Graph Aggregation

(1)

OATAO is an open access repository that collects the work of Toulouse

researchers and makes it freely available over the web where possible

Any correspondence concerning this service should be sent

to the repository administrator:

tech-oatao@listes-diff.inp-toulouse.fr

This is an author’s version published in:

http://oatao.univ-toulouse.fr/2

2155

To cite this version:

Endriss, Ulle and Grandi, Umberto

Graph Aggregation. (2017) Artificial

Intelligence, 245. 86-114. ISSN 0004-3702.

Official URL:

https://doi.org/10.1016/j.artint.2017.01.001

(2)

Graph

aggregation

✩

Ulle Endriss

a

,

Umberto Grandi

b

a_ILLC,_University_of_Amsterdam,_The_Netherlands b_IRIT,_University_of_Toulouse,_France

a

b

s

t

r

a

c

t

Keywords:

Socialchoicetheory Collectiverationality Impossibilitytheorems Graphtheory Modallogic Preferenceaggregation Beliefmerging Consensusclustering Argumentationtheory

Graphaggregation is theprocess ofcomputingasingle output graphthatconstitutesa goodcompromisebetweenseveralinputgraphs,eachprovidedbyadifferentsource.One needstoperformgraphaggregationinawidevarietyofsituations,e.g.,whenapplyinga votingrule(graphs aspreferenceorders),whenconsolidatingconflictingviewsregarding the relationships between arguments in a debate (graphs as abstract argumentation frameworks),orwhen computingaconsensus betweenseveralalternativeclusteringsof agiven dataset (graphs as equivalence relations). Inthis paper, we introduce aformal framework for graph aggregation grounded in social choice theory. Our focus is on understandingwhichpropertiessharedbytheindividualinputgraphswilltransfertothe outputgraphreturnedbyagivenaggregationrule.Weconsiderbothcommonproperties ofgraphs,suchastransitivityandreflexivity,andarbitrarypropertiesexpressibleincertain fragments of modal logic. Ourresults establish several connections between the types ofproperties preserved under aggregation and the choice-theoretic axioms satisfied by therulesused.Themostimportantoftheseresultsis apowerfulimpossibility theorem thatgeneralisesArrow’sseminalresultfortheaggregationofpreferenceorderstoalarge collectionofdifferenttypesofgraphs.

1. Introduction

Suppose each of the members of a group of autonomous agents provides us with a different directed graph that is deﬁned on a common set of vertices. Graph aggregation is the task ofcomputing a single graph over the same set of verticesthat,insomesense,representsagoodcompromisebetweenthevariousindividualviewsexpressedbytheagents. Graphs areubiquitousin computerscience andartiﬁcialintelligence (AI).Forexample,inthecontext ofdecisionsupport systems,anedgefromvertex x tovertex y mightindicatethatalternative x ispreferred toalternative y.Inthecontextof modelling interactionstakingplace onan onlinedebatingplatform, anedge fromx to y mightindicate thatargument x

✩ _This_work_reﬁnes_and_extends_papers_presented_at_COMSOC-2012_[1]_and_ECAI-2014_[2]_._We_are_grateful_for_the_extensive_feedback_received_from

DavideGrossi,SylvieDoutre,WeiweiChen,severalanonymousreviewers,andtheaudiencesattheSSEACWorkshoponSocialChoiceandSocialSoftware heldinKielin2012,theDagstuhlSeminaronComputationandIncentivesinSocialChoicein2012,theKNAWAcademyColloquiumonDependenceLogic heldattheRoyalNetherlandsAcademyofArtsandSciencesinAmsterdamin2014,acourseonlogicalframeworksformultiagentaggregationgivenatthe 26thEuropeanSummerSchoolinLogic,LanguageandInformation(ESSLLI-2014)inTübingenin2014,theLorentzCenterWorkshoponClusters,Games andAxiomsheldinLeidenin2015,theSEGAWorkshoponSharedEvidenceandGroupAttitudesheldinPraguein2016,andlecturesdeliveredatSun Yat-SenUniversityinGuangzhouin2014aswellasÉcoleNormaleSupérieureandPierre&MarieCurieUniversityinParisin2016.Thisworkwaspartly supportedbyCOSTActionIC1205onComputationalSocialChoice.ItwascompletedwhiletheﬁrstauthorwashostedattheUniversityofToulousein2015 aswellasParis-DauphineUniversity,Pierre&MarieCurieUniversity,andtheLondonSchoolofEconomicsin 2016.

(3)

undercuts or otherwise attacks argument y.And in the context of social network analysis, an edge from x to y might

expressthatperson x isinﬂuenced byperson y.Howtobestperformgraphaggregationisarelevantquestioninthesethree domains,aswellasinanyotherdomainwheregraphsare usedasa modellingtool andwhereparticulargraphsmaybe suppliedbydifferentagentsororiginatefromdifferentsources.Forexample,inanelection,i.e.,inagroupdecisionmaking context,wehavetoaggregatethepreferencesofseveralvoters.Inadebate,wesometimeshavetoaggregatetheviewsof theindividualparticipantsinthedebate.Andwhentryingtounderstandthedynamicswithinacommunity,wesometimes havetoaggregateinformationcomingfromseveraldifferentsocialnetworks.

Inthispaper, we introducea formal framework forstudyinggraphaggregationin generalabstractterms andwe dis-cussindetailhowthisgeneralframeworkcanbeinstantiatedtospeciﬁc applicationscenarios.We introduceanumberof concretemethodsforperformingaggregation,butmoreimportantly,ourframeworkprovidestoolsforevaluatingwhat con-stitutesa“good”methodofaggregationanditallows ustoaskquestionsregardingtheexistenceofmethodsthat meeta certainsetofrequirements.Ourapproachisinspiredbyworkinsocialchoicetheory[3],whichoffersarichframeworkfor thestudyofaggregationrules forpreferences—averyspeciﬁcclassofgraphs.Inparticular,weadopttheaxiomaticmethod

usedinsocialchoicetheory,aswellasother partsofeconomictheory,toidentifyintuitivelydesirablepropertiesof aggre-gationmethods,todeﬁnetheminmathematicallypreciseterms,andtosystematicallyexploretheirlogicalconsequences.

Anaggregationrulemapsanygivenproﬁle ofgraphs,oneforeachagent, intoasinglegraph,whichweareoftengoing torefer toasthe collectivegraph. Thecentral conceptwe focusonin thispaperisthecollectiverationality ofaggregation ruleswithrespecttocertainpropertiesofgraphs.Supposeweconsideranagentrationalonlyifthegraphsheprovideshas certainproperties,suchasbeingreﬂexiveortransitive.Then wesaythat agivenaggregationrule F iscollectivelyrational withrespecttothatpropertyofinterestifandonlyif F canguaranteethatthatpropertyispreservedduring aggregation. Forexample, ifwe aggregate individual graphsby computingtheir union (i.e.,ifwe include an edgefrom x to y in our collective graph if atleast one ofthe individual graphs includes that edge), then it is easy to see that the property of

reﬂexivity willalways transfer.Onthe other hand,the propertyoftransitivity will not always transfer.Forexample,ifwe aggregate twographsoverthe setofvertices V

= {

x,y,z

}

,one consistingonly ofthe edge

(x,

y)andone consistingonly oftheedge

(y,

z),thenalthougheachofthesetwo graphsis(vacuously) transitive,their unionisnot,asitismissingthe edge

(x,

z).Thus,theunionruleiscollectivelyrationalwithrespecttoreﬂexivity,butnotwithrespecttotransitivity.

