

Equation (4) implies that Rsystem is positive only when thenontrivial equilibrium point (Eq A6) exists and is stable oth-erwise the system is considered not resilient and denoted byRsystem = 0 Rsystem thus represents the tension between thePIPs being too greedy (high C and low y) whereby they getclose to the stability boundary and ldquonot greedy enoughrdquo (lowC and high y) whereby they get close to the PPC given a par-ticular choice for wP Note that the values of w and wP rep-resent the socioeconomic embedding of the CIS Thereforethe biophysical structure of the CIS along with the socioe-conomic context in which it is embedded co-determine themaximum resilience that can be achieved Given that the non-trivial equilibrium point exists and is stable the following ap-plies If the system is at a greater risk of being abandoned bythe PIPs (and eventually collapsing) Rsystem = RPPC if thesystem is at a greater risk of becoming unstable (and even-tually collapsing) Rsystem = Rstability Figure 1 illustrates therelationships between RPPC Rstability and Rsystem

22 Linking robustness and resilience

As discussed earlier robustness can be thought of as the op-posite of sensitivity A commonly used measure of sensitiv-ity is variance Thus variance of a given function under spe-cific disturbance or uncertainty regimes may be used to in-dicate robustness of that function against those disturbancesor uncertainty regimes (robustness of what to what) In thiscase the system function of interest is the system resilienceRsystem By choosing Rsystem as our function we can usefullylink these concepts If we use a ball and cup metaphor for re-silience the robustness of resilience refers to the degree atwhich the geometry of the cup changes as a result of externaldisturbances andor parameter changes However as we willargue below relating high variance in Rsystem to low robust-ness may be misleading and should not be used in evaluatinga given policy By definition the variance treats ldquogood de-viationsrdquo and ldquobad deviationsrdquo from the mean equally For

functions with preferred values such as resilience or profitvalues greater than the mean and those lower should not betreated in the same way Specifically contribution to a highvariance from a heavy tail in the good direction should notbe translated to less robustness This problem also arises inassessing financial risk what makes an asset risky is the val-ues on the ldquobad tailrdquo of the distribution (ie low or nega-tive profits) This has motivated more and more analyses toswitch to considering other measures of risk such as the con-ditional value at risk in evaluating their portfolios of invest-ment (Rockafellar and Uryasev 2000 Krokhmal et al 2002Sarykalin et al 2014 AitSahlia et al 2011 Zymler et al2013) Intuitively this means that the shape of the ldquocuprdquo canbe asymmetric and we need to take this into account

Following this logic we propose to use a ldquobelow-meanmeanrdquo as a new robustness metric the mean of all resiliencevalues lower than the mean This new definition of the robust-ness metric has several desirable features First it can now beappropriately thought of as a robustness metric in the sensethat the higher the value the more robust the system (unlikethe variance for which low variance means high robustness)Second by using the mean as the threshold value for bad de-viations we remove some arbitrariness associated with pre-scribing a certain quantile (eg 5th or 10th quantile) in cal-culating the conditional value at risk Third it still carriessome information about the sensitivity of the resilience met-ric to outside factors ndash the information that variance conveysthat is the higher the below-mean mean (ie the bad devia-tions from the mean are small and the below-mean mean isclose to the mean) the less sensitive ndash and thus more robustndash the resilience metric

In this study we subject the modeled system to uncertaintyin one natural factor and one social factor namely the natu-ral replenishment rate of the resource g and the payoff thata RU earns from working outside the system w Thus weare computing how the resilience of the system to shocks

Earth Syst Dynam 9 1159ndash1168 2018 wwwearth-syst-dynamnet911592018

M Homayounfar et al Linking resilience and robustness and uncovering their trade-offs 1163

Figure 2 Variation of Rsystem of a CIS with a fixed policy (Cy) over 10 000 settings associated with uncertainty characterized byg isin [75125] w isin [075125] (a)Rsystem surface and (b)Rsystem contours The values ofRsystem are used to calculate the meanmicroRsystem and the below-mean mean microRltmicro In this particular case the resilience does not change much when g is greater than about 100 but becomesmore sensitive to both g and w when g is lower than 100

andor variation in state variables changes as the parametersg and w change (ie we are uncertain about the underlyingsocial-ecological setting of the system) In particular we as-sume that g is uniformly distributed over the range [75 125]and w is uniformly distributed over the range [075 125]A social-ecological setting or setting is defined as a com-bination of g and w For a given policy (a Cndashy combina-tion) we calculate Rsystem for 10 000 settings (ie 10 000gndashw combinations see Fig 2) Then from these 10 000 val-ues of the resilience metric Rsystem we calculate the meanmicroRsystem = E

[Rsystem

] and use it as the resilience metric of

the coupled system with a given policy and the below-meanmean microRltmicro = E

[Rsystem|Rsystem lt microRsystem

] as the metric

for robustness of resilience This metric measures the robust-ness of the capacity of the system to cope with variation instate variables IHM R and U to fundamental uncertaintyabout the underlying setting of the system

3 Results

The surfaces and contours of the system resilience metricmicroRsystem and the associated different policies (Cndashy) over thepolicy space are shown in Fig 3a and b respectively Thepolicies with sustainable outcomes are located in the mid-dle of the policy space with microRsystem peaking in the centerand declining as policies become more extreme in either di-rection Our analysis also shows that microRsystem is more or lessproportional to the fraction of settings (gndashw combinations)under which the system with that particular policy (a Cndashycombination) results in a sustainable outcome (Rsystem gt 0)A similar concept has been used in the robust decision mak-ing literature (eg Groves and Lempert 2007 Bryant andLempert 2010)

The surfaces and contours of the robustness of microRltmicro as-sociated with different policies over the policy space are

shown in Fig 3c and d respectively The microRltmicro ldquolandscaperdquois more irregular having two local maxima with one beingmore dominant than the other The region with high robust-ness appears to be in the same general areas as the regionwith high resilience These features reflect the nonlinear in-terplay between the model parameters and model structureand may affect the nature of the trade-off between robustnessand resilience reported in Figs 4 and 5

We explore the interplay between microRsystem and microRltmicro inFig 4 Figure 4 shows that there are no perfect policies inthe sense that no policies yield both maximum resilience andmaximum robustness Recall that the robustness indicateshow sensitiveRsystem itself is to uncertainty in the underlyingsetting of the system (eg g and w) The best policies arethose along the Pareto frontier in the resiliencendashrobustnessspace among this set of Pareto-optimal policies an increasein resilience is necessarily accompanied by a decrease in ro-bustness clearly illustrating the trade-off between robustnessand resilience Fig 5 illustrates where the Pareto-optimalpolicies are located in the policy space

4 Discussion and conclusions

In this paper we exploit the simplicity of a stylized modelto quantitatively link resilience and robustness by computinghow the CIS resilience to shocks in state variables changeswith parameters In this way we compute the robustness ofCIS resilience to uncertainty in the underlying CIS settingThe resilience metric developed here is a measure of how farthe CIS is from the boundaries beyond which it will collapseThe model affords us with expressions of these boundarieswhich clearly show how social and biophysical factors inter-play to define these boundaries With a concrete definition ofresilience resilience itself can be considered as the ldquoof whatrdquoin the ldquorobustness of what to whatrdquo notion In particular we

wwwearth-syst-dynamnet911592018 Earth Syst Dynam 9 1159ndash1168 2018

1164 M Homayounfar et al Linking resilience and robustness and uncovering their trade-offs

0 02 04 06 08 1C

0

02

04

06

08

1

y

01

02

03

04

05

06

07

0 02 04 06 08 1C

0

02

04

06

08

1

y

01

02

03

04

05

(a) (b)

(d)(c)

Figure 3 The mean microRsystem and the below-mean mean of Rsystem microRltmicro over the entire decision space (a) Surface of the resiliencemetric microRsystem (b) Contours of microRsystem (c) Surface of the robustness the below-mean mean (microRltmicro) (d) Contours of microRltmicro

0 02 04 06 08

R-system

0

01

02

03

04

05

06

(R-s

yste

mlt

R-s

yste

m)

Figure 4 Resiliencendashrobustness trade-off Each point representsmicroRsystem and microRltmicro of the coupled system with a given policy Theblack dots represent a set of Pareto-optimal policies

use the below-mean mean of the quantitatively defined re-silience metric as the metric of robustness Consequently thisenables us to rigorously investigate the interplay between thetwo important but not always well-defined system proper-ties A key finding is the fundamental trade-off between re-silience and robustness there are no perfect policies in gov-erning a CIS only Pareto-optimal ones Specifically policiesdesigned to maximize the resilience of a CIS to shocks ontimescales at which the state variables play out may be verysensitive to being wrong about our understanding of the un-derlying dynamics of the CIS in question

Importantly we hope this work will stimulate further ad-vances in rigorous studies of CISs that address such sub-tle policy-relevant questions a few of which we briefly dis-cussed here More dimensions can be considered in definingPareto-optimality Figure 5 may give an impression that theset of Pareto-optimal policies is confined to a small region inthe policy space which would imply that PIPs do not havethat many choices ndash even in a simple CIS like the one stud-ied But that would be a wrong impression In addition toresilience and robustness (as defined here) a policy makeror a social planner may be interested in other types of ro-bustness with different ldquoof whatrdquo and ldquoto whatrdquo components

Earth Syst Dynam 9 1159ndash1168 2018 wwwearth-syst-dynamnet911592018

M Homayounfar et al Linking resilience and robustness and uncovering their trade-offs 1165

0

02

04

08

1

01

02

03

04

05

06

07

06y

160 800 20 40C

01

02

03

04

05

0

02

04

08

1

06y

160 800 20 40C

(a) (b)

Figure 5 Pareto-optimal policies represented by black dots in the policy space (Cndashy plane) superimposed with resilience (microRsystem ) (a)and robustness (microRltmicro) contours (b)

They may also be concerned about other system propertieseg productivity user participation etc As more dimensionsare considered the set of Pareto-optimal policies grow In thesame spirit as that of the work done here these other dimen-sions should be defined rigorously

This work also lends itself to more rigorous studies ofldquoadaptive governancerdquo In the present study the governancestructure represented by a policy (a combination of C andy) is fixed A natural next step is to explore if a policy isallowed to change how one may improve the resilience androbustness of a CIS andor alter the nature of their inher-ent trade-offs For example if C and y are to be functionsof other factors eg resource availability and outside incen-tives what functional forms should they take to improve thesystemrsquos resilience and robustness Indeed in the absence oftransparent metrics attempts to explore such adaptive poli-cies are severely limited

Additionally agents in a CIS may have the capacity tochange their behavior in response to changes in policy en-vironmental conditions technological changes and so on Inthis study strategic behavior and the decision-making pro-cess are assumed unchanged in the analysis Adaptation instrategic behavior of agents will subsequently alter the natureof resilience its robustness and their trade-offs Capturingsuch effects of adaptation requires structural changes to themodel eg in terms of specification of payoffs or even theformulation of the dynamical equations With such adaptivesocial agents how should one devise adaptive governance toenhance resilience and robustness of a CIS Addressing sucha question is a theoretically intriguing future research direc-tion with great practical implications

