Journal of Ecology. 2019;107:1779–1790. wileyonlinelibrary.com/journal/jec  | 1779© 2019 The Authors. Journal of Ecology © 2019 British Ecological Society

Received:19July2018  |  Accepted:22January2019DOI: 10.1111/1365-2745.13147

R E S E A R C H A R T I C L E

Keystoneness, centrality, and the structural controllability of ecological networks

Edgar Fernando Cagua  | Kate L. Wootton  | Daniel B. Stouffer

CentreforIntegrativeEcology,SchoolofBiologicalSciences,UniversityofCanterbury,Christchurch,NewZealand

CorrespondenceEdgarFernandoCaguaEmail:efc29@uclive.ac.nz

Present AddressKateL.Wootton,DepartmentofEcology,SwedishUniversityofAgriculturalSciences,Uppsala,Sweden

Funding informationRoyalSocietyofNewZealandTeApārangi;EducationNewZealand;UniversityofCanterbury;EuropeanSpaceAgency

HandlingEditor:NicoleRafferty

Abstract1. Animportantdimensionofaspecies'roleisitsabilitytoalterthestateandmain-tain thediversityof itscommunity.Centralitymetricshaveoftenbeenusedtoidentify these species, which are sometimes referred as “keystone” species.However, the relationship between centrality and keystoneness is largely phe-nomenologicalandbasedmostlyonour intuitionregardingwhatconstitutesanimportantspecies.Whilecentralityisusefulwhenpredictingwhichspecies'ex-tinctionscouldcausethelargestchangeinacommunity,itsayslittleabouthowthesespeciescouldbeusedtoattainorpreserveaparticularcommunitystate.

2. Hereweintroducestructuralcontrollability,anapproachthatallowsustoquan-tifytheextenttowhichnetworktopologycanbeharnessedtoachieveadesiredstate.Italsoallowsustoquantifyaspecies'controlcapacity—itsrelativeimpor-tance—andidentifythesetofspeciesthatarecriticalinthiscontextbecausetheyhavethelargestpossiblecontrolcapacity.Weillustratetheapplicationofstruc-tural controllability with ten pairs of uninvaded and invaded plant‐pollinatorcommunities.

3. Wefoundthatthecontrollabilityofacommunityisnotdependentonitsinvasionstatus,butontheasymmetricnatureof itsmutualdependences.Whilecentralspecieswerealsolikelytohavealargecontrolcapacity,centralityfailstoidentifyspecies that, despite being less connected, were critical in their communities.Interestingly, this setofcritical specieswasmostlycomposedofplantsand in-cludedeveryinvasivespeciesinourdataset.Wealsofoundthatspecieswithhighcontrolcapacity,andinparticularcriticalspecies,contributethemosttothesta-blecoexistenceoftheircommunity.Thisresultwastrue,evenwhencontrollingforthespecies'degree,abundance/interactionstrength,andtherelativedepend-enceoftheirpartners.

4. Synthesis.Structuralcontrollabilityisstronglyrelatedtothestabilityofanetworkandmeasuresthedifficultyofmanaginganecologicalcommunity.Italsoidentifiesspeciesthatarecriticaltosustainbiodiversityandtochangeormaintainthestateoftheircommunityandarethereforelikelytobeveryrelevantformanagementandconservation.

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1  | INTRODUC TION

Amajorgoalinecologyistounderstandtherolesplayedbydiffer-ent species in thebioticenvironment.Withincommunityecology,a complex‐systems approachhas led to the development of a va-rietyofanalyticalandsimulationtoolswithwhichtocompareandcontrasttherolesofspeciesembeddedinanetworkofinteractions(Bascompte&Stouffer,2009;Coux,Rader,Bartomeus,&Tylianakis,2016; Guimerà & Amaral, 2005; Stouffer, Sales‐Pardo, Sirer, &Bascompte,2012).Aparticularlyrelevantdimensionofanyspecies'roleisitsabilitytoaltertheabundanceofotherspeciesandthestateofthecommunity—sincechangesofthisnaturecanhaveknock‐oneffects on ecosystem function, diversity, processes, and services(Thompsonetal.,2012;Tylianakis,Didham,Bascompte,&Wardle,2008;Tylianakis,Laliberté,Nielsen,&Bascompte,2010).Thisabilityissometimesreferredtoasaspecies'“keystoneness”(Mills&Doak,1993).

Asignificantproportionof thenetwork toolsusedtoestimatespecies' roles in this context rely on the calculation of a species'centrality—arelativerankingofitspositionalimportancethatorig-inally stems from social‐network research (Friedkin, 1991;MartínGonzález,Dalsgaard,&Olesen,2010).Generally speaking, centralspecies tend to be better connected and consequently are morelikelytoparticipateinthenetwork's“foodchains.”Becausespeciesthatparticipate inmorechainsaremore likely toaffect theabun-dances of other species, centrality metrics have often been usedtoidentifykeystonespeciesinthecommunity(Jordán,Benedek,&Podani,2007).Centralitymetricshavebeenshowntobeusefultoolstorankspeciesinregardtotheirpotentialtoaltertheabundancesof other species, in particular when estimating the probability ofsecondaryextinctionsthatmayfollowthelossofaspecies(Dunne,Williams, & Martinez, 2002; Kaiser‐Bunbury, Muff, Memmott,Müller,&Caflisch,2010).

Despite being conceptually intuitive, the relationship betweencentralityandaspecies'presumedimpactonthestateofthecom-munity is largelyphenomenological.Ontheonehand,substantivechangesinecosystemfunctioningcanalsooccurwithoutcompleteremovalofaspecies(Mouillot,Graham,Villéger,Mason,&Bellwood,2013).Ontheother,weareofteninterestedinaspecificstateofthecommunitythatmightbedesirabletoattain(orpreserve)becauseofitsbiodiversity,resilience,functioning,ortheecosystemservicesitprovides.Inthesecases,itmightbelessusefultounderstandwhichspeciesmay cause any change in the community. Instead,we arebetterservedbyunderstandinghowthestructureof thenetworkcan be harnessed to achieve the desired state andwhich speciesmayplay the largest role in this targetedprocess.Whenthestateofacommunityisunderpinnedbymorethanasinglespecies(often

thecase in real communities) andwemovebeyondsingle‐speciesremovals,wemightexpecttheaccuracyofcentralitytodiminish.Asa result, communityecologycouldarguablybenefit fromanalter-native,perhapsmoremechanistically‐grounded,approachtounder-standhowspeciesaffecteachother'sabundance.

Species' abundances—and consequently the state of the com-munity as a whole—are influenced both by the structure of theirinteractions and the dynamics of these interactions, includingthe mechanisms of self‐regulation (Lever, van Nes, Scheffer, &Bascompte,2014).However, community andpopulationdynamicscan be modelled in innumerable ways, and empirical support foroneversusanother isoftenstillambiguous(Holland,DeAngelis,&Bronstein,2002).Thealternativeapproachshould,therefore,ideallyacknowledgeecosystemdynamics,butwithoutbeingoverlydepen-dentontheparticularchoicesofhowtheyarecharacterised.Amongthevariouspossibilitiesstructural controllability,abranchofcontroltheory, appears to be a strong candidate (Isbell & Loreau, 2013).Controltheoryisawidely‐studiedbranchofengineeringusedtode-termineandsupervisethebehaviourofdynamicalsystems(Motter,2015).Itisinherentlydesignedtodealwithsystemfeedbacksanditsapplicationhasrecentlybeenexpandedtocomplexnetworks (Lin,1974;Liu&Barabási,2016).Consistentwithlong‐standingecologi-calquestions,advancesinstructuralcontrollabilityhaveestablishedaclearlinkbetweenthestructureofthenetworkandthewaynodesaffecteachother.Unlikecentralityindices,however,thislinkisnotbasedonapriori assumptionsbetweennetworkmetrics andkey-stonenessbutisinsteadbasedonwell‐establishedadvancesinbothdynamicalandcomplex‐systemstheory(Motter,2015).