We studycollectiverationality withrespect tosome such well-known andwidely usedproperties ofgraphs,butalso with respect to large families of graph properties that satisfy certain meta-properties. We explore both a semantic and a syntactic approach to deﬁning such meta-properties. In our semantic approach, we identify three high-level features of graphproperties that determine the kindof aggregationrules that are collectivelyrational withrespect to them. For example,transitivity iswhatwecalla“contagious”property:undercertaincircumstances,namelyinthepresenceofedge

(y,

z),inclusionof

(x,

y)spreads to

(x,

z).Transitivityalsosatisﬁesasecondmeta-property,whichwecall“implicativeness”: theinclusionoftwospeciﬁc edges,namely

(x,

y)and

(y,

z),implies theinclusionofathirdedge,namely

(x,

z).Thethird meta-property we introduce, “disjunctiveness”, expresses that, under certain circumstances, at least one of two speciﬁc edges hasto beaccepted.Thisissatisﬁed,forinstance,bythe propertyofcompleteness:everytwo verticesx and y need

tobeconnectedinatleastoneofthetwopossibledirections.Inoursyntacticapproach,weconsidergraphpropertiesthat canbeexpressedinparticularsyntacticfragmentsofalogicallanguage.Tothisend,wemakeuseofthelanguageofmodal logic [4].This allows ustoestablishlinksbetweenthe syntacticproperties ofthelanguage used toexpressthe integrity constraintswewouldliketoseepreservedduringaggregationandtheaxiomaticpropertiesoftherulesused.

Weprove bothpossibility and impossibilityresults. Apossibilityresultestablishes that everyaggregationrule belonging to a certain class ofrules (typically definedin termsof certain axioms) iscollectively rationalwith respectto all graph propertiesthatsatisfya certainmeta-property.Animpossibility result,ontheotherhand,establishesthat itisimpossible to define an aggregation rule belongingto a certain class that would be collectively rational with respectto anygraph property that meets a certain meta-property—or that the onlysuch aggregation rules wouldbe clearly very unattractive for other reasons. Our main result is such an impossibility theorem. It is a generalisation of Arrow’sseminal resultfor preference aggregation[5],which we shall recall inSection 3.1. Ourapproach of workingwithmeta-properties hastwo advantages. First, it permits usto give conceptually simple proofs forpowerful resultswith a highdegree of generality. Second, itmakes it easy to instantiateourgeneral resultsto obtainspecific results forspecific applicationscenarios. For example,Arrow’sTheoremfollowsimmediatelyfromourmoregeneralresultbycheckingthatthepropertiesofgraphsthat representpreference orders (namely transitivity andcompleteness)satisfy the meta-properties featuring in our theorem, yetour proofofthe generaltheoremisarguablysimpler thana directproof ofArrow’sTheorem.Thisis so,becausethe meta-properties weusevery explicitlyexhibit specific featuresrequiredfortheproof,while thosefeaturesare somewhat hiddeninthespecificpropertiesoftransitivityandcompleteness.Similarly, weshowhowalternativeinstantiations ofour generalresulteasilygeneratebothknownandnewresultsinotherdomains,such astheaggregationofplausibilityorders (whichhasapplicationsinnonmonotonicreasoningandbeliefmerging)andtheaggregationofequivalencerelations(which hasapplicationsinclusteringanalysis).

Relatedwork. Our work builds on andis related to contributions in the ﬁeld of social choice theory, starting with the seminalcontributionofArrow [5].Thisconcerns,inparticular,contributionstothetheoryofvotingandpreference aggre-gation[6–10,3],butalsojudgmentaggregation[11–17].Infact,intermsoflevelsofgenerality, graphaggregationmaybe regardedasoccupyingthemiddlegroundbetweenpreferenceaggregation(mostspeciﬁc)andjudgment aggregation(most

(4)

general). Incomputer science, theseframeworksare studiedin thefield ofcomputational social choice [18].As we shall discuss insome detail,graph aggregationis an abstraction ofseveralmore specific forms ofaggregation takingplace in a widerangeofdifferentdomains.Preference aggregationisbutone example.Aggregationofspecifictypesofgraphshas beenstudied,forinstance,innonmonotonicreasoning[19],beliefmerging[20],socialnetworkanalysis[21],clustering[22], andargumentationinmultiagentsystems[23].Asweshallsee,severaloftheresultsobtainedintheseearliercontributions aresimplecorollariesofourgeneralresultsongraphaggregation.

Paperoverview. Theremainder ofthispaperisorganisedasfollows.InSection 2,we introduceour frameworkforgraph aggregation.Thisincludesthediscussionofseveralapplicationscenarios,thedefinitionofanumberofconcreteaggregation rules, andtheformulationofvarious axiomsidentifyingintuitively desirablepropertiesofsuch rules.It alsoincludes the definitionoftheconceptofcollectiverationality.Finally,weproveanumberofbasicresultsinSection2:characterisation results linking rules and axioms, as well aspossibility results linking axiomsand collective rationality requirements. In Section3,wepresentourimpossibilityresultsforgraphaggregationrulesthatarecollectivelyrationalwithrespecttograph propertiesmeetingcertainsemanticallydefinedmeta-properties.Therearetwosuchresults.Oneidentifiesconditionsunder which theonlyavailable rulesareso-calledoligarchies,underwhichtheoutcome isalways theintersectionofthegraphs provided by a subset ofthe agents (the oligarchs).A second result showsthat, underslightly strongerassumptions, the onlyavailablerulesarethedictatorships,whereasingleagentcompletelydeterminestheoutcomeforeverypossibleprofile. Much ofSection3isdevotedtothedefinitionandillustrationofthemeta-properties featuringintheseresults.Oncethey are in place,the proofsare relativelysimple. InSection 4, we introduceour approachto describing collectiverationality requirementsinsyntacticterms,usingthelanguageofmodallogic.OurresultsinSection4establishsimpleconditionson the syntax ofthespecificationof agraphproperty that aresufficient forguaranteeing thatthe propertyin question will be preservedunderaggregation.Thegroundingofourapproachinmodallogicalsoallowsustoprovideadeeperanalysis oftheconceptofcollectiverationalitybyconsideringthepreservationofpropertiesatthreedifferentlevels,corresponding to the three levels naturally defined by the notions of Kripke frame, Kripke model,and possible world, respectively. In Section5,wediscussfourofourapplicationscenariosinmoredetail,focusingonapplicationscenariospreviouslydiscussed in theAI literature.We show how ourgeneralresults allowusto derive newsimpleproofs ofknownresults, howthey clarify the statusofsome oftheseresults,andhowthey allowustoobtain newresultsinthesedomains ofapplication. Section6,finally,concludeswithabriefsummaryofourresultsandpointerstopossibledirectionsforfuturework.

2. Graph aggregation

In thissection,we introducea simpleframework forgraphaggregation.Thebasicdefinitions aregivenin Section2.1. While thisisageneralframework that isindependentofspecific applicationscenariosandspecific choicesregardingthe aggregation ruleused,we briefly discussseveralsuch specific scenarios inSection 2.2andsuggest definitionsforseveral specific aggregationrulesinSection 2.3.Wethen approachtheanalysisofaggregationrulesfromtwo differentbut com-plementary angles. First,in Section 2.4, we define severalaxiomaticproperties of aggregation rules that a usermaywish to imposeasrequirementswhenlookingfora“fair”or“well-behaved”aggregationruleforaspecificapplication. Wealso proveanumberofsimpleresultsthatshowhowsomeoftheseaxiomsrelatetoeachotherandtosomeoftheaggregation rulesdefinedearlier.Second,inSection2.5,weintroducethecentralconceptofcollectiverationality andweproveanumber ofsimplepositiveresultsthatshowhowenforcingcertainaxiomsallowsustoguaranteecollectiverationalitywithrespect tocertaingraphproperties.