In keeping with the theme of ldquosocial dynamics and plane-tary boundaries in Earth system modellingrdquo our results shedlight on how social and biophysical factors may interplay todefine boundaries of a sustainable coupled infrastructure sys-tem While the modeled system here is admittedly simpleour methodology and results constitute a step toward quanti-tatively and meaningfully combining social and biophysicalfactors into indicators of boundaries of more complex sys-tems Just as in this work once those boundaries are clearlydefined calculation and discussion of resilience and robust-ness can become concrete

Data availability No data sets were used in this article

wwwearth-syst-dynamnet911592018 Earth Syst Dynam 9 1159ndash1168 2018

1166 M Homayounfar et al Linking resilience and robustness and uncovering their trade-offs

Appendix A

Basic model Here we briefly describe the basic model pre-sented by Muneepeerakul and Anderies (2017) The modelshows dynamic behavior of three principal variables namelythe state of the public infrastructure IHM resource level Rand the fraction of time a user makes use of infrastructureU through Eqs (A1 A4 and A5) The schematic diagramof this system of equations is shown in Fig A1

In this context IHM depends on PIPs in term of mainte-nance cost and has a positive relationship with the capacityof users to create resource flows Eq (A1) illustrates the dy-namics of IHM as follows

dIHMdt=M ( )minus δH (IHM ) (A1)

where δ is the infrastructurersquos depreciation rate andH (IHM )states functional relationship of public infrastructure and pro-ductivity of each RU According to Muneepeerakul and An-deries (2017) many shared infrastructures can be modeledby threshold functions Given that H (IHM ) shows thresh-old behavior they used a piecewise linear function to capturesuch behavior through Eq (A2)

H (IHM )=

0IHM lt I0

hIHM minus I0

Imminus I0I0 le IHM le Im

hIHM ge Im

(A2)

where h represents the maximum amount of harvest byeach user under no restriction and I0 and Im are the lowerbound and upper bound thresholds of IHM respectivelyAlso M ( ) is the maintenance function (Eq A3) and de-pends on the social structure of the system

M ( )= micro2yCpRUNH (IHM ) (A3)

In Eq (A3) given the number of users N RUNH (IHM ) isthe total harvest from the natural infrastructure The RUs selltotal harvest at price p to generate revenue Subsequentlythey assign a proportion C of revenue to PIPrsquos for their con-tribution Meanwhile the PIPrsquos spend proportion y of C onmaintaining public infrastructure through the maintenancefunction M ( ) Also micro2 is the maintenance effectivenessof PIPrsquos investment

The second variable is the resource levelR They assumedthe dynamics of resource to be

dRdt=G (R)minusRUNH (IHM ) (A4)

Natural infrastructure is assumed to invoke the conservationlaw comprising of regenerating capacity (G (R)= gminus dR)and total unit of harvest RUNH (IHM ) The definition pre-sented for G is the simplest model for natural infrastructurewhere g and d are the natural replenishment and the lossrates respectively

HM IHM

Resource users

N

Natural infrastructure

Public infrastructure

R G(R)wp

infrastructurePublic

providersy

w

M()

UNI

minusδ

HMUNRH( )

IHMH( ) IHMH( )

IHMH( )

IHMCpUNRH( )

I

Figure A1 Schematic diagram of the dynamical system modelTaken from Muneepeerakul and Anderies (2017)

The strategic behavior of the resource users (RUs) is cap-tured by employing a replicator equation Indeed replica-tor dynamics provide modelers with simple realistic so-cial mechanism where agents follow and replicate better-offstrategies The two possible strategies considered for RUsare staying inside the system with the associated payoff ofπU = (1minusC)pRH (IHM ) or leaving the system with thepayoff of πw = w According to the replicator equation

dUdt= rU (1minusU ) (πU minusw) (A5)

The replicator equation represents the fraction of time thatRUs assign to working inside system given C and y LikeRUs there is also two alternatives for PIPs working insidethe system or working for another CIS which leads to sys-tem failure Meanwhile C and y characterize the strategyor policy of PIPs The PIPs will participate in this coupledsystem only when πp = (1minus y)pCRUNH (IHM )ge wP Inother words the PIPs maintain the system when they arebetter-off than working outside This condition is termed thePIP Participation Constraint (PPC)

Based on the system of three differential equations(Eqs A1 A4 and A5) the sustainable equilibria ie long-term system outcomes that satisfy the stability condition andPPC can be expressed as follows

ilowastHM =yCUlowastNRlowast

gH(I lowastHM

)Rlowast =

g

d

(1minus

ilowastHM

yC

)

Ulowast =(1minusC)yC

φ1ilowast

HM (A6)

where ilowastHM =IlowastHMδ

micro2pg(indicates dimensionless) and φ1 =

pgwN

a dimensionless group representing the relative lucrativenessof the system namely the ratio of potential income ndash withthe entire resource flow turned into income ndash relative to out-side wage The results reported in this study are based on thefollowing parameter values h= 00005 δ = 01 I0 = 03Im = 3 g = 100 d = 002 N = 1000 r = 015 p = 10w = 1 wp = 100 and micro2 = 0001

Earth Syst Dynam 9 1159ndash1168 2018 wwwearth-syst-dynamnet911592018

M Homayounfar et al Linking resilience and robustness and uncovering their trade-offs 1167

Author contributions MH RM JMA and CPM designed thestudy MH carried out the analysis MH and RM prepared themanuscript with contributions from all co-authors

Competing interests The authors declare that they have no con-flict of interest

Special issue statement This article is part of the special issueldquoSocial dynamics and planetary boundaries in Earth system mod-ellingrdquo It is not associated with a conference

Acknowledgements John M Anderies and Rachata Muneepeer-akul acknowledge the support from the grant NSF GEO-1115054Rachata Muneepeerakul and Mehran Homayounfar acknowl-edge support from the Army Research OfficeArmy ResearchLaboratory The research reported in this grant was supported inwhole or in part by the Army Research OfficeArmy ResearchLaboratory under award no W911NF1810267 (Multi-UniversityResearch Initiative) The views and conclusions contained in thisdocument are those of the authors and should not be interpreted asrepresenting the official policies either expressed or implied of theArmy Research Office or the US Government The authors alsothank the editor and three anonymous referees for their constructiveand useful comments

Edited by Jonathan DongesReviewed by three anonymous referees

References

AitSahlia F Wang C J Cabrera V E Uryasev S and FraisseC W Optimal crop planting schedules and financial hedgingstrategies under ENSO-based climate forecasts Ann Oper Res190 201ndash220 2011

Anderies J M Janssen M A and Walker B H Graz-ing Management Resilience and the Dynamics of aFire-Driven Rangeland System Ecosystems 5 23ndash44httpsdoiorg101007s10021-001-0053-9 2002

Anderies J M Janssen M and Ostrom E A Frameworkto Analyze the Robustness of Social-Ecological Systems froman Institutional Perspective Ecol Soc 9 18 httpwwwecologyandsocietyorgvol9iss1art18inlinehtml 2004

Anderies J M Ryan P and Walker B H Loss of ResilienceCrisis and Institutional Change Lessons from an Intensive Agri-cultural System in Southeastern Australia Ecosystems 9 865ndash878 httpsdoiorg101007s10021-006-0017-1 2006

Anderies J M Rodriguez A A Janssen M A and CifdalozO Panaceas Uncertainty and the Robust Control Frameworkin Sustainability Science P Natl Acad Sci USA 104 15194ndash15199 httpsdoiorg101073pnas0702655104 2007

Anderies J M Janssen M and Schlager E Institutions andthe performance of coupled infrastructure systems Int J Com-mons 10 495ndash516 httpsdoiorg1018352ijc651 2016

ASCE RCIA Advisory Council Report card for Americarsquos Infras-trcutre Tech Rep American Society of Civil Engineers 2013

Barrett C B and Constas M A Toward a Theoryof Resilience for International Development Applica-tions P Natl Acad Sci USA 111 14625ndash14630httpsdoiorg101073pnas1320880111 2014

Berkes F and Folke C Linking Social and Ecological Systemsfor Resilience and Sustainability Linking Social and Ecologi-cal Systems Management Practices and Social Mechanisms forBuilding Resilience 1 13ndash20 1998

Berkes F Colding J and Folke C Navigating Social-EcologicalSystems Building Resilience for Complexity and change Build-ing p 393 httpsdoiorg101016jbiocon200401010 2003

Biggs R Schluumlter M Biggs D Bohensky E L BurnSil-ver S Cundill G Dakos V Daw T M Evans L SKotschy K Leitch A M Meek C Quinlan A Raudsepp-Hearne C Robards M D Schoon M L Schultz L andWest P C Toward Principles for Enhancing the Resilienceof Ecosystem Services Annu Rev Env Resour 37 421ndash448httpsdoiorg101146annurev-environ-051211-123836 2012

Bode H W Network Analysis and Feedback Amplifier DesignBell Telephone Laboratories Series p 551 1945

Bryant B P and Lempert R J Thinking inside the BoxA Participatory Computer-Assisted Approach to Sce-nario Discovery Technol Forecast Soc 77 34ndash49httpsdoiorg101016jtechfore200908002 2010

Carlson J M and Doyle J Complexity and Ro-bustness P Natl Acad Sci USA 99 2538ndash2545httpsdoiorg101073pnas012582499 2002

Carpenter S Walker B Anderies J M and Abel N FromMetaphor to Measurement Resilience of What to WhatEcosystems 4 765ndash781 httpsdoiorg101007s10021-001-0045-9 2001

Carpenter S R and Brock W A Spatial Complexity Resilienceand Policy Diversity Fishing on Lake-Rich Landscapes EcolSoc 9 8 2004

Carpenter S R Ludwig D and Brock W A Management of Eu-trophication for Lakes Subject to Potentially Irreversible ChangeEcol Appl 9 751ndash771 httpsdoiorg10230726413271999a

Carpenter S Brock W and Hanson P Ecological and SocialDynamics in Simple Models of Ecosystem Management Con-serv Ecol 3 1ndash31 httpsdoiorg105751ES-00122-0302041999b

Carpenter S R Westley F and Turner M G Surrogates for Re-silience of Social-Ecological Systems Ecosystems 8 941ndash944httpsdoiorg101007s10021-005-0170-y 2005

Chang S E and Shinozuka M Measuring Improvements in theDisaster Resilience of Communities Earthq Spectra 20 739ndash755 httpsdoiorg10119311775796 2004

Chang S E McDaniels T Fox J Dhariwal R and LongstaffH Toward Disaster-Resilient Cities Characterizing Resilienceof Infrastructure Systems with Expert Judgments Risk Anal 34416ndash434 httpsdoiorg101111risa12133 2014

Cote M and Nightingale A J Resilience ThinkingMeets Social Theory Prog Hum Geog 36 475ndash489httpsdoiorg1011770309132511425708 2012

Csete M E and Doyle J C Reverse Engineeringof Biological Complexity Science 295 1664ndash1669httpsdoiorg101126science1069981 2002

wwwearth-syst-dynamnet911592018 Earth Syst Dynam 9 1159ndash1168 2018

M Homayounfar et al Linking resilience and robustness and uncovering their trade-offs 1163