At its fundamental level, structural controllability first deter-mineswhetherasystemiscontrollableornot;thatis,itasksifasys-temcouldeverbedriventoadesiredstatewithinafiniteamountoftime.Althoughthecontrollabilityofanetwork isawhole‐systemproperty,ithasrecentlybeenshownthataskingforthecontrolla-bilityofacomplex‐systemisequivalenttofindingaparticularsetofrelevantnodes:thesetwithwhichispossibletocontrolthestateofthewholenetwork(Liu&Barabási,2016).Importantly,thissetofnodesisnotalwaysuniqueforagivennetwork.Thisimpliesthatan examinationof the distinct sets provides ameans to connectnodeswiththeirgeneralabilitytomodifythesystemtowhichtheybelong.

Here,weapplymethodsfromstructuralcontrollabilitytoapar-ticularecologicalproblemandshowhowitcanbeusedtogenerateinsightintotheroleofspeciesinanecologicalnetwork.Specifically,weoutlinetheapproachusingasetoftenpairsofuninvadedandinvaded plant‐pollinator communities. We use invaded communi-tiesbecausethere isstrongempiricalevidenceshowingthat inva-sivespeciesplayanimportantroleshapingtheabundancesofother

K E Y W O R D S

controlcapacity,invasivespecies,managementinterventions,mutualism,networkcontroltheory,plantpopulationandcommunitydynamics,species'importance,structuralstability

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species,somethingwhichisparticularlytrueinthesetennetworks(Bartomeus, Vilà, & Santamaría, 2008; Lopezaraiza‐Mikel, Hayes,Whalley,&Memmott,2007).Thischoicethusoffersusanopportu-nitytoexplicitlycontrastourtheoreticalobservationswithempiricalevidence.Moreover, empirical observations indicate that steeringthestateofsomecommunities—forexampleduringecosystemres-toration or invasive species removal—can be a very difficult task(Woodfordetal.,2016).Therefore,wefirstaskwhethertherearedifferences between the controllability of invaded and uninvadednetworks.Wethenexpandexistingmethodsfromcontroltheorytoeffectivelylinkthecontrollability(Table1)ofanetworkwiththeroleofitsconstituentspecies.Weask—fromacontrol‐theoreticperspec-tive—whethertherearekeydifferencesbetweenspeciesintheroletheyplayatdrivingthestateofthecommunityandexploretheeco-logicalfactorsrelatedtothesedifferences.Thisallowsustoidentifyspeciesthatmightbecriticalfornetworkcontrolandshowthattheyhavealargerthanexpectedimpactonthestablecoexistenceofthecommunity.Finally,wecomparetheproposedapproachtocurrentmethodsbasedonspecies'centralityandshowhowthesemethodsareindeedvaluablebutultimatelypaintalimitedpictureinregardtothe“keystoneness”ofaspecies.

2  | MATERIAL S AND METHODS

Weused tenpairedpollination communities to apply the control‐theoreticapproach.Eachcommunitypairwascomposedofacom-munity invaded by a plant and a community free of the invasivespecies.Fourpairscorrespondtonaturalorsemi‐naturalvegetationcommunities in the city of Bristol, UK (Lopezaraiza‐Mikel, Hayes,Whalley,&Memmott,2007).Thesecommunitiescomprised19–87species(mean55),andnon‐invadedplotswereobtainedbyexperi-mentallyremovingalltheflowersoftheinvasivespecies Impatiens grandulifera.TheothersixpairswereobtainedfromlowerdiversityMediterranean shrublands in Cap de Creus National Park, Spain(Bartomeusetal.,2008).Thesecommunitiescomprised30–57spe-cies (mean 38); in contrast to the above, uninvaded communitieswereobtainedfromplotsthathadnotyetbeencolonisedbyeitherof the invasive speciesCarpobrotus affine acinaciformis or Opuntia stricta.Thestructureofall thesecommunitieswasdefinedby thepollinatorvisitationfrequency,whichhasbeenshowntobeanap-propriatesurrogateforinterspecificeffectsinpollinationnetworks(Bascompte,Jordano,&Olesen,2006;Vázquez,Morris,&Jordano,2005).FulldetailsabouttheempiricalnetworkscanbefoundintheSupportingInformationSectionS1.

Thefirststep inapplyingmethodsofcontrol theory is tocon-struct a directed network that is able to provide an indication oftheextenttowhichspeciesaffecteachother'sabundance.Insomeecologicalnetworks,establishingthedirectionalitycanberelativelystraightforward, for examplewhen links represent biomass trans-ferorenergyflow(Isbell&Loreau,2013). Inpollinationnetworks,however, this directionality is lessobvious asboth species can, intheory,benefitfromtheinteraction.Weovercomethatobstaclebynotingthattheextenttowhichspeciesiaffectsspeciesjrelativetotheextenttowhichjaffectsicanbesummarisedbytheirinteractionasymmetry(Bascompteetal.,2006).Thisasymmetryisgivenby.

wherethedependenceofplantionpollinatorj,dij,isthepro-portion of the visits from pollinator j compared to all pollinatorvisitstoplant i.Previousresearchhasshownthatmutualistic in-teractionsareoftenhighlyasymmetricinnaturalcommunities;inotherwords, ifaplantspecies is largelydependentonapollina-torspecies,thatpollinatortendstodependratherweaklyontheplant (and vice versa).We therefore create a directed link fromspecies itospecies jwhendij−dji≥0toestablishthemostlikelydirectionofcontrolbetweenaspeciespair(Figure1a).Sometimes(2.4% of the observed interactions in our datasets) there is noobserved asymmetry between species pairs (dij = dji), and wecannot infer a dominant direction of control.When this occurs,we deem both species to be equally likely to affect each otherandleaveareciprocal interactionbetweenthem(a linkfrom itojandanotherfrom j to i).Bybasingthedirectionofthe linksonthe asymmetry of their dependence, we are able to generate anetwork that is consistentwith the dynamics of the community

a(i,j)=a(j,i)=dij−dji

max(

dij, dji)

TA B L E 1  Glossary

Networkcontrol

Anetworkissaidtobecontrollableifitispossibletosteeritfromaninitialtoanarbitraryfinalstatewithinfinitetime.

Controllability

Theintrinsicdifficultyofcontrollinganecologicalcommunity.Itismeasuredbytherelativesizeoftheminimumdriver‐nodeset,nD.Italsoindicatestheextenttowhichnetworkstructurecanbeharnessedfornetworkcontrol.