2.1. Basicnotationandterminology

Fix a ﬁniteset of vertices V . A (directed) graph G

=

V

,

E

based on V isdeﬁned by a set ofedges E

⊆

V

×

V . We write xE y for

(x,

y)

∈

E.As V isﬁxed, G isinfact fullydeterminedby E. Wethereforeidentifysetsofedges E

⊆

V

×

V

with thegraphs G

=

V

,

E

they deﬁne. Forany kindofset S, we use2S todenote the powersetof S. So 2V×V is the set ofallgraphs. Weuse E

(x)

:= {

y

∈

V

| (

x,y)

∈

E

}

todenotetheset ofsuccessors ofavertex x in aset ofedges E and

E−1

_(y)

_{:= {}

_x

_∈

_V

_{| (}

_x,_y)

_∈

_E

_}

_to_denote_the_set_of_{predecessors of y in E.}

Agivengraphmayormaynotsatisfyaspecificproperty,suchastransitivityorreflexivity.Table 1recallsthedefinitions ofseveralsuchproperties.1Weareoftengoingtobeinterestedinfamiliesofgraphsthatallsatisfyseveralofthese proper-ties. Forinstance,aweakorder isadirectedgraphthatisreflexive,transitive,andcomplete.Itwilloftenbeusefultothink ofa graphproperty P ,suchastransitivity,asa subsetof2V×V _(the_set_of_all_graphs _over_the_set_of_{vertices V ).}_For_two

1 _Some_of_these_may_be_less_well_known_than_others,_so_let_us_brieﬂy_review_the_less_familiar_{deﬁnitions.}_The_two_Euclidean_properties_encode_Euclid’s

ideathat“thingswhichequalthesamethingalsoequaloneanother”.Negativetransitivity,apropertycommonlyassumedintheeconomicsliteratureon preferences,mayequivalentlybeexpressedas∀xyz.[(¬xE y∧ ¬y E z)→ ¬xE z],whichexplainsthenameoftheproperty.Completenessrequiresanytwo distinctverticestoberelatedonewayortheother.Connectednessonlyrequirestwo(notnecessarilydistinct)verticestoberelatedonewayortheother iftheyarebothreachablefromsomecommonpredecessor(theterm“connectedness”iscommonlyusedinthemodallogicliterature[4]).Nontriviality excludestheemptygraph,whileseriality(alsoatermusedinthemodallogicliterature)requireseveryvertextohaveatleastonesuccessor.

(5)

Table 1

Commonpropertiesofdirectedgraphs.

Property First-order condition Reﬂexivity ∀x.xEx

Irreﬂexivity ¬∃x.xEx

Symmetry ∀xy.(xE y→y Ex)

Antisymmetry ∀xy.(xE y∧y Ex→x=y)

Right Euclidean ∀xyz.[(xE y∧xE z)→y E z]

Left Euclidean ∀xyz.[(xE y∧zE y)→zEx]

Transitivity ∀xyz.[(xE y∧y E z)→xE z]

Negative Transitivity ∀xyz.[xE y→ (xE z∨zE y)]

Connectedness ∀xyz.[(xE y∧xE z)→ (y E z∨zE y)]

Completeness ∀xy.[x=y→ (xE y∨y Ex)]

Nontriviality ∃xy.xE y

Seriality ∀x.∃y.xE y

disjointsetsofedges S+and S− andagraphproperty P

⊆

2V×V,let P

[

S+

,

S−

]

= {

E

∈

P

|

S+

⊆

E and S−

∩

E

= ∅}

denote thesetofgraphsinP thatincludealloftheedgesin S+andnoneofthosein S−.

Let

N = {

1

,

. . . ,

n

}

be a ﬁnite set of (two ormore) individuals (or agents).We are oftengoing to refer to subsets of

N

as coalitions of individuals. Suppose every individual i

∈

N

speciﬁes a graph Ei

⊆

V

×

V .This gives rise to a proﬁle E

= (

E1,

. . . ,

En

)

.WeuseNeE

:= {

i

∈

N |

e

∈

Ei}todenotethecoalitionofindividualsacceptingedge e underproﬁle E. Deﬁnition 1. An aggregation

rule is

afunction F

: (

2V×V

₎

n

_→

₂V×V_,_mapping_any_given_proﬁle_of_individual_graphs_into_a

singlegraph.

We are sometimes going to denote theoutcome F(E

)

obtainedwhen applying an aggregation rule F to a proﬁle E

simplyasE andrefertoitasthecollectivegraph.Anexampleforanaggregationruleisthemajorityrule, acceptingagiven edgeifandonlyifmorethanhalfoftheindividualsacceptit.MoreexamplesaregoingtobeprovidedinSection2.3.

2.2. Examplesofapplicationscenarios

Directedgraphsareubiquitousincomputerscienceandbeyond.Theyhavebeenusedasmodellingdevicesforawide range of applications. We now sketch a number of differentapplication scenarios forgraph aggregation, each requiring differenttypesofgraphs(satisfyingdifferentproperties)tomodelrelevantobjectsofinterest,andeachrequiringdifferent typesofaggregationrules.

Example 1 (Preferences).Ourmainexampleforagraphaggregationproblemisgoingtobepreferenceaggregationas classi-callystudiedinsocialchoicetheory [5].Inthiscontext,verticesareinterpretedasalternativesavailableinanelectionand thegraphs consideredareweak ordersonthesealternatives, interpretedaspreferenceorders.Ouraggregationrulesthen reduce toso-calledsocial welfarefunctions. Socialwelfare functions,which returnapreferenceorder foreveryprofile of individual preferenceorders, aresimilarobjects asvotingrules,which onlyreturnawinningalternativeforeveryprofile. Whilethetypesofpreferencestypicallyconsideredinclassicalsocialchoicetheoryarerequiredtobecomplete,recentwork inAIhasalsoaddressedtheaggregationofpartialpreferenceorders[10],correspondingtoalarger familyofgraphsthan theweakorders.Inthecontextofaggregatingcomplexpreferencesdefinedovercombinatorialdomains,graphaggregation canalsobeusedtodecidewhichpreferentialdependenciesbetweendifferentvariablesoneshouldtrytorespect,basedon thedependenciesreportedbytheindividualdecisionmakers[24].

Example 2 (Knowledge).IfwethinkofV asasetofpossibleworlds,thenagraphonV thatisreﬂexiveandtransitive(and possiblyalso symmetric)can beused tomodelan agent’sknowledge:

(x,

y) beinganedge meansthat, ifx is theactual world,then ouragent will consider y apossibleworld [25].Ifwe aggregatethe graphsofseveralagentsby takingtheir intersection,thentheresultingcollectivegraphrepresentsthedistributedknowledgeofthegroup,i.e.,theknowledgethe membersofthegroupcaninferbypoolingalltheirindividualresources.If,ontheotherhand,weaggregatebytakingthe unionoftheindividualgraphs,thenweobtainwhatissometimescalledthesharedormutualknowledgeoftheindividual agents, i.e., the part of the knowledge available to each andevery individual on their own. Finally, if we aggregate by computingthe transitive closureof the unionof the individual graphs,then we obtaina model ofthe group’s common knowledge [26,p. 512].These concepts play arole in disciplines asdiverseasepistemology [27], gametheory [28], and distributedsystems[29].

Example 3 (Nonmonotonicreasoning).When an intelligent agent attempts toupdate her beliefsor to decide what action totake,shemayresorttoseveralpatternsofcommon-senseinference thatwillsometimesbeinconﬂictwitheach other. Totake afamous example,wemaywish toinferthat Nixonisa paciﬁst,because heis aQuakerandQuakersbydefault

(6)

are pacifists, and we may at the same time wish to infer that Nixon is not a pacifist, because heis a Republican and Republicansbydefaultarenotpacifists.InapopularapproachtononmonotonicreasoninginAI,suchdefaultinferencerules aremodelledasgraphsthatencodetherelativeplausibilityofdifferentconclusions[30].Thus,herethepossibleconclusions aretheverticesandweobtainagraphbylinkingonevertexwithanother,iftheformerisconsideredatleastasplausible asthelatter.Conflictresolutionbetweendifferentrulesofinference thenrequiresustoaggregatesuchplausibilityorders, tobeabletodeterminewhattheultimatelymostplausiblestateoftheworldmightbe[19].

Example 4 (Socialnetworks).Wemayalsothinkofeachofthegraphsinaproﬁleasadifferentsocialnetworkrelating mem-bersofthesamepopulation.Oneofthesenetworksmightdescribe workrelations,anothermightmodelfamilyrelations, andathirdmighthavebeeninducedfromsimilarities inonlinepurchasingbehaviour.Socialnetworksareoftenmodelled using undirectedgraphs, whichwe can simulatein ourframework by requiring all graphsto be symmetric.Aggregating individualgraphsthenamountstoﬁndingasinglemeta-networkthatdescribesrelationshipsatagloballevel.Alternatively, wemaywishtoaggregateseveralgraphsrepresentingsnapshotsofthesamesocialnetworkatdifferentpointsintime.The meta-networkobtainedcanbehelpfulwhenstudyingthesocialstructureswithinthepopulationunderscrutiny[21].