Figure 2 Variation of Rsystem of a CIS with a fixed policy (Cy) over 10 000 settings associated with uncertainty characterized byg isin [75125] w isin [075125] (a)Rsystem surface and (b)Rsystem contours The values ofRsystem are used to calculate the meanmicroRsystem and the below-mean mean microRltmicro In this particular case the resilience does not change much when g is greater than about 100 but becomesmore sensitive to both g and w when g is lower than 100

andor variation in state variables changes as the parametersg and w change (ie we are uncertain about the underlyingsocial-ecological setting of the system) In particular we as-sume that g is uniformly distributed over the range [75 125]and w is uniformly distributed over the range [075 125]A social-ecological setting or setting is defined as a com-bination of g and w For a given policy (a Cndashy combina-tion) we calculate Rsystem for 10 000 settings (ie 10 000gndashw combinations see Fig 2) Then from these 10 000 val-ues of the resilience metric Rsystem we calculate the meanmicroRsystem = E

[Rsystem

] and use it as the resilience metric of

the coupled system with a given policy and the below-meanmean microRltmicro = E

[Rsystem|Rsystem lt microRsystem

] as the metric

for robustness of resilience This metric measures the robust-ness of the capacity of the system to cope with variation instate variables IHM R and U to fundamental uncertaintyabout the underlying setting of the system

3 Results

The surfaces and contours of the system resilience metricmicroRsystem and the associated different policies (Cndashy) over thepolicy space are shown in Fig 3a and b respectively Thepolicies with sustainable outcomes are located in the mid-dle of the policy space with microRsystem peaking in the centerand declining as policies become more extreme in either di-rection Our analysis also shows that microRsystem is more or lessproportional to the fraction of settings (gndashw combinations)under which the system with that particular policy (a Cndashycombination) results in a sustainable outcome (Rsystem gt 0)A similar concept has been used in the robust decision mak-ing literature (eg Groves and Lempert 2007 Bryant andLempert 2010)

The surfaces and contours of the robustness of microRltmicro as-sociated with different policies over the policy space are

shown in Fig 3c and d respectively The microRltmicro ldquolandscaperdquois more irregular having two local maxima with one beingmore dominant than the other The region with high robust-ness appears to be in the same general areas as the regionwith high resilience These features reflect the nonlinear in-terplay between the model parameters and model structureand may affect the nature of the trade-off between robustnessand resilience reported in Figs 4 and 5

We explore the interplay between microRsystem and microRltmicro inFig 4 Figure 4 shows that there are no perfect policies inthe sense that no policies yield both maximum resilience andmaximum robustness Recall that the robustness indicateshow sensitiveRsystem itself is to uncertainty in the underlyingsetting of the system (eg g and w) The best policies arethose along the Pareto frontier in the resiliencendashrobustnessspace among this set of Pareto-optimal policies an increasein resilience is necessarily accompanied by a decrease in ro-bustness clearly illustrating the trade-off between robustnessand resilience Fig 5 illustrates where the Pareto-optimalpolicies are located in the policy space

4 Discussion and conclusions

In this paper we exploit the simplicity of a stylized modelto quantitatively link resilience and robustness by computinghow the CIS resilience to shocks in state variables changeswith parameters In this way we compute the robustness ofCIS resilience to uncertainty in the underlying CIS settingThe resilience metric developed here is a measure of how farthe CIS is from the boundaries beyond which it will collapseThe model affords us with expressions of these boundarieswhich clearly show how social and biophysical factors inter-play to define these boundaries With a concrete definition ofresilience resilience itself can be considered as the ldquoof whatrdquoin the ldquorobustness of what to whatrdquo notion In particular we

wwwearth-syst-dynamnet911592018 Earth Syst Dynam 9 1159ndash1168 2018

1164 M Homayounfar et al Linking resilience and robustness and uncovering their trade-offs

0 02 04 06 08 1C

0

02

04

06

08

1

y

01

02

03

04

05

06

07

0 02 04 06 08 1C

0

02

04

06

08

1

y

01

02

03

04

05

(a) (b)

(d)(c)

Figure 3 The mean microRsystem and the below-mean mean of Rsystem microRltmicro over the entire decision space (a) Surface of the resiliencemetric microRsystem (b) Contours of microRsystem (c) Surface of the robustness the below-mean mean (microRltmicro) (d) Contours of microRltmicro

0 02 04 06 08

R-system

0

01

02

03

04

05

06

(R-s

yste

mlt

R-s

yste

m)

Figure 4 Resiliencendashrobustness trade-off Each point representsmicroRsystem and microRltmicro of the coupled system with a given policy Theblack dots represent a set of Pareto-optimal policies

use the below-mean mean of the quantitatively defined re-silience metric as the metric of robustness Consequently thisenables us to rigorously investigate the interplay between thetwo important but not always well-defined system proper-ties A key finding is the fundamental trade-off between re-silience and robustness there are no perfect policies in gov-erning a CIS only Pareto-optimal ones Specifically policiesdesigned to maximize the resilience of a CIS to shocks ontimescales at which the state variables play out may be verysensitive to being wrong about our understanding of the un-derlying dynamics of the CIS in question

Importantly we hope this work will stimulate further ad-vances in rigorous studies of CISs that address such sub-tle policy-relevant questions a few of which we briefly dis-cussed here More dimensions can be considered in definingPareto-optimality Figure 5 may give an impression that theset of Pareto-optimal policies is confined to a small region inthe policy space which would imply that PIPs do not havethat many choices ndash even in a simple CIS like the one stud-ied But that would be a wrong impression In addition toresilience and robustness (as defined here) a policy makeror a social planner may be interested in other types of ro-bustness with different ldquoof whatrdquo and ldquoto whatrdquo components

Earth Syst Dynam 9 1159ndash1168 2018 wwwearth-syst-dynamnet911592018

M Homayounfar et al Linking resilience and robustness and uncovering their trade-offs 1165

0

02

04

08

1

01

02

03

04

05

06

07

06y

160 800 20 40C

01

02

03

04

05

0

02

04

08

1

06y

160 800 20 40C

(a) (b)

Figure 5 Pareto-optimal policies represented by black dots in the policy space (Cndashy plane) superimposed with resilience (microRsystem ) (a)and robustness (microRltmicro) contours (b)

They may also be concerned about other system propertieseg productivity user participation etc As more dimensionsare considered the set of Pareto-optimal policies grow In thesame spirit as that of the work done here these other dimen-sions should be defined rigorously

This work also lends itself to more rigorous studies ofldquoadaptive governancerdquo In the present study the governancestructure represented by a policy (a combination of C andy) is fixed A natural next step is to explore if a policy isallowed to change how one may improve the resilience androbustness of a CIS andor alter the nature of their inher-ent trade-offs For example if C and y are to be functionsof other factors eg resource availability and outside incen-tives what functional forms should they take to improve thesystemrsquos resilience and robustness Indeed in the absence oftransparent metrics attempts to explore such adaptive poli-cies are severely limited

Additionally agents in a CIS may have the capacity tochange their behavior in response to changes in policy en-vironmental conditions technological changes and so on Inthis study strategic behavior and the decision-making pro-cess are assumed unchanged in the analysis Adaptation instrategic behavior of agents will subsequently alter the natureof resilience its robustness and their trade-offs Capturingsuch effects of adaptation requires structural changes to themodel eg in terms of specification of payoffs or even theformulation of the dynamical equations With such adaptivesocial agents how should one devise adaptive governance toenhance resilience and robustness of a CIS Addressing sucha question is a theoretically intriguing future research direc-tion with great practical implications

In keeping with the theme of ldquosocial dynamics and plane-tary boundaries in Earth system modellingrdquo our results shedlight on how social and biophysical factors may interplay todefine boundaries of a sustainable coupled infrastructure sys-tem While the modeled system here is admittedly simpleour methodology and results constitute a step toward quanti-tatively and meaningfully combining social and biophysicalfactors into indicators of boundaries of more complex sys-tems Just as in this work once those boundaries are clearlydefined calculation and discussion of resilience and robust-ness can become concrete

Data availability No data sets were used in this article

wwwearth-syst-dynamnet911592018 Earth Syst Dynam 9 1159ndash1168 2018

1166 M Homayounfar et al Linking resilience and robustness and uncovering their trade-offs

Appendix A

Basic model Here we briefly describe the basic model pre-sented by Muneepeerakul and Anderies (2017) The modelshows dynamic behavior of three principal variables namelythe state of the public infrastructure IHM resource level Rand the fraction of time a user makes use of infrastructureU through Eqs (A1 A4 and A5) The schematic diagramof this system of equations is shown in Fig A1

In this context IHM depends on PIPs in term of mainte-nance cost and has a positive relationship with the capacityof users to create resource flows Eq (A1) illustrates the dy-namics of IHM as follows

dIHMdt=M ( )minus δH (IHM ) (A1)

where δ is the infrastructurersquos depreciation rate andH (IHM )states functional relationship of public infrastructure and pro-ductivity of each RU According to Muneepeerakul and An-deries (2017) many shared infrastructures can be modeledby threshold functions Given that H (IHM ) shows thresh-old behavior they used a piecewise linear function to capturesuch behavior through Eq (A2)

H (IHM )=

0IHM lt I0

hIHM minus I0

Imminus I0I0 le IHM le Im

hIHM ge Im

(A2)

where h represents the maximum amount of harvest byeach user under no restriction and I0 and Im are the lowerbound and upper bound thresholds of IHM respectivelyAlso M ( ) is the maintenance function (Eq A3) and de-pends on the social structure of the system

M ( )= micro2yCpRUNH (IHM ) (A3)

In Eq (A3) given the number of users N RUNH (IHM ) isthe total harvest from the natural infrastructure The RUs selltotal harvest at price p to generate revenue Subsequentlythey assign a proportion C of revenue to PIPrsquos for their con-tribution Meanwhile the PIPrsquos spend proportion y of C onmaintaining public infrastructure through the maintenancefunction M ( ) Also micro2 is the maintenance effectivenessof PIPrsquos investment

The second variable is the resource levelR They assumedthe dynamics of resource to be

dRdt=G (R)minusRUNH (IHM ) (A4)

Natural infrastructure is assumed to invoke the conservationlaw comprising of regenerating capacity (G (R)= gminus dR)and total unit of harvest RUNH (IHM ) The definition pre-sented for G is the simplest model for natural infrastructurewhere g and d are the natural replenishment and the lossrates respectively

HM IHM

Resource users

N

Natural infrastructure

Public infrastructure

R G(R)wp

infrastructurePublic

providersy

w

M()

UNI

minusδ

HMUNRH( )

IHMH( ) IHMH( )

IHMH( )

IHMCpUNRH( )

I

Figure A1 Schematic diagram of the dynamical system modelTaken from Muneepeerakul and Anderies (2017)

The strategic behavior of the resource users (RUs) is cap-tured by employing a replicator equation Indeed replica-tor dynamics provide modelers with simple realistic so-cial mechanism where agents follow and replicate better-offstrategies The two possible strategies considered for RUsare staying inside the system with the associated payoff ofπU = (1minusC)pRH (IHM ) or leaving the system with thepayoff of πw = w According to the replicator equation

dUdt= rU (1minusU ) (πU minusw) (A5)

The replicator equation represents the fraction of time thatRUs assign to working inside system given C and y LikeRUs there is also two alternatives for PIPs working insidethe system or working for another CIS which leads to sys-tem failure Meanwhile C and y characterize the strategyor policy of PIPs The PIPs will participate in this coupledsystem only when πp = (1minus y)pCRUNH (IHM )ge wP Inother words the PIPs maintain the system when they arebetter-off than working outside This condition is termed thePIP Participation Constraint (PPC)