Minimumdriver‐nodeset

Oneofthesetsofspecieswhoseabundancesneedtobedirectlymanagedinordertoachievefullcontrolofthecommunity.Theminimumdriver‐nodesetscanbeobtainedbyfindingallmaximummatchingsinanetwork.

Maximummatching

Amatchingisasetoflinksthatdonotshareanycommonstartorendnodes;thelargestpossiblematchingiscalledamaximummatching.

Controlconfiguration

Oneofthespeciescombinationswithwhichispossibletoachievenetworkcontrol.Optimalcontrolconfigurationsaregivenbytheminimumdriver‐nodesets.

Controlcapacity

Therelativefrequencyϕwhichwithaspeciesispartoftheoptimalcontrolconfigurationsofanetwork.

Criticalspecies

Aspecieswithamaximalcontrolcapacityϕ = 1.

Superiornode

Aspeciesisasuperiornodeifitcaninternallyaffecttheabundanceofotherspeciesinthenetwork.Superiornodesmakeupthechainsthatpropagatethecontrolsignalsthroughthenetwork.

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whilesatisfyingtherequirementsofstructuralcontrollability.Thisallowsus tocalculate thecontrollabilityof thenetworksand in-vestigatewhethertherearedifferencesbetweeninvadedandun-invadedcommunities.

2.1 | Controllability

Asystemissaidtobecontrollableifitispossibletosteeritfroman initial to an arbitrary final state within finite time (Kalman,1963). A simple version of such a system can be described bydx

dt= Ax + Bu (t),where thechangeof its stateover time

(

dx

dt

)

de-

pendsonitscurrentstatex(forexamplethespecies'abundances),anexternaltime‐varyinginputu(t)(thecontrolsignal),andtwoma-trices A and B, which encode information about the network

structure and how species respond to external inputs, respec-tively.Inclassiccontroltheory,determiningwhetherthissystemiscontrollable can be achieved by checking that its controllabilitymatrixR = [ B AB A2B … An−1B ]hasfullrank.Incomplexsystems,however,employingthisrankcondition,ornumericalap-proximationsofitisinfeasiblebecauseitishardtofullyparame-teriseA and B(eitherbecausetheweightofthelinkschangesovertime or because they are difficult tomeasure).Here,we use anapproach based on the structural controllability theorem (Lin,1974),whichassumesthatweareconfidentaboutwhichelementsofA and Bhaveeithernon‐zeroorzerovalues(thereisaninterac-tionornot),butthatwearelesssureabouttheprecisemagnitudeofthenon‐zerovalues.Usingthisstructuralapproach,wecanfindoutthecontrollabilityofasystemforeverynon‐zerorealisationoftheparameters.

WeareoftenabletoestimateA inecologicalnetworks,asthismatrixrepresentstheinteractionsbetweenspecies.Partofthecon-trolproblemthusresidesinestimatingasupportableestimationofB,whichrepresentsthelinksbetweenexternalinputsandspecies.Naively,anyecologicalcommunity(andanysystemforthatmatter)could be controlled ifwe control the state of every species inde-pendently,but suchanapproach is typically impractical.Here,weareinterestedinfindingaminimumdriver‐nodeset(effectivelyfind-ingB)withwhichtomakethesystemcontrollable.Thebrute‐forcesearchforthisminimumdriver‐nodesetiscomputationallyprohib-itive formost networks as it involves the evaluation of 2N differ-entcontrollabilitymatriceswhereNisthenumberofspeciesinthecommunity.Wethereforeinsteademployarecently‐developedap-proachthatshowsthatthecontrolproblemoffindingtheminimumdriver‐nodesetcanbemappedintoagraph‐theoreticproblem:max-imummatching(Liu&Barabási,2016;Liu,Slotine,&Barabási,2011).

Maximummatchingisawidelystudiedtopicingraphtheoryandiscommonlyusedinmultipleapplications,rangingfromdatingappsandwirelesscommunicationstoorgantransplantallocationandpeer‐to‐peerfilesharing.Amatching inanunweighteddirectedgraph isdefinedasasetoflinksthatdonotsharecommonstartorendnodes;thelargestpossiblematchingiscalledamaximummatching.Forex-ample, inanetworkcomposedof jobsandjobapplicants,amatch-ingisanypairingbetweenapplicantsandpositionsthatsatisfiesonebasicconstraint:anapplicantcanbeassignedtoatmostonepositionandviceversa.Consequently,amaximummatchingisanoptimalpair-ing,onethatmaximisesthenumberofapplicantswithjobsandthenumberofpositions filled.Admittedly, the linkbetweenmatchingsandstructuralcontrollabilitymayappearfarfromstraightforward.

Thislinkbecomesapparentafterexaminingthegraphicalinter-pretation of structural controllability: from a topological perspec-tive,anetworkisstructurallycontrollableiftherearenoinaccessiblenodes—that is, nodeswithout incoming links—or dilations—expan‐sions of the network (Figure 1b; Supporting Information SectionS2). The key is to note that these two fundamental conditions ofstructural controllability imply that there is aone‐to‐one relation-shipbetweensuperior and subordinatenodesjustliketheone‐to‐onerelationship between jobs and applicants (Figure 1b, bottom left).

F I G U R E 1  Thedirectionofcontrolandcontrollabilityconditions.(a)Toestablishthedirectionofcontrol,westartwithaweightedvisitationnetwork(ontheleft).Inthisnetwork,thewidthofthelinkscorrespondstothefrequencyofvisitationbetweenanimalsaiandplantspi,withwiderlinksindicatingmorevisits.Plantp1isvisitedexclusivelybya1butp1representsonlyasmallfractionofthefloralresourcesexploitedbya1.Therefore,thepopulationofp1ismorelikelytobeaffectedbya1thanviceversa.Werepresentthiswithadirectedlinkfroma1top1inthecontrolnetwork(ontheright).Thedirectionofcontrolbetweenallotherspeciespairscanbesimilarlydeterminedbyinspectingthedifferencebetweentheirrelativedependences.(b)Oncewehaveestablishedthedirectionsofcontrol,wecandeterminewhetherthenetworkiscontrollableornot.Anysystemdefinedbyadirectednetwork(withstatenodesxi;species'populationsinanecologicalcontext)andexternalcontrolinputs(nodesui,orangelinks)isstructurallycontrollableifitsatisfiestwoconditions:ithasnodilations(expansionsinthenetwork)andnoinaccessiblenodes.Thesystemonthetopleftisnotcontrollablebecausethereisadilationsincenodex2isbeingusedtocontroltwonodessimultaneously;inotherwords,therearefewersuperiors(x2)thansubordinates(x1 and x3).Thenetworkonthetoprightisnotcontrollablebecausenodex3isinaccessiblefortheonlyinputnodeu1inthesystem.Bothsystemscanbemadecontrollablebyaddinganextrainputnode(u2inbothbottomnetworks)[Colourfigurecanbeviewedatwileyonlinelibrary.com]

Visitation network(a) Direction of control

p1 p2 p3

a1 a2

p1 p2 p3

a1 a2

Dilation(b) Inaccessible node

Not

cont

rolla

ble

Con

trol

labl

e

x1 x2 x3

u1

x1 x2 x3

u1

x1 x2 x3

u1 u2

x1 x2 x3

u1 u2

Control input

     |  1783Journal of EcologyCAGUA et Al.