Example 5 (Clustering).Clusteringistheattemptofpartitioningagivensetofdatapointsintoseveralclusters.Theintention isthatthedatapointsinthesameclustershouldbemoresimilartoeachotherthaneachofthemistodatapoints belong-ing to oneofthe other clusters.This isusefulinmanydisciplines, includinginformationretrievalandmolecular biology, tonamebuttwoexamples.However,thefieldislackingaprecisedefinitionofwhatconstitutesa“correct”partitioningof the dataandtherearemanydifferentclustering algorithms, suchask-meansorsingle-linkageclustering, andeven more parameterisations ofthosebasic algorithms [31]. Observethat every partitioningthat mightget returned bya clustering algorithm induces an equivalence relation (i.e.,a graph that is reflexive, symmetric,and transitive):two data points are equivalentifandonlyiftheybelongtothesamecluster.Findingacompromisebetweenthesolutionssuggestedbyseveral clustering algorithmsiswhatisknownasconsensusclustering [32].Thisthusamountstoaggregatingseveralgraphsthat areequivalencerelations.

Example 6 (Argumentation).Inaso-calledabstractargumentationframework,argumentsaretakentobeverticesinagraph and attacksbetweenarguments are modelled asdirected edges betweenthem [33]. Agraph property ofinterest inthis context is acyclicity, asthat makes it easier to decide which arguments to ultimately accept. If we think of V as the collectionofarguments proposedina debate,aproﬁle E

= (

E1,

. . . ,

En

)

speciﬁesan attackrelationforeach ofanumber

of agents that we may wish to aggregate into a collective attack relation before attempting to determine which of the arguments might be acceptable to the group. Recent work has addressed the challenge of aggregating several abstract argumentation frameworksfrom anumber ofangles, e.g., by proposingconcrete aggregation methodsgrounded inwork onbeliefmerging[34],byinvestigatingthecomputationalcomplexity ofaggregation[35],andbyanalysingwhatkindsof proﬁleswemayreasonablyexpecttoencounterinthiscontext[36].

Example 7 (Logic). Graph aggregationis also atthe core of recentwork on the aggregation ofdifferent logics [37]. The central ideahereisthatevery logicisdeﬁnedbya consequencerelationbetweenformulas.Thus,givenasetofformulas, we can think ofa logic

L

asthe graphcorresponding to the consequence relationdeﬁning

L

. Aggregating severalsuch graphsthengivesrisetoanewlogic.Thus,thisisaninstanceofourgraphaggregationproblem,exceptthatforthecaseof logicaggregationitismorenaturaltomodelthesetofverticesasbeinginﬁnite.2

Recallthatwehaveassumedthateveryindividual speciﬁesagraphonthesame setofvertices V .Thisisanatural as-sumptiontomakeinallofourexamplesabove,butingeneralwemightalsobeinterestedinaggregatinggraphsdeﬁnedon differentsetsofvertices.Forinstance,Coste-Marquisetal.[34] havearguedthat,inthecontextofmergingargumentation frameworks, the caseof agents who are not all aware of the exact same setof arguments is ofgreat practical interest. Observethatalsointhiscaseourframeworkisapplicable,aswemaythinkof V astheunionofalltheindividualsetsof vertices(witheachindividualonlyprovidingedgesinvolving“her”vertices).

WearegoingtoreturntoseveraloftheseapplicationscenariosingreaterdetailinSection5.

2.3. Aggregationrules

Next,we deﬁne anumber ofconcreteaggregationrules. Webeginwiththree thatare particularlysimple,theﬁrst of whichwehavealreadyintroducedinformally.

Deﬁnition 2. The(strict) majority

rule is

theaggregationFmaj withFmaj

:

E

→ {

e

∈

V

×

V

: |

NeE

|

>

n2

}

.

2 _All _results_reported_in_this_paper _remain_true_if_we _permit_graphs_with _inﬁnite_{sets V of}_vertices. _However,_for_ease _of_exposition_and _as_most

applicationsaremorenaturallymodelledusingﬁnitegraphs,wedonotexplorethisgeneralisationhere.TheﬁnitenessofthesetNofagents,however,is crucial.ItisgoingtobeexploitedintheproofsofLemmas 9 and 10below,onwhichallofourtheoremsrely.

(7)

Deﬁnition 3. The intersection

rule is

theaggregationrule F_∩withF_∩

:

E

→

E1

∩ · · · ∩

En. Deﬁnition 4. The union

rule is

theaggregationrule F_∪ withF_∪

:

E

→

E1

∪ · · · ∪

En.

Inrelatedcontexts,theintersectionruleisalsoknownastheunanimityrule,asitrequiresunanimousapprovalfromall individualsforanedgetobeaccepted.Similarly,theunionruleisanominationrule,asnominationbyjustoneindividualis enoughforanedgetogetaccepted.

Undera quotarule, anedge willbe includedin thecollective graphifthenumberof individualsacceptingitmeets a certainquota.Auniform quotaruleusesthesamequotaforeveryedge.

Deﬁnition 5. A quota

rule is

an aggregationrule Fq deﬁnedvia a function q

:

V

×

V

→ {

0

,

1

,

. . . ,

n

+

1

}

, associatingeach

edgewithaquota,bystipulating Fq

:

E

→ {

e

∈

V

×

V

: |

NeE

|

≥

q(e)

}

.Fqiscalled uniform incaseq isaconstantfunction.

Theclassofuniformquotarulesincludesthethreesimpleruleswehaveseenearlierasspecialcases:the(strict)majority rule Fmaj istheuniformquotarule withq

=

n+21

,theintersectionrule F∩is theuniformquotarulewithq

=

n,andthe unionrule F_∪ isthe uniformquota rule withq

=

1.We callthe uniformquota rules withq

=

0 and q

=

n

+

1 the trivial

quotarules; q

=

0 means thatall edgeswillbe includedinthecollectivegraphandq

=

n

+

1 means thatno edgewillbe included(independentlyoftheproﬁleencountered).Theideaofusingquotarulesisnaturalandwidespread.Forexample, quotaruleshavealsobeenstudiedinjudgmentaggregation[13].

We now introduce a newclass ofaggregation rules speciﬁcally designedforgraphs that is inspired by approval vot-ing [38]. Imaginewe associate each vertex withan election in which all the possible successors of that vertex are the candidates(andinwhichtheremaybemorethanonewinner).Eachagentvotesbystatingwhichverticessheconsiders ac-ceptablesuccessorsand,basedonthisinformation,achoicefunctionselectswhichedgestoincludeinthecollective graph.

Deﬁnition 6. Letv

: (

2V

)

n

→

2V beafunction associatinganygivenvector ofsetsofverticeswithasingle setofvertices. Then the successor-approval

rule based

on v isthe aggregationrule Fv deﬁnedby stipulating Fv

:

E

→ {(

x,y)

∈

V

×

V

|

y

∈

v(E1(x),

. . . ,

En

(x))

}

.

Example 8 (Successor-approvalbasedonclassicalapprovalvoting).Consideragraphwithfourvertices:V

= {

x,y,z,w

}

.Suppose two individualsreport the graphs E1

= {(

x,y),

(x,

z)

}

and E2

= {(

x,z)

}

.When deciding which verticesto connect fromx

usingasuccessor-approvalrule,welookatE1(x)

= {

y,z

}

andE2(x)

= {

z

}

asapprovalballots,andweusev todecidewhich vertexisthewinner.If v is theclassical approvalvotingrule,whichselectsthe candidatewiththemostapprovals,then

z is thewinnerwitha scoreoftwoapprovals,followedby y withone approval,andx and w withnone. Sinceall other verticeshavenooutgoingedgesatall,wehavethat Fv

(

E

)

= {(

x,z)

}

.