Based on the system of three differential equations(Eqs A1 A4 and A5) the sustainable equilibria ie long-term system outcomes that satisfy the stability condition andPPC can be expressed as follows

ilowastHM =yCUlowastNRlowast

gH(I lowastHM

)Rlowast =

g

d

(1minus

ilowastHM

yC

)

Ulowast =(1minusC)yC

φ1ilowast

HM (A6)

where ilowastHM =IlowastHMδ

micro2pg(indicates dimensionless) and φ1 =

pgwN

a dimensionless group representing the relative lucrativenessof the system namely the ratio of potential income ndash withthe entire resource flow turned into income ndash relative to out-side wage The results reported in this study are based on thefollowing parameter values h= 00005 δ = 01 I0 = 03Im = 3 g = 100 d = 002 N = 1000 r = 015 p = 10w = 1 wp = 100 and micro2 = 0001

Earth Syst Dynam 9 1159ndash1168 2018 wwwearth-syst-dynamnet911592018

M Homayounfar et al Linking resilience and robustness and uncovering their trade-offs 1167

Author contributions MH RM JMA and CPM designed thestudy MH carried out the analysis MH and RM prepared themanuscript with contributions from all co-authors

Competing interests The authors declare that they have no con-flict of interest

Special issue statement This article is part of the special issueldquoSocial dynamics and planetary boundaries in Earth system mod-ellingrdquo It is not associated with a conference

Acknowledgements John M Anderies and Rachata Muneepeer-akul acknowledge the support from the grant NSF GEO-1115054Rachata Muneepeerakul and Mehran Homayounfar acknowl-edge support from the Army Research OfficeArmy ResearchLaboratory The research reported in this grant was supported inwhole or in part by the Army Research OfficeArmy ResearchLaboratory under award no W911NF1810267 (Multi-UniversityResearch Initiative) The views and conclusions contained in thisdocument are those of the authors and should not be interpreted asrepresenting the official policies either expressed or implied of theArmy Research Office or the US Government The authors alsothank the editor and three anonymous referees for their constructiveand useful comments

Edited by Jonathan DongesReviewed by three anonymous referees

References

AitSahlia F Wang C J Cabrera V E Uryasev S and FraisseC W Optimal crop planting schedules and financial hedgingstrategies under ENSO-based climate forecasts Ann Oper Res190 201ndash220 2011

Anderies J M Janssen M A and Walker B H Graz-ing Management Resilience and the Dynamics of aFire-Driven Rangeland System Ecosystems 5 23ndash44httpsdoiorg101007s10021-001-0053-9 2002

Anderies J M Janssen M and Ostrom E A Frameworkto Analyze the Robustness of Social-Ecological Systems froman Institutional Perspective Ecol Soc 9 18 httpwwwecologyandsocietyorgvol9iss1art18inlinehtml 2004

Anderies J M Ryan P and Walker B H Loss of ResilienceCrisis and Institutional Change Lessons from an Intensive Agri-cultural System in Southeastern Australia Ecosystems 9 865ndash878 httpsdoiorg101007s10021-006-0017-1 2006

Anderies J M Rodriguez A A Janssen M A and CifdalozO Panaceas Uncertainty and the Robust Control Frameworkin Sustainability Science P Natl Acad Sci USA 104 15194ndash15199 httpsdoiorg101073pnas0702655104 2007

Anderies J M Janssen M and Schlager E Institutions andthe performance of coupled infrastructure systems Int J Com-mons 10 495ndash516 httpsdoiorg1018352ijc651 2016

ASCE RCIA Advisory Council Report card for Americarsquos Infras-trcutre Tech Rep American Society of Civil Engineers 2013

Barrett C B and Constas M A Toward a Theoryof Resilience for International Development Applica-tions P Natl Acad Sci USA 111 14625ndash14630httpsdoiorg101073pnas1320880111 2014

Berkes F and Folke C Linking Social and Ecological Systemsfor Resilience and Sustainability Linking Social and Ecologi-cal Systems Management Practices and Social Mechanisms forBuilding Resilience 1 13ndash20 1998

Berkes F Colding J and Folke C Navigating Social-EcologicalSystems Building Resilience for Complexity and change Build-ing p 393 httpsdoiorg101016jbiocon200401010 2003

Biggs R Schluumlter M Biggs D Bohensky E L BurnSil-ver S Cundill G Dakos V Daw T M Evans L SKotschy K Leitch A M Meek C Quinlan A Raudsepp-Hearne C Robards M D Schoon M L Schultz L andWest P C Toward Principles for Enhancing the Resilienceof Ecosystem Services Annu Rev Env Resour 37 421ndash448httpsdoiorg101146annurev-environ-051211-123836 2012

Bode H W Network Analysis and Feedback Amplifier DesignBell Telephone Laboratories Series p 551 1945

Bryant B P and Lempert R J Thinking inside the BoxA Participatory Computer-Assisted Approach to Sce-nario Discovery Technol Forecast Soc 77 34ndash49httpsdoiorg101016jtechfore200908002 2010

Carlson J M and Doyle J Complexity and Ro-bustness P Natl Acad Sci USA 99 2538ndash2545httpsdoiorg101073pnas012582499 2002

Carpenter S Walker B Anderies J M and Abel N FromMetaphor to Measurement Resilience of What to WhatEcosystems 4 765ndash781 httpsdoiorg101007s10021-001-0045-9 2001

Carpenter S R and Brock W A Spatial Complexity Resilienceand Policy Diversity Fishing on Lake-Rich Landscapes EcolSoc 9 8 2004

Carpenter S R Ludwig D and Brock W A Management of Eu-trophication for Lakes Subject to Potentially Irreversible ChangeEcol Appl 9 751ndash771 httpsdoiorg10230726413271999a

Carpenter S Brock W and Hanson P Ecological and SocialDynamics in Simple Models of Ecosystem Management Con-serv Ecol 3 1ndash31 httpsdoiorg105751ES-00122-0302041999b

Carpenter S R Westley F and Turner M G Surrogates for Re-silience of Social-Ecological Systems Ecosystems 8 941ndash944httpsdoiorg101007s10021-005-0170-y 2005

Chang S E and Shinozuka M Measuring Improvements in theDisaster Resilience of Communities Earthq Spectra 20 739ndash755 httpsdoiorg10119311775796 2004

Chang S E McDaniels T Fox J Dhariwal R and LongstaffH Toward Disaster-Resilient Cities Characterizing Resilienceof Infrastructure Systems with Expert Judgments Risk Anal 34416ndash434 httpsdoiorg101111risa12133 2014

Cote M and Nightingale A J Resilience ThinkingMeets Social Theory Prog Hum Geog 36 475ndash489httpsdoiorg1011770309132511425708 2012

Csete M E and Doyle J C Reverse Engineeringof Biological Complexity Science 295 1664ndash1669httpsdoiorg101126science1069981 2002

wwwearth-syst-dynamnet911592018 Earth Syst Dynam 9 1159ndash1168 2018

1164 M Homayounfar et al Linking resilience and robustness and uncovering their trade-offs

0 02 04 06 08 1C

0

02

04

06

08

1

y

01

02

03

04

05

06

07

0 02 04 06 08 1C

0

02

04

06

08

1

y

01

02

03

04

05

(a) (b)

(d)(c)

Figure 3 The mean microRsystem and the below-mean mean of Rsystem microRltmicro over the entire decision space (a) Surface of the resiliencemetric microRsystem (b) Contours of microRsystem (c) Surface of the robustness the below-mean mean (microRltmicro) (d) Contours of microRltmicro

0 02 04 06 08

R-system

0

01

02

03

04

05

06

(R-s

yste

mlt

R-s

yste

m)

Figure 4 Resiliencendashrobustness trade-off Each point representsmicroRsystem and microRltmicro of the coupled system with a given policy Theblack dots represent a set of Pareto-optimal policies

use the below-mean mean of the quantitatively defined re-silience metric as the metric of robustness Consequently thisenables us to rigorously investigate the interplay between thetwo important but not always well-defined system proper-ties A key finding is the fundamental trade-off between re-silience and robustness there are no perfect policies in gov-erning a CIS only Pareto-optimal ones Specifically policiesdesigned to maximize the resilience of a CIS to shocks ontimescales at which the state variables play out may be verysensitive to being wrong about our understanding of the un-derlying dynamics of the CIS in question

Importantly we hope this work will stimulate further ad-vances in rigorous studies of CISs that address such sub-tle policy-relevant questions a few of which we briefly dis-cussed here More dimensions can be considered in definingPareto-optimality Figure 5 may give an impression that theset of Pareto-optimal policies is confined to a small region inthe policy space which would imply that PIPs do not havethat many choices ndash even in a simple CIS like the one stud-ied But that would be a wrong impression In addition toresilience and robustness (as defined here) a policy makeror a social planner may be interested in other types of ro-bustness with different ldquoof whatrdquo and ldquoto whatrdquo components

Earth Syst Dynam 9 1159ndash1168 2018 wwwearth-syst-dynamnet911592018

M Homayounfar et al Linking resilience and robustness and uncovering their trade-offs 1165

0

02

04

08

1

01

02

03

04

05

06

07

06y

160 800 20 40C

01

02

03

04

05

0

02

04

08

1

06y

160 800 20 40C

(a) (b)

Figure 5 Pareto-optimal policies represented by black dots in the policy space (Cndashy plane) superimposed with resilience (microRsystem ) (a)and robustness (microRltmicro) contours (b)

They may also be concerned about other system propertieseg productivity user participation etc As more dimensionsare considered the set of Pareto-optimal policies grow In thesame spirit as that of the work done here these other dimen-sions should be defined rigorously

This work also lends itself to more rigorous studies ofldquoadaptive governancerdquo In the present study the governancestructure represented by a policy (a combination of C andy) is fixed A natural next step is to explore if a policy isallowed to change how one may improve the resilience androbustness of a CIS andor alter the nature of their inher-ent trade-offs For example if C and y are to be functionsof other factors eg resource availability and outside incen-tives what functional forms should they take to improve thesystemrsquos resilience and robustness Indeed in the absence oftransparent metrics attempts to explore such adaptive poli-cies are severely limited

Additionally agents in a CIS may have the capacity tochange their behavior in response to changes in policy en-vironmental conditions technological changes and so on Inthis study strategic behavior and the decision-making pro-cess are assumed unchanged in the analysis Adaptation instrategic behavior of agents will subsequently alter the natureof resilience its robustness and their trade-offs Capturingsuch effects of adaptation requires structural changes to themodel eg in terms of specification of payoffs or even theformulation of the dynamical equations With such adaptivesocial agents how should one devise adaptive governance toenhance resilience and robustness of a CIS Addressing sucha question is a theoretically intriguing future research direc-tion with great practical implications

In keeping with the theme of ldquosocial dynamics and plane-tary boundaries in Earth system modellingrdquo our results shedlight on how social and biophysical factors may interplay todefine boundaries of a sustainable coupled infrastructure sys-tem While the modeled system here is admittedly simpleour methodology and results constitute a step toward quanti-tatively and meaningfully combining social and biophysicalfactors into indicators of boundaries of more complex sys-tems Just as in this work once those boundaries are clearlydefined calculation and discussion of resilience and robust-ness can become concrete