We thus use themaximum‐matching algorithm to find an optimalpairingofsuperior (those thatcancontrolanothernode)andsub-ordinatenodes(thosethatcanbecontrolledbyanothernode)inamanner consistent with the controllability conditions (SupportingInformationSectionS3.1).Giventheresult,wecanfurtherdecom-posethematchingintoasetofpathsthatrevealhowacontrolsignalcanflowacrossthelinksinanetworktoreacheverynodewithinit.Asrecentlyshown(Liuetal.,2011),theminimumdriver‐nodeset—thosetowhichanexternalcontrolinputshouldbeappliedtomakethe system controllable—corresponds exactly to the unmatched nodesinthenetwork(Figure2).

2.2 | Differences between invaded and uninvaded networks

Our first objective is to investigatewhether the controllability ofacommunity isassociatedwith invasionstatusornot.Findingoutexactlyhowdifficultitistocontrolanetworkdependsstronglyontheparticularitiesofthedesiredcontroltrajectory(i.e.thepathtothedesiredfinalstate)aswellasthedynamicalrelationshipbetweennodes. However, we are interested in understanding the control-lability of a network in amore general sense, such that it can beapplied evenwhen the precise control scenario is knownonly in-completely.Tothisend,wechoseanindicatorthatfollowsdirectly

fromour approach: the sizeof theminimumdriver‐node set. Thissimplemetricprovidesageneralindicationofhowdifficultcontrol-linganetworkmightbe,assystemsthatrequirealargenumberofexternalinputstobefullycontrolledareintuitivelymoredifficultorcostlytomanage.Forinstance,achievingfullcontrolina“network”inwhichspeciesdonotinteractatallisrelativelymoredifficultaswewouldrequireaninterventionforeverysinglespecies.Conversely,thestructureofalineartrophicchaincanbeharnessedtoachievefullcontrolusingjustoneinterventiontargetedatthetopspecies;asuitablecontrolsignalcouldthencascadethroughthetrophiclevelsandreachotherspeciesinthecommunity.Specifically,drawingfromthestructural‐controllabilityliterature,weusethesizeofthemini-mumdriver‐nodesetrelativetothetotalnumberofspeciesnD=

D

N

asameasureofthecontrollabilityofanetwork—theextenttowhichthenetworkstructurecanbeharnessedtocontrolthecommunity.ThelowernDthemorecontrollablethecommunity.Inanecologicalcontext,externalinputscanbethoughtofasmanagementinterven-tionsthatmodifytheabundanceofaparticularspecies.

After finding theminimumdriver‐node set in eachof our net-works,wewantedtotestwhetherinvasionstatusorotherpredictorsarecorrelatedtocontrollability.WedothisusingasetofgeneralisedlinearmodelswithGaussianerrorsandalogitlinkfunction.There-sponse variablewas the relative size of theminimumdriver‐nodesetnDofthetwentyempiricalnetworks(teninvadedandtenunin-vaded),andweincludedinvasionstatusasapredictor.Aspredictors,wealsoincludethenetworkconnectance,thenetworknestedness(NODF), thenumberofspecies (sinceonemightnaivelyexpecttosee a negative relationship between richness and controllability;Menge,1995),thenetworkasymmetry(anindicationofthebalancebetweenplantandpollinatordiversity),andtheinteractionstrengthasymmetry (the asymmetry on the dependences between trophiclevels;Blüthgen,Menzel,Hovestadt,Fiala,&Blüthgen,2007).Wecomparedmodels using theAkaike information criterion for smallsamplesizes(AICc).

In addition, we also explored whether real networks differ intheirarchitecturefromrandomonesinaconcertedwaythatcouldimpacttheseresults.Specifically,weusedtwonullmodelseachwith99randomisationspernetwork.Inthefirst,wefollowedVázquezetal.(2007)andmaintainedtheconnectanceofthenetworkbutran-domisedthevisitsacrossspeciessuchthattherelativeprobabilitiesofinteractionsweremaintained.Wethenre‐estimatedthedirectionofcontrolandthecorrespondingsizeof theminimumdriver‐nodeset,nD.Forthesecondnullmodel,weusedtheempiricaldirectednetwork described above and randomly shuffled the direction ofcontrolbetweenaspeciespairpriortore‐estimatingthesizeoftheminimumdriver‐nodeset.

2.3 | Species' roles

Oursecondobjective is relatedtohowspeciesdiffer in theirabil-itytodrivethepopulationdynamicsofthecommunity.Weinturnexamine whether these differences are also reflected in the rolespeciesplayat supporting thestablecoexistenceofotherspecies

F I G U R E 2  Maximummatchingsandcontrolconfigurations.Indirectednetworks,amaximummatchingisthelargestpossiblesetoflinksthatdonotsharestartorendnodes(darkpurple).Maximummatchingsarenotnecessarilyunique;instead,eachofthemisrelatedtoapossibleminimumdriver‐nodesetinthenetwork(thenodestowhichanexternalcontrolinput,inorange,shouldbeappliedinordertoensurecontrollability).Thesizeoftheminimumdriver‐nodesetD correspondsexactlytothenumberofunmatchednodes(thenumberofnodesinthenetworkNminusthematchingsize).Toaccountfornetworksize,weusethesizeoftheminimumdriver‐nodesetrelativetothetotalnumberofnodesnD = D/Nasameasureoftheextenttowhichthenetworkstructurecanbeharnessedtocontrolthesystem[Colourfigurecanbeviewedatwileyonlinelibrary.com]

Directed network Maximum matchings / Control configurations

N = 4 Matching size = 3 D = 1 nD = 0.25

N = 4 Matching size = 1 D = 3 nD = 0.75

N = 5 Matching size = 3 D = 2 nD = 0.40

Matched link Unmatched link Control inputMatched node Unmatched node

1784  |    Journal of Ecology CAGUA et Al.

in the community. Ecologically, these differences are relevant be-cause resources anddata are limited, and therefore full control isinfeasible.Whilecalculatingthesizeoftheminimumdrive‐nodesetcanmeasurethecontrollabilityofanecologicalcommunity,itdoesnotprovideinformationabouttherolesthatparticularspeciesplay.

Toanswerthisquestion,weharnessthefacttheremaybemul-tiplemaximummatchings for a given network, and each of thesemaximummatchingsindicatesauniquecombinationofspecieswithwhichitispossibletocontrolthenetwork.Moreover,somespeciesbelong to thesecombinationsmoreoften thandoothers.Wecallthispropertyaspecies' “controlcapacity,”ϕ.Thehigheraspecies'controlcapacity,thegreaterthelikelihoodthatitwouldneedtobedirectlymanagedtochange(ormaintain)theecologicalstateoftheircommunity.Therefore,aspecies'controlcapacityprovidesanesti-mationofitsrelativeimportanceatdrivingthestateofthecommu-nity(Jia&Barabási,2013).