Wecallv thechoicefunction associatedwithFv.Ittakesavectorofsetsofvertices,oneforeachagent,andreturns

an-othersuchset.Forexample,theclassicalapprovalvotingruleisformallydeﬁnedasv

: (

S1,

. . . ,

Sn

)

→

argmaxx∈S1∪···∪Sn

|{

i

∈

N :

x

∈

Si}|.Notehowtheargmax-operatorrangesovertheunionofallsuccessorsmentionedbyanyoftheagentsrather thanthefullsetofvertices V .Thisensuresthat,incasenoneoftheagentsapproveanyvertexasasuccessor,wedonotend upacceptingallverticesforallhavingthesame“maximal”support.Weareonlygoingtobeinterestedinchoicefunctions v thatare(i) anonymous and(ii) neutral,i.e.,forwhich(i) v

(S1,

. . . ,

Sn

)

=

v(Sπ(1)

,

. . . ,

Sπ(n)

)

foranypermutation

π

:

N → N

andforwhich (ii)

{

i

∈

N |

x

∈

Si}

= {

i

∈

N |

y

∈

Si}entails x

∈

v(S1,

. . . ,

Sn

)

⇔

y

∈

v(S1,

. . . ,

Sn

)

. There area number of

naturalchoicesfor v.Apartfromtheclassicalapprovalvotingrulementionedbefore,wemightwanttoacceptalledges re-ceivingabove-averagesupport.Whileclassicalapprovalvotingwilltypicallyresultinvery“sparse”outputgraphs,intuitively thelatterrulewillreturngraphsthathavesimilarattributesastheinputgraphs.Athirdoptionistouse“even-and-equal” cumulative voting with v

: (

S1,

. . . ,

Sn

)

→

argmaxx∈S1∪···∪Sn

i|x∈Si

1

|Si|, i.e., to let each individual distribute her weight

evenlyoverthesuccessorssheapprovesof.Thiswouldbeattractive,forinstance,underanepistemicinterpretation,where agentsspecifyingfeweredgesmightbeconsideredmorecertainaboutthoseedges.Finally,observethattheuniformquota rules(but notthegeneralquotarules)area specialcaseofthesuccessor-approvalrules.We obtain Fq withtheconstant

functionq

:

e

→

k,mappinganygivenedgetotheﬁxedquota k,byusingv

: (

S1,

. . . ,

Sn

)

→ {

x

∈

V

: |{

i

∈

N :

x

∈

Si}|

≥

k

}

.

Whilewe are notgoing todoso inthispaper,itisalsopossible toadaptthe distance-basedrules—familiarfrom pref-erenceaggregation,beliefmerging,andjudgmentaggregation[39–41]—tothecaseofgraphaggregation.Suchrules select acollective graphthatsatisﬁescertain propertiesandthat minimisesthedistancetotheindividual graphs(fora suitable notionofdistanceandasuitable formofaggregatingsuch distances).Adownside ofthisapproachisthatdistance-based rulesaretypicallycomputationallyintractable[42–45],whilequotaandsuccessor-approvalruleshaveverylowcomplexity. Wecanalsoadapttherepresentative-voterrules[46] tothecaseofgraphaggregation.Here,theideaistoreturnoneof theinputgraphsastheoutput,andforeveryproﬁletopicktheinputgraphthatinsomesenseis“mostrepresentative”of theviewsofthegroup.

(8)

Deﬁnition 7. A representative-voter

rule is

an aggregation rule F thatis such that for every proﬁle E there exists an individual i

_∈

_N

_such_that _F(_E

₎

₌

_E

i.

For instance,we mightpick the input graph that is closest to the outcome ofthe majority rule. Thismajority-based representative-voter rule alsohasverylow complexity.While we arenot going tostudyanyspeciﬁcrepresentative-voter ruleinthispaper,inSection4.4wearegoingtobrieﬂydiscussthisclassofrulesasawhole.

Weconcludeourpresentationofconcrete(familiesof)aggregationruleswithanumberofrulesthat,intuitivelyspeaking, arenotveryattractive.

Deﬁnition 8. The dictatorship ofindividuali

_∈

_N

_is_the_aggregation_rule_F

i withF_i

:

E

→

E_i.

Thus, foranygivenproﬁleofinputgraphs, Fi always simplyreturnsthegraphsubmitted bythedictator i.Notethat

everydictatorshipisarepresentative-voterrule,buttheconverseisnottrue.

Deﬁnition 9. The oligarchy ofcoalitionC

_⊆

_N

_,_with_C_being_nonempty,_is_the_aggregation_rule_F

C withF_C

:

E

→

i∈C

Ei.

Thus, FC alwaysreturnstheintersectionofthegraphssubmitted bytheoligarchsinthecoalition C.Soanindividual

inC_can_veto_the_acceptance_of_any_given_edge,_but_she_cannot_enforce_its_acceptance._In_case_C_is_a_singleton,_we_obtain adictatorship.IncaseC

₌

_N

_,_we_obtain_the_intersection_rule.

2.4. Axiomaticpropertiesandbasiccharacterisationresults

When choosing an aggregation rule, we need to consider its properties. In social choice theory, such properties are calledaxioms [9].We nowintroduceseveralbasicaxiomsforgraphaggregation.Theﬁrstsuch axiomisan independence conditionthatrequiresthatthedecisionofwhetherornotagivenedge e shouldbepartofthecollectivegraphshouldonly depend onwhichoftheindividual graphsinclude e. Thiscorrespondstowell-known axiomsinpreferenceandjudgment aggregation[5,17].

Deﬁnition 10. An aggregationrule F is calledindependent of irrelevant edges (IIE) if NE

e

=

NE

e impliese

∈

F(E

)

⇔

e

∈

F

(

E

)

.

Thatis,ifexactlythesameindividualsaccept e underproﬁles E and E,thene shouldbe partofeitherbothornone ofthecorrespondingcollectivegraphs.Thedeﬁnitionaboveappliestoall edges e

∈

V

×

V andall pairsofproﬁles E

,

E

∈

(

2V×V

)

n.Forthesakeofreadability,weshallleavethiskindofuniversalquantificationimplicitalsoinlaterdefinitions. IEE isa desirable property,because—if itcan be satisfied—itgreatly simplifiesaggregation,in bothcomputational and conceptual terms. As we shall see,some ofthe arguablymost naturalaggregation rules, thequota rules defined earlier, satisfy IIE. At the same time, as we shall also see, IEE is a very demanding property that is hard to satisfy if we are interestedinricherformsofaggregation.Indeed,IIEwillturnouttobeattheverycentreofourimpossibilityresults.

While verymuch a standardaxiom, wemightbe dissatisfiedwithIIE fornotmaking referencetothe factthat edges are definedinterms ofvertices.Ournext twoaxiomsare much moregraph-specificanddonot havecloseanalogues in preferenceorjudgmentaggregation.The firstofthemrequiresthatthedecisionofwhetherornottocollectivelyaccepta givenedgee

= (

x,y)shouldonlydependonwhichedges withthesamesource x are acceptedbytheindividuals. Thatis, acceptanceofanedgemaybeinﬂuencedbywhatagentsthinkaboutotheredges,butnotthoseedgesthataresuﬃciently unrelatedtotheedgeunderconsideration.Belowwewrite F

(

E

)(x)

forthesetofsuccessorsofvertex x inthesetofedges inthecollectivegraph F

(

E

)

,andsimilarly F(E

)

−1

(

y)forthepredecessorsof y in F(E

)

.

Deﬁnition 11. Anaggregation F iscalledindependent ofirrelevant sources (IIS) if Ei

(x)

=

E_i

(x)

forall individualsi

∈

N

implies F(E

)(x)

=

F

(

E

)(x)

.

Deﬁnition 12. Anaggregationrule F iscalledindependentofirrelevanttargets(IIT)if E_i−1

(y)

=

E_i−1

(y)

forallindividuals

i

∈

N

impliesF

(

E

)

−1

(y)

=

F(E

)

−1

(y)

.

Both IIS andIIT are strictly weaker than IIE. That is, we obtain the following result, which is easy to verify(simple counterexamplescanbedevisedtoshowthattheconversedoesnothold):

Proposition 1. IfanaggregationruleisIIE,thenitisalsobothIISandIIT.

The fundamental economic principle of unanimity requires that an edge should be accepted by a group in case all individualsinthatgroupacceptit.

(9)

Deﬁnition 13. AnaggregationruleF iscalled unanimous ifitisalwaysthecasethat F(E

)

⊇

E1

∩ · · · ∩

En.

Arequirementthat, insome sense, isdual tounanimity isto askthat the collectivegraphshould onlyinclude edges thatarepartofatleastoneoftheindividualgraphs.Inthecontextofontologyaggregationthisaxiomhasbeenintroduced underthenamegroundedness[47].