Data availability No data sets were used in this article

wwwearth-syst-dynamnet911592018 Earth Syst Dynam 9 1159ndash1168 2018

1166 M Homayounfar et al Linking resilience and robustness and uncovering their trade-offs

Appendix A

Basic model Here we briefly describe the basic model pre-sented by Muneepeerakul and Anderies (2017) The modelshows dynamic behavior of three principal variables namelythe state of the public infrastructure IHM resource level Rand the fraction of time a user makes use of infrastructureU through Eqs (A1 A4 and A5) The schematic diagramof this system of equations is shown in Fig A1

In this context IHM depends on PIPs in term of mainte-nance cost and has a positive relationship with the capacityof users to create resource flows Eq (A1) illustrates the dy-namics of IHM as follows

dIHMdt=M ( )minus δH (IHM ) (A1)

where δ is the infrastructurersquos depreciation rate andH (IHM )states functional relationship of public infrastructure and pro-ductivity of each RU According to Muneepeerakul and An-deries (2017) many shared infrastructures can be modeledby threshold functions Given that H (IHM ) shows thresh-old behavior they used a piecewise linear function to capturesuch behavior through Eq (A2)

H (IHM )=

0IHM lt I0

hIHM minus I0

Imminus I0I0 le IHM le Im

hIHM ge Im

(A2)

where h represents the maximum amount of harvest byeach user under no restriction and I0 and Im are the lowerbound and upper bound thresholds of IHM respectivelyAlso M ( ) is the maintenance function (Eq A3) and de-pends on the social structure of the system

M ( )= micro2yCpRUNH (IHM ) (A3)

In Eq (A3) given the number of users N RUNH (IHM ) isthe total harvest from the natural infrastructure The RUs selltotal harvest at price p to generate revenue Subsequentlythey assign a proportion C of revenue to PIPrsquos for their con-tribution Meanwhile the PIPrsquos spend proportion y of C onmaintaining public infrastructure through the maintenancefunction M ( ) Also micro2 is the maintenance effectivenessof PIPrsquos investment

The second variable is the resource levelR They assumedthe dynamics of resource to be

dRdt=G (R)minusRUNH (IHM ) (A4)

Natural infrastructure is assumed to invoke the conservationlaw comprising of regenerating capacity (G (R)= gminus dR)and total unit of harvest RUNH (IHM ) The definition pre-sented for G is the simplest model for natural infrastructurewhere g and d are the natural replenishment and the lossrates respectively

HM IHM

Resource users

N

Natural infrastructure

Public infrastructure

R G(R)wp

infrastructurePublic

providersy

w

M()

UNI

minusδ

HMUNRH( )

IHMH( ) IHMH( )

IHMH( )

IHMCpUNRH( )

I

Figure A1 Schematic diagram of the dynamical system modelTaken from Muneepeerakul and Anderies (2017)

The strategic behavior of the resource users (RUs) is cap-tured by employing a replicator equation Indeed replica-tor dynamics provide modelers with simple realistic so-cial mechanism where agents follow and replicate better-offstrategies The two possible strategies considered for RUsare staying inside the system with the associated payoff ofπU = (1minusC)pRH (IHM ) or leaving the system with thepayoff of πw = w According to the replicator equation

dUdt= rU (1minusU ) (πU minusw) (A5)

The replicator equation represents the fraction of time thatRUs assign to working inside system given C and y LikeRUs there is also two alternatives for PIPs working insidethe system or working for another CIS which leads to sys-tem failure Meanwhile C and y characterize the strategyor policy of PIPs The PIPs will participate in this coupledsystem only when πp = (1minus y)pCRUNH (IHM )ge wP Inother words the PIPs maintain the system when they arebetter-off than working outside This condition is termed thePIP Participation Constraint (PPC)

Based on the system of three differential equations(Eqs A1 A4 and A5) the sustainable equilibria ie long-term system outcomes that satisfy the stability condition andPPC can be expressed as follows

ilowastHM =yCUlowastNRlowast

gH(I lowastHM

)Rlowast =

g

d

(1minus

ilowastHM

yC

)

Ulowast =(1minusC)yC

φ1ilowast

HM (A6)

where ilowastHM =IlowastHMδ

micro2pg(indicates dimensionless) and φ1 =

pgwN

a dimensionless group representing the relative lucrativenessof the system namely the ratio of potential income ndash withthe entire resource flow turned into income ndash relative to out-side wage The results reported in this study are based on thefollowing parameter values h= 00005 δ = 01 I0 = 03Im = 3 g = 100 d = 002 N = 1000 r = 015 p = 10w = 1 wp = 100 and micro2 = 0001

Earth Syst Dynam 9 1159ndash1168 2018 wwwearth-syst-dynamnet911592018

M Homayounfar et al Linking resilience and robustness and uncovering their trade-offs 1167

Author contributions MH RM JMA and CPM designed thestudy MH carried out the analysis MH and RM prepared themanuscript with contributions from all co-authors

Competing interests The authors declare that they have no con-flict of interest

Special issue statement This article is part of the special issueldquoSocial dynamics and planetary boundaries in Earth system mod-ellingrdquo It is not associated with a conference

Acknowledgements John M Anderies and Rachata Muneepeer-akul acknowledge the support from the grant NSF GEO-1115054Rachata Muneepeerakul and Mehran Homayounfar acknowl-edge support from the Army Research OfficeArmy ResearchLaboratory The research reported in this grant was supported inwhole or in part by the Army Research OfficeArmy ResearchLaboratory under award no W911NF1810267 (Multi-UniversityResearch Initiative) The views and conclusions contained in thisdocument are those of the authors and should not be interpreted asrepresenting the official policies either expressed or implied of theArmy Research Office or the US Government The authors alsothank the editor and three anonymous referees for their constructiveand useful comments

Edited by Jonathan DongesReviewed by three anonymous referees

References

AitSahlia F Wang C J Cabrera V E Uryasev S and FraisseC W Optimal crop planting schedules and financial hedgingstrategies under ENSO-based climate forecasts Ann Oper Res190 201ndash220 2011

Anderies J M Janssen M A and Walker B H Graz-ing Management Resilience and the Dynamics of aFire-Driven Rangeland System Ecosystems 5 23ndash44httpsdoiorg101007s10021-001-0053-9 2002

Anderies J M Janssen M and Ostrom E A Frameworkto Analyze the Robustness of Social-Ecological Systems froman Institutional Perspective Ecol Soc 9 18 httpwwwecologyandsocietyorgvol9iss1art18inlinehtml 2004

Anderies J M Ryan P and Walker B H Loss of ResilienceCrisis and Institutional Change Lessons from an Intensive Agri-cultural System in Southeastern Australia Ecosystems 9 865ndash878 httpsdoiorg101007s10021-006-0017-1 2006

Anderies J M Rodriguez A A Janssen M A and CifdalozO Panaceas Uncertainty and the Robust Control Frameworkin Sustainability Science P Natl Acad Sci USA 104 15194ndash15199 httpsdoiorg101073pnas0702655104 2007

Anderies J M Janssen M and Schlager E Institutions andthe performance of coupled infrastructure systems Int J Com-mons 10 495ndash516 httpsdoiorg1018352ijc651 2016

ASCE RCIA Advisory Council Report card for Americarsquos Infras-trcutre Tech Rep American Society of Civil Engineers 2013

Barrett C B and Constas M A Toward a Theoryof Resilience for International Development Applica-tions P Natl Acad Sci USA 111 14625ndash14630httpsdoiorg101073pnas1320880111 2014

Berkes F and Folke C Linking Social and Ecological Systemsfor Resilience and Sustainability Linking Social and Ecologi-cal Systems Management Practices and Social Mechanisms forBuilding Resilience 1 13ndash20 1998

Berkes F Colding J and Folke C Navigating Social-EcologicalSystems Building Resilience for Complexity and change Build-ing p 393 httpsdoiorg101016jbiocon200401010 2003

Biggs R Schluumlter M Biggs D Bohensky E L BurnSil-ver S Cundill G Dakos V Daw T M Evans L SKotschy K Leitch A M Meek C Quinlan A Raudsepp-Hearne C Robards M D Schoon M L Schultz L andWest P C Toward Principles for Enhancing the Resilienceof Ecosystem Services Annu Rev Env Resour 37 421ndash448httpsdoiorg101146annurev-environ-051211-123836 2012

Bode H W Network Analysis and Feedback Amplifier DesignBell Telephone Laboratories Series p 551 1945

Bryant B P and Lempert R J Thinking inside the BoxA Participatory Computer-Assisted Approach to Sce-nario Discovery Technol Forecast Soc 77 34ndash49httpsdoiorg101016jtechfore200908002 2010

Carlson J M and Doyle J Complexity and Ro-bustness P Natl Acad Sci USA 99 2538ndash2545httpsdoiorg101073pnas012582499 2002

Carpenter S Walker B Anderies J M and Abel N FromMetaphor to Measurement Resilience of What to WhatEcosystems 4 765ndash781 httpsdoiorg101007s10021-001-0045-9 2001

Carpenter S R and Brock W A Spatial Complexity Resilienceand Policy Diversity Fishing on Lake-Rich Landscapes EcolSoc 9 8 2004

Carpenter S R Ludwig D and Brock W A Management of Eu-trophication for Lakes Subject to Potentially Irreversible ChangeEcol Appl 9 751ndash771 httpsdoiorg10230726413271999a

Carpenter S Brock W and Hanson P Ecological and SocialDynamics in Simple Models of Ecosystem Management Con-serv Ecol 3 1ndash31 httpsdoiorg105751ES-00122-0302041999b

Carpenter S R Westley F and Turner M G Surrogates for Re-silience of Social-Ecological Systems Ecosystems 8 941ndash944httpsdoiorg101007s10021-005-0170-y 2005

Chang S E and Shinozuka M Measuring Improvements in theDisaster Resilience of Communities Earthq Spectra 20 739ndash755 httpsdoiorg10119311775796 2004

Chang S E McDaniels T Fox J Dhariwal R and LongstaffH Toward Disaster-Resilient Cities Characterizing Resilienceof Infrastructure Systems with Expert Judgments Risk Anal 34416ndash434 httpsdoiorg101111risa12133 2014

Cote M and Nightingale A J Resilience ThinkingMeets Social Theory Prog Hum Geog 36 475ndash489httpsdoiorg1011770309132511425708 2012

Csete M E and Doyle J C Reverse Engineeringof Biological Complexity Science 295 1664ndash1669httpsdoiorg101126science1069981 2002

wwwearth-syst-dynamnet911592018 Earth Syst Dynam 9 1159ndash1168 2018

M Homayounfar et al Linking resilience and robustness and uncovering their trade-offs 1165

0

02

04

08

1

01

02

03

04

05

06

07

06y

160 800 20 40C

01

02

03

04

05

0

02

04

08

1

06y

160 800 20 40C

(a) (b)

Figure 5 Pareto-optimal policies represented by black dots in the policy space (Cndashy plane) superimposed with resilience (microRsystem ) (a)and robustness (microRltmicro) contours (b)

They may also be concerned about other system propertieseg productivity user participation etc As more dimensionsare considered the set of Pareto-optimal policies grow In thesame spirit as that of the work done here these other dimen-sions should be defined rigorously

This work also lends itself to more rigorous studies ofldquoadaptive governancerdquo In the present study the governancestructure represented by a policy (a combination of C andy) is fixed A natural next step is to explore if a policy isallowed to change how one may improve the resilience androbustness of a CIS andor alter the nature of their inher-ent trade-offs For example if C and y are to be functionsof other factors eg resource availability and outside incen-tives what functional forms should they take to improve thesystemrsquos resilience and robustness Indeed in the absence oftransparent metrics attempts to explore such adaptive poli-cies are severely limited