To calculate a species' control capacity ϕ, we must first enu-merate all possible maximum matchings (Supporting InformationSectionS3.2).Unfortunately,enumeratingallmaximummatchingsisextremelyexpensivefromacomputationalperspective—anetworkwithacoupledozenspecieshasseveralhundredmillionuniquemax-imummatchings. To solve this problem,we employ a recently‐de-velopedalgorithmthatrevealsthecontrolcorrelationsbetweenthenodesinthegraphwhilerequiringconsiderablylesscomputationalresources(Zhang,Lv,&Pu,2016).Usingthisalgorithm,weareabletoidentifyspeciesthatarepossiblecontrolinputs—thosethatbelongtotheminimumdriver‐nodesetinatleastoneofthepossiblecontrolconfigurations.Here,weextendthisalgorithmsuchthatitispossibletocalculateahighlyaccurateapproximationofthecontrolcapacityϕofeveryspecies inthenetwork (Supporting InformationSectionS3.3).Inthenetworksthatcontainedreciprocallinks(becausetherewasnoasymmetry inthedependencesofaspeciespair),weaver-agedaspecies'controlcapacityϕacrosseverypossible“non‐recip-rocal”versionofthenetwork(SupportingInformationSectionS3.4).

Wethenexaminedhowspecies‐levelpropertieswererelatedtocontrolcapacityusingasetofcandidategeneralisedlinearmodelswithbinomialerrorstructure.Thesemodelsincludedfivepredictorvariablesthatmirrorthenetwork‐levelpredictors.First,thespecies'contributiontonestedness,whichhasbeenproposedasakeyfea-turethatpromotesstabilityandrobustnessinmutualisticnetworks(Saavedra,Stouffer,Uzzi,&Bascompte,2011).Second,thespecies'strength(thesumofaspecies'visits),whichquantifiesthestrengthofaspecies'associationsandis indirectlyrelatedtoitsabundance(Poisot,Canard,Mouquet,&Hochberg,2012).Third,thedirectionofasymmetrywhichquantifiesthenetbalanceindependencies;thatis,itindicatesifaspeciesaffectsotherspeciesmorethanwhattheyaffectitornot(Vázquezetal.,2007).Fourth,thespecies'degreeinordertoaccountfortheintrinsiccentralityofaspecies.Finally,weincludeda categorical variable for the species' trophic level (plantorpollinator)andaninteractiontermbetweentrophiclevelandtheprevious four variables. To facilitate comparison between predic-tors, degree and visitation strength were log‐transformed and allfourcontinuousvariableswerescaledtohaveameanofzeroand

astandarddeviationofone. Inthesemodels,speciesfromallnet-workswereanalysedtogether.Weinitiallyincludedrandomeffectsto account for possible variation across communities. Specifically,wetestedstructuresthatallowedforarandominterceptforthenet-work,site,andthestudyitcomesfrom.However,wefoundthatinallcasestheamong‐groupvariancewaseffectivelyzero,andthereforewedidnotincludeanyrandomeffectinfurtheranalyses.Wethengeneratedallpossiblecandidatemodelsacrossthespaceofmodelswithall,some,andnoneofthepredictorvariables.Toidentifythemodelsthatwerebestsupportedbythedata,wefirstdeterminedthemostparsimoniousrandomstructureusingtheAICc.Therela-tive importanceofvariableswasthenassessedby lookingattheireffectsizesinthetop‐rankedmodelsandthecumulativeweightofthemodelsinwhichtheyarepresent.

Inaddition,wewantedtounderstandhowaspecies'controlca-pacityϕdescribedaboverelatestometricsofkeystonenessbasedoncentrality.Specifically, ineachnetwork,wecalculatedthespe-cies'degree,betweenness,closenesscentrality(MartínGonzálezetal.,2010),pagerank(McDonald‐Maddenetal.,2016),andEigencen-trality (Jordano,Bascompte,&Olesen, 2006).We then calculatedthespearmancorrelationcoefficientbetweencontrolcapacityandeachofthesecentralitymetrics.

Ouranalysisrevealedthatsomespecieshaveacontrolcapacityϕ=1.Thesespeciesarecriticaltocontrollingtheircommunitybe-causetheyarepartoftheminimumdriver‐nodesetineverycontrolscenario. Inotherwords, it is theoretically impossible todrive thestateofthecommunitytoadesiredstatewithoutdirectlymanag-ingtheabundanceofthesespecies.Wethusanticipatethatthesespecies have a disproportionally large impact on the communitydynamics.To test thishypothesis,we identified thesecritical spe-ciesineachofthenetworksandinvestigatedwhethertheyhavealargerthanaverage impactonthestablecoexistenceofspecies inthe community.Withinmutualistic networks, one usefulmeasureofstablecoexistenceiscalledstructuralstability(Rohr,Saavedra,&Bascompte,2014).Mathematically,thestructuralstabilityofanet-workrepresentsthesizeoftheparameterspace(i.e.,growthrates,carryingcapacities,etc.)underwhichallspeciescansustainpositiveabundances(Saavedra,Rohr,Olesen,&Bascompte,2016).Thecon-tribution of any given species i to stable coexistence can be esti-matedbycalculatingthestructuralstabilityofthecommunitywhenthefocalspeciesiisremoved.Toallowcomparisonacrosscommuni-ties,thestructuralstabilityvalueswerescaledwithineachnetworktohaveameanofzeroandastandarddeviationofone.Giventhesespecies‐specific estimatesof structural stability,we thenused a t testtocomparethecontributiontostablecoexistenceofcriticalandnon‐criticalspecies.MoredetailsaboutthecalculationofstructuralstabilitycanbefoundintheSupportingInformationSectionS4.

2.4 | Testing assumptions

Justlikethecentralitymetrics,theinformationobtainedbyapplyingstructural controllabilitydependson the abilityof thenetwork toaccuratelyrepresenttheecologicalcommunity.Wethustestedthe

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sensitivityofourapproachtotwofundamentalassumptions.First,wetestedthatvisitationisanappropriateproxytoinferinterspecificeffectsbycomparingtheresultsobtainedusingvisitationtotwoal-ternativemetricsinaseparatedatasetthatlackedinvasivespecies(Ballantyne,Baldock,&Willmer,2015).Specifically,wealsocalcu-lated the controllability (the size of theminimumdriver node‐set)and the control capacityofnetworks constructedusingpollinatorefficiency(whichmeasuresthepollendepositionofaninteraction)andpollinator importance (whichaccountsforbothpollendeposi-tionandvisitationandhenceisregardedasamoreaccurateestima-tionofthepollinationservicereceivedbyplants;Ne'eman,Jürgens,Newstrom‐Lloyd,Potts,&Dafni,2010).SeeSupportingInformationSectionS5formoredetails.

Second,becauseinterspecificdependenciesthemselvesdependonthenetworktopologyandconsequentlyontheaccuratesamplingofinteractions,wetestedtherobustnessofstructuralcontrollabilitytotheuncertaintyinvolvedwiththesamplingofinteractions.Here,wecomparedtheresultsobtainedwhenusingthefullnetworkandwhenrandomlyremovinginteractionsfromtheweakestlinksinthenetwork. This effectively removed the rare interactions from thenetworks(moredetailsintheSupportingInformationSectionS6).

3  | RESULTS

3.1 | Controllability

Thesizeoftheminimumdriver‐nodesetrelativetothenumberofspeciesineachnetworknDrangedbetweennD = 0.58 and nD = 0.88 (median0.74).