Deﬁnition 14. Anaggregation F iscalled grounded ifitisalwaysthecasethatF

(

E

)

⊆

E1

∪ · · · ∪

En.

Thenext axiomexpresses abasicsymmetryrequirement, namelythattheaggregationruleshould treatallindividuals thesame.Itisusedintheexactsameforminbothpreferenceandjudgmentaggregation[5,17].

Deﬁnition 15. Anaggregation rule F is called anonymous if F(E1,

. . . ,

En

)

=

F

(

Eπ(1)

,

. . . ,

Eπ(n)

)

foranypermutation

π

:

N → N

.

Theaxiomofneutrality,looselyspeaking,postulatessymmetrywithrespecttodifferentpartsofthegraphstobe aggre-gated.Wearegoingtomostly workwiththefollowingformalisationofthisintuitiveidea,whichisinspiredbythewayin whichneutralityisoftendeﬁnedinjudgmentaggregation[48].

Deﬁnition 16. AnaggregationruleF iscalled neutral ifNeE

=

NeE impliese

∈

F

(

E

)

⇔

e

∈

F

(

E

)

.

Thus,thisaxiom saysthat, iftwo edges areacceptedby thesamecoalitionofindividuals, theneitherboth orneither should be included in the collective graph. (Observe that, while IIE speaks about one edge and two profiles, neutrality speaksabouttwoedges withinthe sameprofile.)Whenwe restrictattentiontographs thatcanbe interpretedas prefer-enceorders,e.g., weakorders,thisnotionofneutrality,however,isdifferentfromhowneutralityisusuallydefinedinthe preference aggregationliterature [3],where it is takento representsymmetry with respectto alternatives (i.e.,vertices) ratherthanpairwisepreferences(i.e.,edges).Thefollowingalternativedefinitiongeneralisesthisideatoarbitrarygraphs.It isformulatedintermsofapermutation

π

:

V

→

V onvertices.Anysuch

π

naturallyextendstoedges e

= (

x,y),graphs E, andproﬁles E:

π

((x,

y))

= (

π

(x),

π

(y))

,

π

(E

)

= {

π

(e)

|

e

∈

E

}

,and

π

(

E

)

= (

π

(E1),

. . . ,

π

(E

n

))

.

Deﬁnition 17. AnaggregationruleF iscalled permutation-neutral ifF

(

π

(

E

))

=

π

(F

(

E

))

foranypermutation

π

:

V

→

V .

The following two examplesshow that thereare neutral aggregationrules that are not permutation-neutral andthat therearepermutation-neutralaggregationrulesthatarenotneutral. However,asweshallseenext,inthepresenceofIIE, thetwodeﬁnitionshavethesamelogicalstrength.

Example 9 (Neutralyetnotpermutation-neutralrule). Let V

= {

x,y

}

and consider the aggregationrule F that returns the empty graph

∅

incaseagent 1 acceptsedge

(x,

y)andthat returns thecompletegraph

{(

x,x),

(x,

y),

(y,

x),

(y,

y)

}

inall other cases. This rule is easily seen to be neutral, as the output graph always agrees on all edges. However, F is not permutation-neutral:ifwe swapx and y inaproﬁlewhereagent 1acceptsonly

(x,

y),thentheoutputwillchangefrom theemptytothecompletegraph.

Example 10 (Permutation-neutralyetnotneutralrule).Let V

= {

x,y,z

}

andconsider theaggregationrule F thatﬁrst com-putestheintersectionofallindividual graphsandthen,incertainspecialcases,removesone furtheredge:namely,ifthe intersectiongraph happenstobe exactly

π

(

{(

x,y),

(y,

z)

})

,forsome permutation

π

:

V

→

V ,then the edge

π

((y,

z))is removed.Inotherwords:iftheintersectiongraphisa“line”oflength 2,thenthesecondhalfofthatlineisremoved.This ruleispermutation-neutralbydeﬁnition.However,itisnotneutral.Forinstance,ifallagentsacceptboth

(x,

y)and

(y,

z), andnootheredges,thenthesetwoedgesneverthelessarenottreatedsymmetricallyintheoutput.

Proposition 2. LetF beanaggregationrulethatisIIE.ThenF isneutralifandonlyifitispermutation-neutral.

Proof. It suﬃces toobservethat both(i) aggregation rulesthat areIIE andneutraland(ii) aggregationrules that areIIE andpermutation-neutralhavethefollowingpropertyincommon. Anysuchrulecanbecompletelydescribedbyspecifying whichcoalitions C ofagentsaresuchthatitisthecasethatagivenedgewillgetacceptedbytheruleifandonlyifexactly theagentsinC acceptit.3

2

Thefollowingmonotonicityaxiomexpressesthatadditionalsupportforacollectivelyacceptededgeshouldnevercause thatedgetoberejected.Itappliesincaseproﬁles E and E areidentical,exceptthatsome individualswhodonotaccept

(10)

edge e in the former proﬁle do accept it in thelatter. Its deﬁnitionis closelymodelled on its counterpart in judgment aggregation[17].

Deﬁnition 18. Anaggregationrule F iscalled monotonic ifeitherE_i

=

Eior Ei

=

Ei

∪ {

e

}

holdingforallindividualsi

∈

N

impliese

∈

F(E

)

⇒

e

∈

F

(

E

)

.

The linkbetweenaggregationrulesandaxiomaticpropertiesisexpressedinso-calledcharacterisationresults.Foreach rule (orclassofrules), theaimisto ﬁndasetofaxiomsthatuniquely deﬁnethisrule(or classofrules,respectively). A simpleadaptationofaresultbyDietrichandList[13]yieldsthefollowingcharacterisationoftheclassofquotarules:

Proposition 3. Anaggregationruleisaquotaruleifandonlyifitisanonymous,monotonic,and IIE.

Proof. To provetheleft-to-rightdirectionwesimplyhavetoverifythatthequotarulesallhavethesethreeproperties.For the right-to-left direction,observe that,to accept a givenedge

(x,

y) inthe collectivegraph,an IIE aggregationrule will only lookattheset ofindividuals i such that xEiy.Iftherule isalsoanonymous,then theacceptancedecisionis based

only on the number ofindividuals accepting the edge.Finally, by monotonicity, there will be some minimal number of individual acceptancesrequiredtotriggercollectiveacceptance.Thatnumberisthequotaassociatedwiththe edgeunder consideration.

2

Ifweaddtheaxiomofneutrality,thenweobtaintheclassofuniformquotarules.Ifwefurthermoreimposeunanimity andgroundedness,then thisexcludesthetrivialquotarules.Similarly, itiseasytoverifythat IISessentially characterises theclassofsuccessor-approvalrules:

Proposition 4. Anaggregationruleisasuccessor-approvalrule(withananonymousandneutralchoicefunction)ifandonlyifitis anonymous,neutral,andIIS.

Anextremeformofviolatinganonymityistouseadictatorial oran oligarchic aggregationrule,i.e., arulethatiseither adictatorshiporanoligarchy(unlesstheoligarchyinquestionisthefullset

N

).

Sometimes weare onlygoing tobeinterested inthepropertiesofan aggregationruleasfarasthenonreﬂexive edges e

= (

x,y)withx

=

y areconcerned.Speciﬁcally,wecallF neutralonnonreﬂexiveedges (orjustNR-neutral)ifN₍E_x_,_y₎

=

N₍E_x,y) implies

(x,

y)

∈

F

(

E

)

⇔ (

x

,

y

)

∈

F

(

E

)

forall x

=

y andx

=

y.Analogously, wecall F dictatorialonnonreﬂexiveedges (or NR-dictatorial) if there exists an individual i

_∈

_N

_such _that

_(x,

_y)

_∈

_F

₍

_E

₎

_{⇔ (}

_x,_y)

_∈

_E

i forall x

=

y.Finally, we call F

oligarchiconnonreﬂexiveedges (or NR-oligarchic) ifthere exists a nonempty coalition C

_⊆

_N

_such _that

_(x,

_y)

_∈

_F(_E

₎

_⇔

(x,

y)

∈

_i_∈_CEi forallx

=

y.