Additionally agents in a CIS may have the capacity tochange their behavior in response to changes in policy en-vironmental conditions technological changes and so on Inthis study strategic behavior and the decision-making pro-cess are assumed unchanged in the analysis Adaptation instrategic behavior of agents will subsequently alter the natureof resilience its robustness and their trade-offs Capturingsuch effects of adaptation requires structural changes to themodel eg in terms of specification of payoffs or even theformulation of the dynamical equations With such adaptivesocial agents how should one devise adaptive governance toenhance resilience and robustness of a CIS Addressing sucha question is a theoretically intriguing future research direc-tion with great practical implications

In keeping with the theme of ldquosocial dynamics and plane-tary boundaries in Earth system modellingrdquo our results shedlight on how social and biophysical factors may interplay todefine boundaries of a sustainable coupled infrastructure sys-tem While the modeled system here is admittedly simpleour methodology and results constitute a step toward quanti-tatively and meaningfully combining social and biophysicalfactors into indicators of boundaries of more complex sys-tems Just as in this work once those boundaries are clearlydefined calculation and discussion of resilience and robust-ness can become concrete

Data availability No data sets were used in this article

wwwearth-syst-dynamnet911592018 Earth Syst Dynam 9 1159ndash1168 2018

1166 M Homayounfar et al Linking resilience and robustness and uncovering their trade-offs

Appendix A

Basic model Here we briefly describe the basic model pre-sented by Muneepeerakul and Anderies (2017) The modelshows dynamic behavior of three principal variables namelythe state of the public infrastructure IHM resource level Rand the fraction of time a user makes use of infrastructureU through Eqs (A1 A4 and A5) The schematic diagramof this system of equations is shown in Fig A1

In this context IHM depends on PIPs in term of mainte-nance cost and has a positive relationship with the capacityof users to create resource flows Eq (A1) illustrates the dy-namics of IHM as follows

dIHMdt=M ( )minus δH (IHM ) (A1)

where δ is the infrastructurersquos depreciation rate andH (IHM )states functional relationship of public infrastructure and pro-ductivity of each RU According to Muneepeerakul and An-deries (2017) many shared infrastructures can be modeledby threshold functions Given that H (IHM ) shows thresh-old behavior they used a piecewise linear function to capturesuch behavior through Eq (A2)

H (IHM )=

0IHM lt I0

hIHM minus I0

Imminus I0I0 le IHM le Im

hIHM ge Im

(A2)

where h represents the maximum amount of harvest byeach user under no restriction and I0 and Im are the lowerbound and upper bound thresholds of IHM respectivelyAlso M ( ) is the maintenance function (Eq A3) and de-pends on the social structure of the system

M ( )= micro2yCpRUNH (IHM ) (A3)

In Eq (A3) given the number of users N RUNH (IHM ) isthe total harvest from the natural infrastructure The RUs selltotal harvest at price p to generate revenue Subsequentlythey assign a proportion C of revenue to PIPrsquos for their con-tribution Meanwhile the PIPrsquos spend proportion y of C onmaintaining public infrastructure through the maintenancefunction M ( ) Also micro2 is the maintenance effectivenessof PIPrsquos investment

The second variable is the resource levelR They assumedthe dynamics of resource to be

dRdt=G (R)minusRUNH (IHM ) (A4)

Natural infrastructure is assumed to invoke the conservationlaw comprising of regenerating capacity (G (R)= gminus dR)and total unit of harvest RUNH (IHM ) The definition pre-sented for G is the simplest model for natural infrastructurewhere g and d are the natural replenishment and the lossrates respectively

HM IHM

Resource users

N

Natural infrastructure

Public infrastructure

R G(R)wp

infrastructurePublic

providersy

w

M()

UNI

minusδ

HMUNRH( )

IHMH( ) IHMH( )

IHMH( )

IHMCpUNRH( )

I

Figure A1 Schematic diagram of the dynamical system modelTaken from Muneepeerakul and Anderies (2017)

The strategic behavior of the resource users (RUs) is cap-tured by employing a replicator equation Indeed replica-tor dynamics provide modelers with simple realistic so-cial mechanism where agents follow and replicate better-offstrategies The two possible strategies considered for RUsare staying inside the system with the associated payoff ofπU = (1minusC)pRH (IHM ) or leaving the system with thepayoff of πw = w According to the replicator equation

dUdt= rU (1minusU ) (πU minusw) (A5)

The replicator equation represents the fraction of time thatRUs assign to working inside system given C and y LikeRUs there is also two alternatives for PIPs working insidethe system or working for another CIS which leads to sys-tem failure Meanwhile C and y characterize the strategyor policy of PIPs The PIPs will participate in this coupledsystem only when πp = (1minus y)pCRUNH (IHM )ge wP Inother words the PIPs maintain the system when they arebetter-off than working outside This condition is termed thePIP Participation Constraint (PPC)

Based on the system of three differential equations(Eqs A1 A4 and A5) the sustainable equilibria ie long-term system outcomes that satisfy the stability condition andPPC can be expressed as follows

ilowastHM =yCUlowastNRlowast

gH(I lowastHM

)Rlowast =

g

d

(1minus

ilowastHM

yC

)

Ulowast =(1minusC)yC

φ1ilowast

HM (A6)

where ilowastHM =IlowastHMδ

micro2pg(indicates dimensionless) and φ1 =

pgwN

a dimensionless group representing the relative lucrativenessof the system namely the ratio of potential income ndash withthe entire resource flow turned into income ndash relative to out-side wage The results reported in this study are based on thefollowing parameter values h= 00005 δ = 01 I0 = 03Im = 3 g = 100 d = 002 N = 1000 r = 015 p = 10w = 1 wp = 100 and micro2 = 0001

Earth Syst Dynam 9 1159ndash1168 2018 wwwearth-syst-dynamnet911592018

M Homayounfar et al Linking resilience and robustness and uncovering their trade-offs 1167

Author contributions MH RM JMA and CPM designed thestudy MH carried out the analysis MH and RM prepared themanuscript with contributions from all co-authors

Competing interests The authors declare that they have no con-flict of interest

Special issue statement This article is part of the special issueldquoSocial dynamics and planetary boundaries in Earth system mod-ellingrdquo It is not associated with a conference

Acknowledgements John M Anderies and Rachata Muneepeer-akul acknowledge the support from the grant NSF GEO-1115054Rachata Muneepeerakul and Mehran Homayounfar acknowl-edge support from the Army Research OfficeArmy ResearchLaboratory The research reported in this grant was supported inwhole or in part by the Army Research OfficeArmy ResearchLaboratory under award no W911NF1810267 (Multi-UniversityResearch Initiative) The views and conclusions contained in thisdocument are those of the authors and should not be interpreted asrepresenting the official policies either expressed or implied of theArmy Research Office or the US Government The authors alsothank the editor and three anonymous referees for their constructiveand useful comments

Edited by Jonathan DongesReviewed by three anonymous referees

References

AitSahlia F Wang C J Cabrera V E Uryasev S and FraisseC W Optimal crop planting schedules and financial hedgingstrategies under ENSO-based climate forecasts Ann Oper Res190 201ndash220 2011

Anderies J M Janssen M A and Walker B H Graz-ing Management Resilience and the Dynamics of aFire-Driven Rangeland System Ecosystems 5 23ndash44httpsdoiorg101007s10021-001-0053-9 2002

Anderies J M Janssen M and Ostrom E A Frameworkto Analyze the Robustness of Social-Ecological Systems froman Institutional Perspective Ecol Soc 9 18 httpwwwecologyandsocietyorgvol9iss1art18inlinehtml 2004

Anderies J M Ryan P and Walker B H Loss of ResilienceCrisis and Institutional Change Lessons from an Intensive Agri-cultural System in Southeastern Australia Ecosystems 9 865ndash878 httpsdoiorg101007s10021-006-0017-1 2006

Anderies J M Rodriguez A A Janssen M A and CifdalozO Panaceas Uncertainty and the Robust Control Frameworkin Sustainability Science P Natl Acad Sci USA 104 15194ndash15199 httpsdoiorg101073pnas0702655104 2007

Anderies J M Janssen M and Schlager E Institutions andthe performance of coupled infrastructure systems Int J Com-mons 10 495ndash516 httpsdoiorg1018352ijc651 2016

ASCE RCIA Advisory Council Report card for Americarsquos Infras-trcutre Tech Rep American Society of Civil Engineers 2013

Barrett C B and Constas M A Toward a Theoryof Resilience for International Development Applica-tions P Natl Acad Sci USA 111 14625ndash14630httpsdoiorg101073pnas1320880111 2014

Berkes F and Folke C Linking Social and Ecological Systemsfor Resilience and Sustainability Linking Social and Ecologi-cal Systems Management Practices and Social Mechanisms forBuilding Resilience 1 13ndash20 1998

Berkes F Colding J and Folke C Navigating Social-EcologicalSystems Building Resilience for Complexity and change Build-ing p 393 httpsdoiorg101016jbiocon200401010 2003

Biggs R Schluumlter M Biggs D Bohensky E L BurnSil-ver S Cundill G Dakos V Daw T M Evans L SKotschy K Leitch A M Meek C Quinlan A Raudsepp-Hearne C Robards M D Schoon M L Schultz L andWest P C Toward Principles for Enhancing the Resilienceof Ecosystem Services Annu Rev Env Resour 37 421ndash448httpsdoiorg101146annurev-environ-051211-123836 2012

Bode H W Network Analysis and Feedback Amplifier DesignBell Telephone Laboratories Series p 551 1945

Bryant B P and Lempert R J Thinking inside the BoxA Participatory Computer-Assisted Approach to Sce-nario Discovery Technol Forecast Soc 77 34ndash49httpsdoiorg101016jtechfore200908002 2010

Carlson J M and Doyle J Complexity and Ro-bustness P Natl Acad Sci USA 99 2538ndash2545httpsdoiorg101073pnas012582499 2002

Carpenter S Walker B Anderies J M and Abel N FromMetaphor to Measurement Resilience of What to WhatEcosystems 4 765ndash781 httpsdoiorg101007s10021-001-0045-9 2001

Carpenter S R and Brock W A Spatial Complexity Resilienceand Policy Diversity Fishing on Lake-Rich Landscapes EcolSoc 9 8 2004

Carpenter S R Ludwig D and Brock W A Management of Eu-trophication for Lakes Subject to Potentially Irreversible ChangeEcol Appl 9 751ndash771 httpsdoiorg10230726413271999a

Carpenter S Brock W and Hanson P Ecological and SocialDynamics in Simple Models of Ecosystem Management Con-serv Ecol 3 1ndash31 httpsdoiorg105751ES-00122-0302041999b

Carpenter S R Westley F and Turner M G Surrogates for Re-silience of Social-Ecological Systems Ecosystems 8 941ndash944httpsdoiorg101007s10021-005-0170-y 2005

Chang S E and Shinozuka M Measuring Improvements in theDisaster Resilience of Communities Earthq Spectra 20 739ndash755 httpsdoiorg10119311775796 2004

Chang S E McDaniels T Fox J Dhariwal R and LongstaffH Toward Disaster-Resilient Cities Characterizing Resilienceof Infrastructure Systems with Expert Judgments Risk Anal 34416ndash434 httpsdoiorg101111risa12133 2014