3.2 | Differences between invaded and uninvaded networks

Wefoundthattherelativesizeoftheminimumdriver‐nodesetofin-vadedcommunitieswasnotsignificantlydifferentfromthatofcom-munitiesthathavenotbeeninvaded(Figure3a).Incontrast,therewasalargenegativerelationshipbetweennDandthenetworkasymmetry

(Figure3b).Furthermore,therewerealsonegative,albeitweaker,rela-tionshipsbetweennDandconnectance,nestednessandspeciesrich-ness(TableS3).Therelativesizeoftheminimumdriver‐nodesetnDofempiricalnetworksdidnotdifferfromthatofanullmodelthatroughlypreserved the degree distribution and fully preserved the networkconnectance(p=0.66;Figure3c).However,empiricalnetworkshadalargernDthannullmodelsthatpreservedtheinteractionsbutshuffledthedirectionofcontroloftheempiricalnetwork(p = 2.4 × 10−7).

3.3 | Species' roles

Speciesvariedwidelyintheircontrolcapacity(Figure4).Pollinatorshad, inaverage, largercontrolcapacitiesthanplants.Thatsaid,al-most nopollinatorwas critical for network control, (where a spe-ciesiscriticalforcontrolifithascontrolcapacityϕ=1).Plantshadamultimodaldistributionofcontrol capacitywithmaximaatbothextremesofthedistribution(Figure4a).Intriguingly,everyinvasivespecieswascritical fornetworkcontrol ineachof theircommuni-ties.Thespecies‐levelmodels identifiedapositiverelationshipbe-tweencontrolcapacityϕandaspecies'contributiontonestedness,visitationstrength,andtheasymmetryofitsdependences(Table2;Figure5;TableS4).Comparatively,species'degreewasonlyweaklyassociatedwith control capacity (Table S5). In fact, many specieswith a lowdegree, especially pollinators, exhibited a large controlcapacityintheircommunities(FigureS10a).

Species'controlcapacityϕwasonlyweaklycorrelatedwithcom-monly‐usedcentralitymetrics.TheSpearmancorrelationbetweentheserangedbetween−0.14(withbetweeness‐centrality)and0.42(withEigen‐centrality),seeFigureS11a.Thecorrelationcoefficientwithdegreewas−0.13,howevermostspecieswithhighdegreealsotendedtoattainahighcontrolcapacity(FigureS10a).

Finally,wefoundthatcriticalspecieshaveaparticularlylargeim-pactonspeciescoexistencewhencomparedtonon‐criticalspecies.Indeed,thestructuralstabilityofthenetworkswherecriticalspe-cieswereremovedwasconsiderablylowerthanthosewherenon‐critical species were removed (p = 2 × 10−15; Figure 6; SupportingInformationS4).

F I G U R E 3  Driversofnetworkcontrollability.(a)Probabilitydensityoftherelativesizeoftheminimumdriver‐nodesetnDintheinvaded(light)anduninvaded(dark)empiricalnetworks.(b)Relationshipbetweentheasymmetryplant/pollinatorrichnessandnD.(c)Probabilitydensityofthedifferencebetweentherelativesizeoftheminimumdriver‐nodesetofrandomnetworksandthatofempiricalnetworks.Werandomisedeitherthespeciesvisitationpatterns(lightline)orrandomisedthedirectionofcontrolbetweenaspeciespair(darkline).Theverticaldashedlinesin(a)and(c)indicatethemedianvaluesofthedistributions

2

4

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(a)

0.65

0.67

0.69

0.71

−0.8 −0.7 −0.6 −0.5 −0.4

Network asymmetry

n D

(b)

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(c)

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3.4 | Testing assumptions

Wefoundthatusingvisitationasaproxyforthestrengthofspecies'interactions leads to similar results than those obtainedusing pol-linatorimportance(regardedasanaccuratemeasureofthepollina-tionservicetoplants;SupportingInformationSectionS5;Ne'eman,Jürgens,Newstrom‐Lloyd,Potts,&Dafni,2010).Importantly,wealsofoundthatstructuralstabilityisrobusttoincompletesamplingofin-teractions.Indeed,wefoundstrongagreementbetweenresultsob-tainedusingthecompleteempiricalnetworksandthoseobtainedbyrandomlyremovingtheweakestinteractions(SupportingInformationSectionS6).Despiteremovingrareinteractionsandspecies,therela-tivesizeof theminimumdriver‐nodeset, thesuperiorspecies,andtherelativerankingsofcontrolcapacityweregenerallymaintained.Ofparticularnote,wefoundthatcriticalspeciesinthefullnetworkwerealsocriticalinthevastmajorityofrarefiednetworks.

4  | DISCUSSION

Ourmaingoalwastounderstandtherolethatspeciesplayinbothmodifyingtheabundanceofthespeciestheyinteractwithandthestateofthecommunityasawhole.Toachievethatgoalweapplied

structural controllability,afieldatthe intersectionbetweencontroland complex theory that allowed us to obtain two key pieces ofinformation: the controllability of a network and a species' control capacity (Table 1).We found that the controllability of a networkdoesnotdependonitsinvasionstatusandthatthespeciesthatarecriticaltoalteringthestateofthecommunityarealsotheonesthatmostsustainthestablecoexistenceofspeciesintheircommunities.

Ourresultsindicatethatfullycontrollingecologicalnetworksmightcurrentlybeoutofreachforallbutthesmallestcommuni-ties (Motter, 2015). Indeed, themedian sizeof the relativemin-imumdriver‐node set inourdatasetwasnD =0.74, ahighvaluewhen compared to other complex systems in which controlla-bility has been investigated (the lowernD themore controllablethe community). For instance, only gene regulation networksappear to achieve similar levels of controllabilitywhilemost so-cial, power transmission, Internet, neuronal, and evenmetabolicnetworks seem to be “easier” to control (0.1<nD<0.35) (Liu etal.,2011).Structural controllabilityprovides solid theoretical ra-tionaleforthemanydifficultiesencounteredinthemanagementand restoration of natural communities (Woodford et al., 2016).Nevertheless, structural controllabilitymight be helpful at iden-tifyingcommunities inwhichchanges in theecological statearemorelikelytooccur.

TA B L E 2  Selectiontableofthebinomialgeneralisedlinearmodelsofspecies'controlcapacity,ϕ.Onlymodelswithaweightlargerorequalto0.01areshown

Model terms

df ΔAICc Weightint. k l a n s k:l l:a l:n l:s

−1.20 + 0.80 0.15 0.29 + + 7 0.00 0.48

−1.19 + 0.76 0.13 0.35 + + + 8 1.52 0.22

−1.26 −1.24 + 1.44 0.39 1.07 + + + 9 4.09 0.06

−1.37 −0.66 + 1.03 1.06 + + + 8 4.39 0.05

−1.27 −1.15 + 1.37 0.33 1.07 + + + + 10 4.92 0.04

−1.37 −0.10 + 0.90 0.43 + + 7 6.36 0.02

−1.25 −0.28 + 1.24 0.40 + + 7 6.47 0.02

−1.24 −0.62 + 1.29 0.38 0.40 + + 8 6.50 0.02

−1.39 0.30 + 0.83 + + 6 6.72 0.02

−1.28 −0.17 + 1.16 0.32 + + + 8 7.03 0.01

−1.26 −0.53 + 1.23 0.32 0.39 + + + 9 7.42 0.01

−1.02 + 0.69 0.30 0.31 + 6 7.48 0.01

Note.Terms:intercept(int),degree(k),trophiclevel(l),asymmetry(a),contributiontonestedness(n),visitationstrength(s).