2.5. Collectiverationalityandbasicpossibilityresults

Towhatextentcanagivenaggregationruleensurethatagivenpropertythatissatisfiedbyeachoftheindividualinput graphs willbe preservedduring aggregation?Thisquestionrelatestoawell-studiedconceptinsocialchoicetheory,often referred toascollectiverationality [5,11].Intheliterature,collectiverationalityisusually definedwithrespecttoaspecific propertythatshouldbepreserved(e.g.,thetransitivityofpreferencesorthelogicalconsistencyofjudgments).Here,instead, weformulateadefinitionthatisparametricwithrespecttoagivengraphproperty.4

Deﬁnition 19. Anaggregationrule F iscalled collectively

rational with

respectto agraphproperty P if F(E

)

satisﬁes P

wheneveralloftheindividualgraphsinE

= (

E1,

. . . ,

En

)

do.

Toillustrate theconcept,let usconsider twoexamples.Both concernthemajorityrule,butdifferentgraphproperties. Theﬁrstisapurelyabstractexample,whilethesecondhasanaturalinterpretationofgraphsaspreferencerelations.

Example 11 (Collectiverationality).SupposethreeindividualsprovideuswiththreegraphsoverthesamesetV

= {

x,y,z,w

}

offourvertices,asshowntotheleftofthedashedlinebelow:

4 _In_previous_work_on_binary_aggregation,_a_variant_of_judgment_aggregation,_we_have_used_the_term_collective_rationality_in_the_same_sense,_with_the

(11)

x y z w x y z w x y z w x y z w

If we apply the majorityrule, then we obtain the graph to the rightof the dashed line. Thus, the majority rule is not collectivelyrationalwithrespecttoseriality,aseachindividualgraphisserial,butthecollectivegraphisnot.Symmetry,on theotherhand,ispreservedinthisexample.

Example 12 (Condorcetparadox). Now suppose threeindividuals provide uswith thethree graphs on the set ofvertices

V

= {

x,y,z

}

shownonthelefthandsideofthedashedlinebelow:

x y z x y z x y z x y z

Thegraphontherighthandsideisonceagaintheresultofapplyingthemajorityrule.Observethateachofthethreeinput graphsistransitiveandcomplete.Sowemayinterpretthesegraphsas(strict)preferenceordersonthecandidatesx, y,and

z. Forexample,thepreferencesoftheﬁrstagentwouldbex

y

z.Theoutputgraph,ontheotherhand,isnottransitive (althoughitiscomplete).Itdoesnotcorrespondtoa“rational”preference,asunderthatpreferenceweshouldprefer x to y and y toz,butalsoz tox.ThisisthefamousCondorcetparadoxdescribedbytheMarquisdeCondorcetin 1785[49].

Sothemajorityruleisnotcollectivelyrationalwithrespecttoeitherserialityortransitivity.Ontheotherhand,wesaw thatbothsymmetryandcompletenesswerepreservedunderthemajorityrule—atleastforthespeciﬁcexamplesconsidered here.Infact,itisnotdiﬃculttoverifythatthiswasnocoincidence,andthatthemajorityruleiscollectivelyrationalwith respecttoanumberofpropertiesofinterest.

Fact 5. Themajorityruleiscollectivelyrationalwithrespecttoreﬂexivity,irreﬂexivity,symmetry,andantisymmetry.Incase

n,thenumberofindividuals,isodd,themajorityrulefurthermoreiscollectivelyrationalwithrespecttocompletenessand connectedness.

Proof sketch. We givetheproofsforsymmetryandcompleteness.Theotherproofsareverysimilar.First,iftheinputgraphs aresymmetric,thenthesetofsupportersofedge

(x,

y)isalwaysidenticaltothesetofsupportersoftheedge

(

y,x).Thus, eitherbothorneitherhaveastrictmajority.Second,iftheinputgraphsare complete,then eachofthemmustincludeat leastoneof

(x,

y)and

(y,

x).Thus,bythepigeonholeprinciple,whenn isodd,atleastoneofthesetwoedgesmusthave astrictmajority.

2

Ratherthanestablishingfurthersuchresultsforspeciﬁcaggregationrules,ourmaininterestinthispaperisthe connec-tionbetweentheaxiomssatisﬁedbyanaggregationruleandtherangeofgraphpropertiespreservedbythesamerule.For somegraphproperties,collectiverationalityiseasytoachieve,asthefollowingsimplepossibilityresults demonstrate.

Proposition 6. Anyunanimousaggregationruleiscollectivelyrationalwithrespecttoreﬂexivity.

Proof. If every individual graphincludesall edges ofthe form

(x,

x), thenunanimity ensures thesame forthecollective graph.

2

Proposition 7. Anygroundedaggregationruleiscollectivelyrationalwithrespecttoirreﬂexivity.

Proof. If noindividualgraphincludes

(x,

x),thengroundednessensuresthesameforthecollectivegraph.

2

Proposition 8. Anyneutralaggregationruleiscollectivelyrationalwithrespecttosymmetry.

Proof. If edges

(x,

y)and

(y,

x)havethesamesupport,thenneutralityensuresthateitherbothorneitherwillgetaccepted forthecollectivegraph.

2

(12)

Unfortunately, as we are going to see next, things do not always work out that harmoniously, and certain axiomatic requirementsareinconﬂictwithcertaincollectiverationalityrequirements.

3. Impossibility results

Insocial choicetheory,an impossibilitytheorem statesthatitisnot possibletodeviseanaggregationrulethat satisfies certain axioms andthat isalso collectivelyrationalwith respectto a certain combinationofpropertiesof thestructures beingaggregated(whichinourcasearegraphs).Inthissection,wearegoingtoprovetwopowerfulimpossibilitytheorems forgraphaggregation,theOligarchyTheorem andtheDictatorshipTheorem.Thelatteridentifiesasetofrequirementsthatare impossibletosatisfy inthesensethat theonlyaggregationrulesthat meetthemarethedictatorships. Theformerdrives on somewhat weaker requirements(specifically,regarding collectiverationality)andpermits asomewhat larger—but still decidedlyunattractive—setofaggregationrules,namelytheoligarchies.

Ourresultsareinspiredby—andsigniﬁcantlygeneralise—theseminalimpossibilityresultforpreferenceaggregationdue to Arrow, ﬁrst published in 1951 [5]. We recall Arrow’s Theorem in Section 3.1. The following subsections are devoted to developing the framework in which to present and then prove our results. Section 3.2 introduces winningcoalitions,

i.e., setsofindividualswho canforce theacceptanceorrejectionofa givenedge,discussesunderwhatcircumstances an aggregationrulecanbedescribedintermsofafamilyofwinningcoalitions,andwhatstructuralpropertiesofsuchafamily correspondtoeitherdictatorialoroligarchicaggregationrules.Sections3.3and3.4introducethreeso-calledmeta-properties

for classifyinggraph properties andestablish fundamental results forthese meta-properties. Ourimpossibility theorems, which are formulated andproved in Section 3.5,apply to aggregation rules that are collectivelyrational withrespect to graph propertiesthat arecovered by some ofthesemeta-properties. Section 3.6,ﬁnally,discusses severalvariants ofour theoremsandprovidesaﬁrstillustrationoftheiruse.

3.1. Background:Arrow’sTheoremforpreferenceaggregation

The primeexampleofan impossibility resultisArrow’sTheoremforpreferenceaggregation,withpreferencerelations beingmodelledasweakordersonsomesetofalternatives[5].WecanreformulateArrow’sTheoreminourframeworkfor graphaggregationasfollows:

For

|

V

|

≥

3,everyunanimous,grounded,andIIEaggregationrulethatiscollectivelyrationalwithrespecttoreﬂexivity,transitivity, andcompletenessmustbeadictatorship.