Cote M and Nightingale A J Resilience ThinkingMeets Social Theory Prog Hum Geog 36 475ndash489httpsdoiorg1011770309132511425708 2012

Csete M E and Doyle J C Reverse Engineeringof Biological Complexity Science 295 1664ndash1669httpsdoiorg101126science1069981 2002

wwwearth-syst-dynamnet911592018 Earth Syst Dynam 9 1159ndash1168 2018

1166 M Homayounfar et al Linking resilience and robustness and uncovering their trade-offs

Appendix A

Basic model Here we briefly describe the basic model pre-sented by Muneepeerakul and Anderies (2017) The modelshows dynamic behavior of three principal variables namelythe state of the public infrastructure IHM resource level Rand the fraction of time a user makes use of infrastructureU through Eqs (A1 A4 and A5) The schematic diagramof this system of equations is shown in Fig A1

In this context IHM depends on PIPs in term of mainte-nance cost and has a positive relationship with the capacityof users to create resource flows Eq (A1) illustrates the dy-namics of IHM as follows

dIHMdt=M ( )minus δH (IHM ) (A1)

where δ is the infrastructurersquos depreciation rate andH (IHM )states functional relationship of public infrastructure and pro-ductivity of each RU According to Muneepeerakul and An-deries (2017) many shared infrastructures can be modeledby threshold functions Given that H (IHM ) shows thresh-old behavior they used a piecewise linear function to capturesuch behavior through Eq (A2)

H (IHM )=

0IHM lt I0

hIHM minus I0

Imminus I0I0 le IHM le Im

hIHM ge Im

(A2)

where h represents the maximum amount of harvest byeach user under no restriction and I0 and Im are the lowerbound and upper bound thresholds of IHM respectivelyAlso M ( ) is the maintenance function (Eq A3) and de-pends on the social structure of the system

M ( )= micro2yCpRUNH (IHM ) (A3)

In Eq (A3) given the number of users N RUNH (IHM ) isthe total harvest from the natural infrastructure The RUs selltotal harvest at price p to generate revenue Subsequentlythey assign a proportion C of revenue to PIPrsquos for their con-tribution Meanwhile the PIPrsquos spend proportion y of C onmaintaining public infrastructure through the maintenancefunction M ( ) Also micro2 is the maintenance effectivenessof PIPrsquos investment

The second variable is the resource levelR They assumedthe dynamics of resource to be

dRdt=G (R)minusRUNH (IHM ) (A4)

Natural infrastructure is assumed to invoke the conservationlaw comprising of regenerating capacity (G (R)= gminus dR)and total unit of harvest RUNH (IHM ) The definition pre-sented for G is the simplest model for natural infrastructurewhere g and d are the natural replenishment and the lossrates respectively

HM IHM

Resource users

N

Natural infrastructure

Public infrastructure

R G(R)wp

infrastructurePublic

providersy

w

M()

UNI

minusδ

HMUNRH( )

IHMH( ) IHMH( )

IHMH( )

IHMCpUNRH( )

I

Figure A1 Schematic diagram of the dynamical system modelTaken from Muneepeerakul and Anderies (2017)

The strategic behavior of the resource users (RUs) is cap-tured by employing a replicator equation Indeed replica-tor dynamics provide modelers with simple realistic so-cial mechanism where agents follow and replicate better-offstrategies The two possible strategies considered for RUsare staying inside the system with the associated payoff ofπU = (1minusC)pRH (IHM ) or leaving the system with thepayoff of πw = w According to the replicator equation

dUdt= rU (1minusU ) (πU minusw) (A5)

The replicator equation represents the fraction of time thatRUs assign to working inside system given C and y LikeRUs there is also two alternatives for PIPs working insidethe system or working for another CIS which leads to sys-tem failure Meanwhile C and y characterize the strategyor policy of PIPs The PIPs will participate in this coupledsystem only when πp = (1minus y)pCRUNH (IHM )ge wP Inother words the PIPs maintain the system when they arebetter-off than working outside This condition is termed thePIP Participation Constraint (PPC)

Based on the system of three differential equations(Eqs A1 A4 and A5) the sustainable equilibria ie long-term system outcomes that satisfy the stability condition andPPC can be expressed as follows

ilowastHM =yCUlowastNRlowast

gH(I lowastHM

)Rlowast =

g

d

(1minus

ilowastHM

yC

)

Ulowast =(1minusC)yC

φ1ilowast

HM (A6)

where ilowastHM =IlowastHMδ

micro2pg(indicates dimensionless) and φ1 =

pgwN

a dimensionless group representing the relative lucrativenessof the system namely the ratio of potential income ndash withthe entire resource flow turned into income ndash relative to out-side wage The results reported in this study are based on thefollowing parameter values h= 00005 δ = 01 I0 = 03Im = 3 g = 100 d = 002 N = 1000 r = 015 p = 10w = 1 wp = 100 and micro2 = 0001

Earth Syst Dynam 9 1159ndash1168 2018 wwwearth-syst-dynamnet911592018

M Homayounfar et al Linking resilience and robustness and uncovering their trade-offs 1167

Author contributions MH RM JMA and CPM designed thestudy MH carried out the analysis MH and RM prepared themanuscript with contributions from all co-authors

Competing interests The authors declare that they have no con-flict of interest

Special issue statement This article is part of the special issueldquoSocial dynamics and planetary boundaries in Earth system mod-ellingrdquo It is not associated with a conference

Acknowledgements John M Anderies and Rachata Muneepeer-akul acknowledge the support from the grant NSF GEO-1115054Rachata Muneepeerakul and Mehran Homayounfar acknowl-edge support from the Army Research OfficeArmy ResearchLaboratory The research reported in this grant was supported inwhole or in part by the Army Research OfficeArmy ResearchLaboratory under award no W911NF1810267 (Multi-UniversityResearch Initiative) The views and conclusions contained in thisdocument are those of the authors and should not be interpreted asrepresenting the official policies either expressed or implied of theArmy Research Office or the US Government The authors alsothank the editor and three anonymous referees for their constructiveand useful comments

Edited by Jonathan DongesReviewed by three anonymous referees

References

AitSahlia F Wang C J Cabrera V E Uryasev S and FraisseC W Optimal crop planting schedules and financial hedgingstrategies under ENSO-based climate forecasts Ann Oper Res190 201ndash220 2011

Anderies J M Janssen M A and Walker B H Graz-ing Management Resilience and the Dynamics of aFire-Driven Rangeland System Ecosystems 5 23ndash44httpsdoiorg101007s10021-001-0053-9 2002

Anderies J M Janssen M and Ostrom E A Frameworkto Analyze the Robustness of Social-Ecological Systems froman Institutional Perspective Ecol Soc 9 18 httpwwwecologyandsocietyorgvol9iss1art18inlinehtml 2004

Anderies J M Ryan P and Walker B H Loss of ResilienceCrisis and Institutional Change Lessons from an Intensive Agri-cultural System in Southeastern Australia Ecosystems 9 865ndash878 httpsdoiorg101007s10021-006-0017-1 2006

Anderies J M Rodriguez A A Janssen M A and CifdalozO Panaceas Uncertainty and the Robust Control Frameworkin Sustainability Science P Natl Acad Sci USA 104 15194ndash15199 httpsdoiorg101073pnas0702655104 2007

Anderies J M Janssen M and Schlager E Institutions andthe performance of coupled infrastructure systems Int J Com-mons 10 495ndash516 httpsdoiorg1018352ijc651 2016

ASCE RCIA Advisory Council Report card for Americarsquos Infras-trcutre Tech Rep American Society of Civil Engineers 2013

Barrett C B and Constas M A Toward a Theoryof Resilience for International Development Applica-tions P Natl Acad Sci USA 111 14625ndash14630httpsdoiorg101073pnas1320880111 2014

Berkes F and Folke C Linking Social and Ecological Systemsfor Resilience and Sustainability Linking Social and Ecologi-cal Systems Management Practices and Social Mechanisms forBuilding Resilience 1 13ndash20 1998

Berkes F Colding J and Folke C Navigating Social-EcologicalSystems Building Resilience for Complexity and change Build-ing p 393 httpsdoiorg101016jbiocon200401010 2003

Biggs R Schluumlter M Biggs D Bohensky E L BurnSil-ver S Cundill G Dakos V Daw T M Evans L SKotschy K Leitch A M Meek C Quinlan A Raudsepp-Hearne C Robards M D Schoon M L Schultz L andWest P C Toward Principles for Enhancing the Resilienceof Ecosystem Services Annu Rev Env Resour 37 421ndash448httpsdoiorg101146annurev-environ-051211-123836 2012

Bode H W Network Analysis and Feedback Amplifier DesignBell Telephone Laboratories Series p 551 1945

Bryant B P and Lempert R J Thinking inside the BoxA Participatory Computer-Assisted Approach to Sce-nario Discovery Technol Forecast Soc 77 34ndash49httpsdoiorg101016jtechfore200908002 2010

Carlson J M and Doyle J Complexity and Ro-bustness P Natl Acad Sci USA 99 2538ndash2545httpsdoiorg101073pnas012582499 2002

Carpenter S Walker B Anderies J M and Abel N FromMetaphor to Measurement Resilience of What to WhatEcosystems 4 765ndash781 httpsdoiorg101007s10021-001-0045-9 2001

Carpenter S R and Brock W A Spatial Complexity Resilienceand Policy Diversity Fishing on Lake-Rich Landscapes EcolSoc 9 8 2004

Carpenter S R Ludwig D and Brock W A Management of Eu-trophication for Lakes Subject to Potentially Irreversible ChangeEcol Appl 9 751ndash771 httpsdoiorg10230726413271999a

Carpenter S Brock W and Hanson P Ecological and SocialDynamics in Simple Models of Ecosystem Management Con-serv Ecol 3 1ndash31 httpsdoiorg105751ES-00122-0302041999b

Carpenter S R Westley F and Turner M G Surrogates for Re-silience of Social-Ecological Systems Ecosystems 8 941ndash944httpsdoiorg101007s10021-005-0170-y 2005

Chang S E and Shinozuka M Measuring Improvements in theDisaster Resilience of Communities Earthq Spectra 20 739ndash755 httpsdoiorg10119311775796 2004

Chang S E McDaniels T Fox J Dhariwal R and LongstaffH Toward Disaster-Resilient Cities Characterizing Resilienceof Infrastructure Systems with Expert Judgments Risk Anal 34416ndash434 httpsdoiorg101111risa12133 2014

Cote M and Nightingale A J Resilience ThinkingMeets Social Theory Prog Hum Geog 36 475ndash489httpsdoiorg1011770309132511425708 2012

Csete M E and Doyle J C Reverse Engineeringof Biological Complexity Science 295 1664ndash1669httpsdoiorg101126science1069981 2002

wwwearth-syst-dynamnet911592018 Earth Syst Dynam 9 1159ndash1168 2018

M Homayounfar et al Linking resilience and robustness and uncovering their trade-offs 1167

Author contributions MH RM JMA and CPM designed thestudy MH carried out the analysis MH and RM prepared themanuscript with contributions from all co-authors

Competing interests The authors declare that they have no con-flict of interest

Special issue statement This article is part of the special issueldquoSocial dynamics and planetary boundaries in Earth system mod-ellingrdquo It is not associated with a conference