F I G U R E 4  Probabilitydensityofthecontrolcapacityϕof(a)plantsand(b)pollinatorsacrossallnetworks.Thecontrolcapacityofallinvasivespeciesisϕ=1andisdepictedwithsolidcircles[Colourfigurecanbeviewedatwileyonlinelibrary.com]

0.0

2.5

5.0

7.5

0.00 0.25 0.50 0.75 1.00Control capacity (φ)

Pro

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dens

ity(a) Control capacity of plants

0.00 0.25 0.50 0.75 1.00Control capacity (φ)

(b) Control capacity of pollinators

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Thedifferencesbetweenthecontrollabilityacrossnetworksarelikely to arise fromdifferences in their structure rather than theirinvasionstatus.Specifically,whencontrollingfornetworkstructure,wefoundnodifferencebetweenthecontrollabilityofinvadedanduninvaded networks. Instead controllability is almost completelyconstrainedbythepatternsofspeciesrichnessateachtrophicguildandtheirdegreedistributions (Blüthgen,Menzel,Hovestadt,Fiala,&Blüthgen,2007;Melián&Bascompte,2002).Thesetwofactorsare particularly relevant because they govern the asymmetric na-ture ofmutual dependences,which themselves provide the foun-dationofstructureandstabilityinmutualisticnetworks(Astegiano,Massol,Vidal,Cheptou,&Guimarães,2015;Bascompteetal.,2006;Memmott,Waser,&Price,2004).

Accordingly, our results suggest that structural controllability iscloselyrelatedtothepersistenceofanecologicalcommunitybasedontwolinesofevidence.First,wefoundacomparativelysmallbutthought‐provoking negative relationship between the controllabil-ity of a network and its nestedness. Previous studies indicate thatnestednesspromotesspeciescoexistenceandconfersrobustnesstoextinction(Bastollaetal.,2009;Memmottetal.,2004)evenattheex-penseofthedynamicstabilityofthemutualisticcommunity(Saavedraetal.,2016).Theseobservationsareinagreementwithourresults,aswewouldexpectthedynamicstability(theabilitytoreturntoequi-libriumafteraperturbationinspeciesabundances)ofacommunitytobecorrelatedtothedifficultytocontrol it.Second,species'controlcapacitywasstronglycorrelatedtotheircontributiontonestednessandcriticalspecieshadthelargestimpacttothestablecoexistenceofspeciesintheircommunities.Therefore,speciesthatplayakeyroleatdeterminingthestateofthecommunitymightalsobemorekeyto“maintaintheorganizationanddiversityoftheirecologicalcommuni-ties,”oneofthehallmarksofkeystonespecies(Mills&Doak,1993).

Whencontrolling for a species' visitation strength (the sumofa species' visits),which is indirectlyaproxyof itsabundance, andthenetbalanceofitsdependencies,wefoundthatcontrolcapacitycouldnotbeeasilypredictedbyspecies'degreeorothermetricsofcentrality. For instance, some specieswith a lowdegree achievedthemaximumcontrolcapacityandwerecriticalforcontrolintheircommunities.Atfirstglance,ourfindingschallengenumerousstud-iesthathighlighttherolethatcentralspeciesplayinthedynamicsoftheircommunitiesandtheirutilityatpredictingspeciesextinctions(Jordan,2009).However,furtherinspectionshowsthatourresultsdonotcontradictthesefindings;mostspecieswitha largedegreealsohavealargecontrolcapacityandallofthemwereclassifiedassuperior nodeswhich corroborates the utility of classic centralitymetrics.Puttingtheseobservationstogether,ourresultsthereforetakepreviousfindingsonestepfurtherandsuggestthatcentralitymightpaintanincompletepictureoftherelevanceofspecies.

Other conceptual differences between structural controllabilityandcentralitymetricsprovidethreekeyinsightsintotheconservationof ecological networks. First, structural controllability emphasizesthattheeffectaspecieshasonotherspeciesisnotindependentof

F I G U R E 5  Partial‐residualplotsfortheindependentvariables:(a)contributiontonestedness,(b)visitationstrength,(c)asymmetryofdependences,and(d)degree.Partial‐residualplotsshowtherelationshipbetweencontrolcapacityandeachoftheindependentvariableswhileacccountingforallotherremainingvariables.PlotedvaluescorrepondtothepredictionsoftheweightedaverageofthemodelsshowninTable2[Colourfigurecanbeviewedatwileyonlinelibrary.com]

−10

0

10

20

−2.5 0.0 2.5 5.0

Contribution to nestedness

Parti

al re

sidu

als Plants

Pollinators

(a)

1 10 100Visitation strength

PlantsPollinators

(b)

−1.0 −0.5 0.0 0.5 1.0Dependence asymmetry

PlantsPollinators

(c)

1 10Degree

PlantsPollinators

(d)

F I G U R E 6  Probabilitydensityofthestructuralstabilityofthecommunitiesafterasinglefocalspeciesisremoved.Mathematically,thestructuralstabilityofanetworkrepresentsthesizeoftheparameterspace(i.e.,growthrates,carryingcapacities,etc.)underwhichallspeciescansustainpositiveabundances.Thestructuralstabilityofcommunitiesinwhichcriticalspecieshavebeenremoved(darkerline)isconsiderablysmallerthanthatofcommunitiesinwhichnon‐criticalspecieshavebeenremoved.Thisindicatesthatcriticalspeciescontributemoretothestablecoexistenceoftheircommunities.Toallowcomparisonacrosscommunities,thestructuralstabilityvalueswerescaledwithineachnetworktohaveameanofzeroandastandarddeviationofone.Here,weassumevaluesofthemutualistictrade‐offandmeaninterspecificcompetitionofδ = 0 and ρ=0.01respectively(Saavedraetal.,2016).However,thechoiceoftheseparametersdoesnotaffecttheresults(SupportingInformationS4)

0.0

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−5.0 −2.5 0.0Scaled structural stability

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Critical species (φ = 1)Redundant (φ < 1)

1788  |    Journal of Ecology CAGUA et Al.

theeffectsthattheotherspecieshaveinthecommunity.Therankingsprovidedbycentralitymetricsandotherheuristicsfailtoaccountforthecollectiveinfluenceofseveralspeciesatonce.Second,itdemon-stratesthattoensurethepersistenceofacommunityitisoftennec-essarytoconsidertheabundancesofmorethanasinglespecies,evenwhenfullcontrolisinfeasibleorundesired(forexample90%ofourcommunitiescontainedmorethanonecriticalspecies).Third,struc-turalcontrollabilityexplicitlyacknowledgestheexistenceofmultiplemanagementstrategiesandsomewillbebetterthanothersdepend-ingonthecontext.Approachestoprioritisespeciesforconservationandreintroductionbasedontraitsorcentralityarestillusefulandarelikelytooverlapwithspecies'controlcapacity(Devoto,Bailey,Craze,&Memmott, 2012; Pires,Marquitti, &Guimarães, 2017). Steppingback,ourresultsalsoprovidesupporttotheideathatmanagementdecisionsshouldnotbebasedonasingletechniquebutindicatethatfocusingonecosystemprocessesand interactionsmaybemoreef-fectivethantraditionalranking‐basedapproaches(Harvey,Gounand,Ward,&Altermatt,2017).Asmuchpotentialasanymetricormetricstosummarisespecies'importancemightappeartohave,it'sclearthatwealsoneedmoreempiricalstudiesindifferenttypesofnetworksinordertobuildintuitionandgroundtruththeirusefulness.