Thus,Arrow’sTheoremappliestothefollowingscenario.Wewishtoaggregatethepreferencesofseveralagentsregardinga setofthreeormorealternatives.Theagentsareassumedtoexpresstheirpreferencesbyrankingthealternativesfrombest toworst(withindifferencesbeingallowed),i.e.,byeachprovidinguswithaweakorder(agraphthatisreﬂexive,transitive, andcomplete),andwewantouraggregationruletocomputeasinglesuchweakorderrepresentingasuitablecompromise. Furthermore,wewantouraggregationruletorespectthebasicaxiomsofunanimity(ifallagentsagreethatx isatleastas goodas y,thenthecollectivepreferenceordershouldsayso),groundedness(ifnoagentsaysthatx isatleastasgoodas

y,thenthecollectivepreferenceordershouldnotsaysoeither),andIIE(itshouldbepossibletocomputetheoutcomeon an edge-by-edgebasis).Arrow’sTheoremtellsusthatthisisimpossible—unlesswearewillingtouseadictatorshipasour aggregationrule.

Thisresultnotonlyissurprisingbutalsodeeplytroubling.Itthereforeisimportanttounderstandtowhatextentsimilar phenomenaariseinotherareasofgraphaggregation.WearegoingtorevisitArrow’sTheoreminSection3.6,whereweare alsogoingtobeinapositiontoexplainwhythestandardformulationofthetheorem,giveninthatsectionasTheorem 19, isindeedimpliedbythevariantgivenhere.

In thesequel,weare sometimesgoing torefertoaggregationrules thatare unanimous,grounded,andIIEasArrovian

aggregationrules.

3.2. Winningcoalitions,ﬁlters,andultraﬁlters

Asiswellunderstoodinsocialchoicetheory,impossibilitytheoremsinpreferenceaggregationheavilyfeedon indepen-denceaxioms(inourcaseIIE).Observethatanaggregationrule F satisﬁesIIEifandonlyifforeachedgee

∈

V

×

V there

exists a setofwinningcoalitions

W

e

⊆

2N such thate

∈

F

(

E

)

⇔

NeE

∈

W

e.Thatis, F accepts e ifandonly ifexactly the

individualsinone ofthewinningcoalitionsfore do. Imposingadditionalaxiomson F corresponds torestrictions onthe associatedfamilyofwinningcoalitions

{

W

e}e_∈V×V:

•

If F isunanimous,then

N ∈ W

e foranyedge e (i.e.,thegrandcoalitionisalwaysawinningcoalition).

•

If F isgrounded,then

∅

∈

/

W

e foranyedge e (i.e.,theemptysetisnotawinningcoalition).

•

If F ismonotonic,thenC1

∈

W

e impliesC2

∈

W

e foranyedge e andanysetC2

⊃

C1(i.e.,winningcoalitionsareclosed

(13)

•

If F is(NR-)neutral,then

W

e

=

W

e foranytwo(nonreﬂexive) edgese and e (i.e.,every edgemusthaveexactlythe

samesetofwinningcoalitions).

Thus,anaggregationrulethatisbothIIEandneutralcanbefullydescribedintermsofasingleset

W

ofwinningcoalitions. Any such

W

isasubset ofthepowersetof

N

,thesetofindividuals. Theproofsofourimpossibility resultsaregoing to exploit the special structure of such subsets of the powerset of

N

, enforced by both axioms and collective rationality requirements.Specifically,inourproofswearegoingtoencountertheconceptsoffilters andultrafilters familiarfrommodel theory[50].

Deﬁnition 20. A ﬁlter

W

onaset

N

isacollectionofsubsetsof

N

satisfyingthefollowingthreeconditions: (i)

∅

∈

/

W

;

(ii) C1,C2

∈

W

impliesC1

∩

C2

∈

W

foranytwosetsC1,C2

⊆

N

(closureunderintersection);

(iii) C1

∈

W

impliesC2

∈

W

foranysetC2

⊆

N

withC2

⊃

C1(closureundersupersets).

Deﬁnition 21. An ultraﬁlter

W

onaset

N

isacollectionofsubsetsof

N

satisfyingthefollowingthreeconditions: (i)

∅

∈

/

W

(ii) C1,C2

∈

W

impliesC1

∩

C2

∈

W

foranytwosetsC1,C2

⊆

N

(closureunderintersection); (iii) C or

N \

C isin

W

foranysetC

⊆

N

(maximality).

Everyultrafilterisafilter;inparticular,theultrafilterconditionsimplyclosureundersupersets.Notethatthecondition

∅

∈

/

W

directlycorrespondstogroundedness,whileclosureundersupersetscorrespondstomonotonicity.

The useofultrafiltersinsocial choicetheory goesback tothe workofFishburn [6]andKirmanandSondermann [7], whoemployedultrafilterstoproveArrow’sTheoremanditsgeneralisationtoaninfinitenumberofindividuals. The ultra-filtermethodalsohasfoundapplicationsinjudgmentaggregation[15],andalsofiltershavebeenusedinboth preference aggregation[8]andjudgmentaggregation[12].Therelevanceoffiltersandultrafilterstoaggregationproblemsisduetothe followingsimpleresults,whichinterpretwell-knownfactsfrommodeltheoryinourspecificcontext.

Lemma 9 (FilterLemma).LetF beanIIEandNR-neutralaggregationruleandlet

W

bethecorrespondingsetofwinningcoalitions fornonreﬂexiveedges,i.e.,

(x,

y)

∈

F(E

)

⇔

N₍E_x_,_y₎

∈

W

forallx

=

y

∈

V .ThenF isNR-oligarchicifandonlyif

W

isaﬁlter.

Proof.

(

⇒)

Recallthat F beingNR-oligarchicmeansthatthereexistsanonemptycoalitionCsuchthatagivennonreﬂexive edgeisacceptedifandonlyifalltheagentsin C_accept_it._Thus,_the_winning_coalitions_are_exactly_C _and_its_supersets. Thisfamilyofsetsdoesnotincludetheemptysetandisclosedunderbothintersectionandsupersets.

(

⇐)

Suppose F isdetermined bythe ﬁlter

W

asfarasnonreﬂexiveedges are concerned.Let C

_:=

C∈CC ,whichis

well-deﬁneddueto

N

beingfinite.ObservethatC_must_be_nonempty,_due_to_the_first_two_filter_conditions._Now_note_that

F isNR-oligarchicwithrespecttocoalition C_.

₂

Lemma 10 (UltraﬁlterLemma).LetF beanIIEandNR-neutralaggregationruleandlet

W

bethecorrespondingsetofwinning coalitionsfornonreﬂexiveedges,i.e.,

(x,

y)

∈

F(E

)

⇔

N₍E_x_,_y₎

∈

W

forallx

=

y

∈

V .ThenF isNR-dictatorialifandonlyif

W

isan ultraﬁlter.

Proof.

(

⇒)

F being NR-dictatorial means that there exists an i

_∈

_N

_such _that _the _winning _coalitions _for _{nonreﬂexive} edgesareexactly

{

i

_}

_and_its_supersets._This_family_of_sets_does_not_include_the_empty_set,_is_closed_under_{intersection,}_and maximal.

(

⇐)

Suppose F isdeterminedbytheultraﬁlter

W

asfarasnonreﬂexiveedges areconcerned.Takean arbitraryC

∈

W

with

|

C

|

≥

2 and considerany nonemptyC

C . Bymaximality, one of C and

N \

C must be in

W

. Thus, by closure underintersection, oneof C

∩

C

=

C and C

∩ (

N \

C

)

=

C

\

C mustbe in

W

as well.Observe that bothof thesesets arenonemptyandoflowercardinalitythan C .Tosummarise,wehavejustshownforanyC

∈

W

with

|

C

|

≥

2 atleastone nonemptypropersubsetofC isalsoin

W

.Bymaximality,

W

isnotempty.SotakeanyC

∈

W

.Dueto

N

beingﬁnite,we canapplyourreductionruleaﬁnitenumberoftimestoinferthat

W

mustincludesomesingleton

{

i

_}

_{· · ·}

_{C .}_Hence, _F isanNR-dictatorshipwithdictator i_.

₂

3.3. Theneutralityaxiomandcontagiousgraphproperties

Recallthat theneutrality axiom is requiredto beable to workwitha single familyof winningcoalitions asoutlined earlier,yetthisaxiom doesnotfeature inArrow’sTheorem.Aswe shallseesoon, thereasonwedo notneed toassume neutralityisthat,inArrow’ssetting,thesamerestrictiononwinningcoalitionsisalreadyenforcedbycollectiverationality withrespect totransitivity.Thisis an interesting linkbetweena speciﬁc collectiverationality requirementanda speciﬁc