Acknowledgements John M Anderies and Rachata Muneepeer-akul acknowledge the support from the grant NSF GEO-1115054Rachata Muneepeerakul and Mehran Homayounfar acknowl-edge support from the Army Research OfficeArmy ResearchLaboratory The research reported in this grant was supported inwhole or in part by the Army Research OfficeArmy ResearchLaboratory under award no W911NF1810267 (Multi-UniversityResearch Initiative) The views and conclusions contained in thisdocument are those of the authors and should not be interpreted asrepresenting the official policies either expressed or implied of theArmy Research Office or the US Government The authors alsothank the editor and three anonymous referees for their constructiveand useful comments

Edited by Jonathan DongesReviewed by three anonymous referees

References

AitSahlia F Wang C J Cabrera V E Uryasev S and FraisseC W Optimal crop planting schedules and financial hedgingstrategies under ENSO-based climate forecasts Ann Oper Res190 201ndash220 2011

Anderies J M Janssen M A and Walker B H Graz-ing Management Resilience and the Dynamics of aFire-Driven Rangeland System Ecosystems 5 23ndash44httpsdoiorg101007s10021-001-0053-9 2002

Anderies J M Janssen M and Ostrom E A Frameworkto Analyze the Robustness of Social-Ecological Systems froman Institutional Perspective Ecol Soc 9 18 httpwwwecologyandsocietyorgvol9iss1art18inlinehtml 2004

Anderies J M Ryan P and Walker B H Loss of ResilienceCrisis and Institutional Change Lessons from an Intensive Agri-cultural System in Southeastern Australia Ecosystems 9 865ndash878 httpsdoiorg101007s10021-006-0017-1 2006

Anderies J M Rodriguez A A Janssen M A and CifdalozO Panaceas Uncertainty and the Robust Control Frameworkin Sustainability Science P Natl Acad Sci USA 104 15194ndash15199 httpsdoiorg101073pnas0702655104 2007

Anderies J M Janssen M and Schlager E Institutions andthe performance of coupled infrastructure systems Int J Com-mons 10 495ndash516 httpsdoiorg1018352ijc651 2016

ASCE RCIA Advisory Council Report card for Americarsquos Infras-trcutre Tech Rep American Society of Civil Engineers 2013

Barrett C B and Constas M A Toward a Theoryof Resilience for International Development Applica-tions P Natl Acad Sci USA 111 14625ndash14630httpsdoiorg101073pnas1320880111 2014

Berkes F and Folke C Linking Social and Ecological Systemsfor Resilience and Sustainability Linking Social and Ecologi-cal Systems Management Practices and Social Mechanisms forBuilding Resilience 1 13ndash20 1998

Berkes F Colding J and Folke C Navigating Social-EcologicalSystems Building Resilience for Complexity and change Build-ing p 393 httpsdoiorg101016jbiocon200401010 2003

Biggs R Schluumlter M Biggs D Bohensky E L BurnSil-ver S Cundill G Dakos V Daw T M Evans L SKotschy K Leitch A M Meek C Quinlan A Raudsepp-Hearne C Robards M D Schoon M L Schultz L andWest P C Toward Principles for Enhancing the Resilienceof Ecosystem Services Annu Rev Env Resour 37 421ndash448httpsdoiorg101146annurev-environ-051211-123836 2012

Bode H W Network Analysis and Feedback Amplifier DesignBell Telephone Laboratories Series p 551 1945

Bryant B P and Lempert R J Thinking inside the BoxA Participatory Computer-Assisted Approach to Sce-nario Discovery Technol Forecast Soc 77 34ndash49httpsdoiorg101016jtechfore200908002 2010

Carlson J M and Doyle J Complexity and Ro-bustness P Natl Acad Sci USA 99 2538ndash2545httpsdoiorg101073pnas012582499 2002

Carpenter S Walker B Anderies J M and Abel N FromMetaphor to Measurement Resilience of What to WhatEcosystems 4 765ndash781 httpsdoiorg101007s10021-001-0045-9 2001

Carpenter S R and Brock W A Spatial Complexity Resilienceand Policy Diversity Fishing on Lake-Rich Landscapes EcolSoc 9 8 2004

Carpenter S R Ludwig D and Brock W A Management of Eu-trophication for Lakes Subject to Potentially Irreversible ChangeEcol Appl 9 751ndash771 httpsdoiorg10230726413271999a

Carpenter S Brock W and Hanson P Ecological and SocialDynamics in Simple Models of Ecosystem Management Con-serv Ecol 3 1ndash31 httpsdoiorg105751ES-00122-0302041999b

Carpenter S R Westley F and Turner M G Surrogates for Re-silience of Social-Ecological Systems Ecosystems 8 941ndash944httpsdoiorg101007s10021-005-0170-y 2005

Chang S E and Shinozuka M Measuring Improvements in theDisaster Resilience of Communities Earthq Spectra 20 739ndash755 httpsdoiorg10119311775796 2004

Chang S E McDaniels T Fox J Dhariwal R and LongstaffH Toward Disaster-Resilient Cities Characterizing Resilienceof Infrastructure Systems with Expert Judgments Risk Anal 34416ndash434 httpsdoiorg101111risa12133 2014

Cote M and Nightingale A J Resilience ThinkingMeets Social Theory Prog Hum Geog 36 475ndash489httpsdoiorg1011770309132511425708 2012

Csete M E and Doyle J C Reverse Engineeringof Biological Complexity Science 295 1664ndash1669httpsdoiorg101126science1069981 2002

wwwearth-syst-dynamnet911592018 Earth Syst Dynam 9 1159ndash1168 2018

1168 M Homayounfar et al Linking resilience and robustness and uncovering their trade-offs

Cumming G S and Peterson G D Unifying Research on SocialndashEcological Resilience and Collapse Trends Ecol Evol 32 695ndash713 httpsdoiorg101016jtree201706014 2017

Folke C Resilience The Emergence of a Perspective for Socialndashecological Systems Analyses Global Environ Chang 16 253ndash267 httpsdoiorg101016jgloenvcha200604002 2006

Folke C Carpenter S Elmqvist T Gunderson L HollingC S and Walker B Resilience and Sustainable Develop-ment Building Adaptive Capacity in a World of Transfor-mations AMBIO 31 437ndash440 httpsdoiorg1016390044-7447(2002)031[0437RASDBA]20CO2 2002

Folke C Carpenter S R Walker B Scheffer M ChapinT and Rockstroumlm J Resilience Thinking Integrating Re-silience Adaptability and Transformability Ecol Soc 15 20httpsdoiorg105751ES-03610-150420 2010

Folke C Biggs R Norstroumlm A V Reyers B and RockstroumlmJ Social-Ecological Resilience and Biosphere-Based Sustain-ability Science Ecol Soc 21 41 httpsdoiorg105751ES-08748-210341 2016

Groves D G and Lempert R J A New Analytic Method for Find-ing Policy-Relevant Scenarios Global Environ Chang 17 73ndash85 httpsdoiorg101016jgloenvcha200611006 2007

Gunderson L H and Holling C S Barriers and Bridges to theRenewal of Ecosystems and Institutions Columbia UniversityPress 1995

Holling C S Resilience and Stability of Ecolog-ical Systems Annu Rev Ecol Syst 4 1ndash23httpsdoiorg101146annureves04110173000245 1973

Holling C S Engineering Resilience versus Ecological Re-silience Engineering Within Ecological Constraints 1996 31ndash44 httpsdoiorg10172264919 1996

Janssen M A Anderies J M and Walker B H Ro-bust Strategies for Managing Rangelands with Multiple Sta-ble Attractors J Environ Econ Manag 47 140ndash162httpsdoiorg101016S0095-0696(03)00069-X 2004

Kerner D and Thomas J Resilience Attributes of Social-Ecological Systems Framing Metrics for Management Re-sources 3 Multidisciplinary Digital Publishing Institute 672ndash702 httpsdoiorg103390resources3040672 2014

Kot M Elements of Mathematical Ecology Cambridge UniversityPress 2001

Krokhmal P Palmquist J and Uryasev S Portfolio optimizationwith conditional value-at-risk objective and constraints J Risk4 43ndash68 2002

Longstaff P H Armstrong N J Perrin K Parker W Mand Hidek M Building Resilient Communities A PreliminaryFramework for Assessment Homeland Security Affairs 6 1ndash232010

May R M Stability and Complexity in Model Ecosystems Prince-ton University Press 2001

Mitra C Kurths J and Donner R V An Integrative Quantifierof Multistability in Complex Systems Based on Ecological Re-silience Sci Rep 5 1ndash10 httpsdoiorg101038srep161962015

Muneepeerakul R and Anderies J M Strategic Behaviors andGovernance Challenges in Social-Ecological Systems EarthrsquosFuture 5 865ndash76 httpsdoiorg1010022017EF000562 2017

Ostrom E Janssen M A and Anderies J M Going be-yond Panaceas P Natl Acad Sci USA 104 15176ndash15178httpsdoiorg101073pnas0701886104 2007

Redman C L Should Sustainability and Resilience Be Com-bined or Remain Distinct Pursuits Ecol Soc 19 37httpsdoiorg105751ES-06390-190237 2014

Rockafellar R T and Uryasev S Optimization of conditionalvalue-at-risk J Risk 2 21ndash41 2000

Sarykalin S Serraino G and Uryasev S Value-at-Risk vsConditional Value-at-Risk in Risk Management and Optimiza-tion INFORMS TutORials in Operations Research 27000294httpsdoiorg101287educ10800052 2014

Scheffer M Brock W and Westley F Socioeconomic Mecha-nisms Preventing Optimum Use of Ecosystem Services An In-terdisciplinary Theoretical Analysis Ecosystems 3 451ndash471httpsdoiorg101007s100210000040 2000

Scheffer M Bascompte J Brock W A Brovkin V CarpenterS R Dakos V Held H van Nes E H Rietkerk M and Sug-ihara G Early-Warning Signals for Critical Transitions Nature461 7260 httpsdoiorg101038nature08227 2009

Scheffer M Carpenter S R Lenton T M Bascompte JBrock W Dakos V van de Koppel J van de Leemput IA Levin S A van Nes E H Pascual M and VandermeerJ Anticipating Critical Transitions Science 6105 344ndash348httpsdoiorg101126science1225244 2012

Walker B Carpenter S Anderies J Abel N CummingsG Janssen M Lebel L Norberg J Peterson G D andPritchard R Resilience Management in Social-Ecological Sys-tems A Working hypothesis for a Participatory Approach Con-serv Ecol 6 p 14 2002

Walker B Holling C S Carpenter S R and Kinzig A Re-silience Adaptability and Transformability in SocialndashEcologicalSystems Ecol Soc 9 5 httpsdoiorg105751ES-00650-090205 2004

Water Act Chap 21 available at httpwwwlegislationgovukukpga201421contentsenacted (last access April 2018) 2014

Wolpert D H and Macready W G No Free Lunch Theo-rems for Optimization IEEE T Evolut Comput 1 67ndash82httpsdoiorg1011094235585893 1997

Zymler S Kuhn D and Rustem B Worst-Case Value at Riskof Nonlinear Portfolios Management Science 59 172ndash188httpsdoiorg101287mnsc11201615 2013

Earth Syst Dynam 9 1159ndash1168 2018 wwwearth-syst-dynamnet911592018