Ourchoiceofstudyinginvaded/uninvadednetworkswasbasedonadesiretocontrasttheextensiveempiricalevidenceoftheroleof invasiveplantswithourtheoretical results.Wefoundthat inva-siveplantswerealwayscriticalfornetworkcontrolandassuchourresultswereinlinewithourexpectations.Invasiveplantshavebeenpreviouslyfoundtoexacerbatetheasymmetriesintheircommunities(Aizen,Morales,&Morales,2008;Bartomeusetal.,2008;Henriksson,Wardle,Trygg,Diehl,&Englund,2016)andtoattainhighcentralityintheircommunities(Palacio,Valderrama‐Ardila,&Kattan,2016;Vilaetal.,2009).Wefound,however,thatitisnotthatinvasiveplantshavesomedifferentmechanismforinfluencingthecommunitycomparedtotheirnativecounterparts(Emer,Memmott,Vaughan,Montoya,&Tylianakis,2016;Stouffer,Cirtwill,&Bascompte,2014).Bothnativespeciesandinvasiveplantstendedtoattainahighcontrolcapacityiftheywereimportanttonetworkpersistence,wereabundant,anddependedlittleonotherspecies.Furthermore,ourobservationthatchangesintheabundanceofinvasiveplants(andpresumablyallcrit-icalspecies)arecrucialtomodifythestateofthecommunityagreeswithrecentevidenceshowingthatecosystemrestorationfocusedontheeradicationofinvasiveplantscanhavetransformativedesirableeffectsinplant‐pollinatorcommunities(Kaiser‐Bunburyetal.,2017).However,ourresultsalsosuggest thatremovalsmustbeexercisedwithcaution.Notonlyitishardtopredictthedirectioninwhichthesystemwill change, butwe also show that critical species can un-derpinthecoexistenceofspeciesandthereforesomecommunitiesmay be acutely vulnerable to their eradication (Albrecht, Padron,Bartomeus,&Traveset,2014;Travesetetal.,2013).

Structural controllability assumes that the networks can be ap-proximatedusing linear functional responses (Liu&Barabási,2016).Theramificationsofthisassumptionimplythat,whilestructuralcon-trollability isuseful to identifyspecies thatarerelevant fornetworkcontrol,itcannotbeusedtodesigntheexactinterventionsthatshould

beappliedtothesespeciesinordertoachieveadesiredstate.Inanidealscenario,wewouldcompletelyincorporatethespecies'dynam-ics intothecontrollabilityanalysis(Cornelius,Kath,&Motter,2013);therealityisthatsuchinformationisrarelyavailableinmostecologicalscenarios. Incontrast,structuralcontrollabilityonlyrequiresaquan-titativeapproximationof thenetwork's interactionstogainvaluableinsightfromthecommunity.Furthermore,whiletherelationshipbe-tween centrality and keystoneness is based on an intuitive under-standingofwhatakeystonespeciesis,theassumptionsofstructuralcontrollabilityareexplicitandtheestimationofaspecies'importancearisesfromamechanisticunderstandingofthepopulationdynamicsbetweenspecies.Byaccountingfornetworkdynamics(evenifinasim-pleway),structuralstabilityincorporatesmoreecologicalrealism,es-peciallyintheextremescenarioinwhichthestructureofinteractionswithinthecommunityonlymarginallyaffectsthecommunity'sstate.

5  | CONCLUSIONS

Herewe show that structural controllability can be applied in anecological setting togain insight into thestabilityofacommunityandtherolethatspeciesplayinmodifyingtheabundanceofotherspeciesandultimatelythestateofthecommunity.Thesecharacter-isticsmake structural stability an ideal framework toevaluate theeffectsof invasionsandothertypesofperturbations. Importantly,structuralcontrollabilitycanbeusedtoidentifycriticalspeciesinthecommunitythatpromotebiodiversityandunderpinthestablecoex-istenceof species in their community.Collectively, critical specieshavetheabilitytostrongly influencethestateoftheircommunityandthereforearelikelytobehighlyrelevantforecosystemmanage-mentandconservation.Whileuseful,centralitymetrics—whichhaveoftenbeenusedasaproxyforkeystoneness—failtoidentifysomeofthesespecies,highlightingtheir limitationswhenwefullyembracethenotionthatecologicalcommunitiesaredynamicalsystems.Paine(1969)showednearly50yearsagothatasinglespeciescansome-timesfillthisroleonitsown.Structuralcontrollabilitysuggeststhatthissituationmightbetheexceptionratherthantherule.Weseeourstudyasastartingpointtostudythecontrollabilityofecologicalandsocio‐ecologicalsystemswheremanyexcitingquestionslieahead.

ACKNOWLEDG EMENTS

The authors thank JaneMemmott and co‐authors, and everyonethat hasmade their data available to us, Takeuki Uno for the in-sightprovidedtofindthesetofallmaximummatchingalgorithms,and Jason Tylianakis, Bernat Bramon Mora, Guadalupe Peralta,Rogini Runghen,MichelleMarraffini,MarkHerse,Warwick Allen,MatthewHutchinson,andMariliaGaiarsaforfeedbackandvaluablediscussions. E.F.C. acknowledges the support from theUniversityofCanterburyDoctoral Scholarship, theUniversity ofCanterburyMeadow Mushrooms Postgraduate Scholarship, a New ZealandInternationalDoctoralResearchScholarship,andatravelgrantfromtheEuropeanSpaceAgency.D.B.S.acknowledgesthesupportofa

     |  1789Journal of EcologyCAGUA et Al.

Marsden Fast‐Start grant and a Rutherford Discovery Fellowship,administeredbytheRoyalSocietyTeofNewZealandeAparangi.

AUTHORS' CONTRIBUTIONS

D.B.S.conceivedtheidea;allauthorscontributedtothedevelopmentofthetheoreticalframework;E.F.C.performedallanalysis;E.F.C.andD.B.S.wrotethemanuscript.Allauthorscontributedtoitsrevision.

DATA ACCE SSIBILIT Y

All data used in this manuscript have already been published byLopezaraiza‐MikelHayesWhalley andMemmott (2007), BartomeusVilàandSantamaría(2008),andBallantyneBaldockandWillmer(2015).Thereadershouldrefertotheoriginalsourcestoaccessthedata.

ORCID

Edgar Fernando Cagua https://orcid.org/0000‐0001‐5867‐3687

Kate L. Wootton https://orcid.org/0000‐0002‐9033‐8210

Daniel B. Stouffer https://orcid.org/0000‐0001‐9436‐9674

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How to cite this article:CaguaEF,WoottonKL,StoufferDB.Keystoneness,centrality,andthestructuralcontrollabilityofecologicalnetworks.J Ecol. 2019;107:1779–1790. https://doi.org/10.1111/1365‐2745.13147