Maintaining the duality of closeness and betweenness centrality

Maintaining the duality of closeness and betweenness centrality ^夽

Ulrik Brandes

^a,∗

, Stephen P. Borgatti

^b

, Linton C. Freeman

^c

aDepartmentofComputer&InformationScience,UniversityofKonstanz,Germany

bGattonCollegeofBusinessandEconomics,UniversityofKentucky,UnitedStates

cInstituteofMathematicalBehavioralSciences,UniversityofCaliforniaatIrvine,UnitedStates

Keywords:

Betweennesscentrality Closenesscentrality Duality

Dependency Derivedrelations

a b s t r a c t

Betweennesscentralityisgenerallyregardedasameasureofothers’dependenceonagivennode,and thereforeasameasureofpotentialcontrol.Closenesscentralityisusuallyinterpretedeitherasamea- sureofaccessefficiencyorofindependencefrompotentialcontrolbyintermediaries.Betweennessand closenessarecommonlyassumedtoberelatedfortworeasons:first,becauseoftheirconceptualduality withrespecttodependency,andsecond,becausebotharedefinedintermsofshortestpaths.

Weshowthatthefirstoftheseideas–theduality–isnotonlytrueinageneralconceptualsensebut alsoinprecisemathematicalterms.Thisbecomesapparentwhenthetwoindicesareexpressedinterms ofashareddyadicdependencyrelation.Wealsoshowthatthesecondidea–theshortestpaths–isfalse becauseitisnotpreservedwhentheindicesaregeneralizedusingthestandarddefinitionofshortest pathsinvaluedgraphs.Thisunveilsthatcloseness-as-independenceisinfactdifferentfromcloseness- as-efficiency,andweproposeavariantnotionofdistancethatmaintainsthedualityofcloseness-as- independencewithbetweennessalsoonvaluedrelations.

1. Introduction

A number of attempts have been made to bring order to theuniverseof centralitymeasures,includingSabidussi(1966), Koschützkietal.(2005),andBorgattiandEverett(2006).Byfarthe mostinfluentialofthesehasbeenFreeman(1979).Sincethepubli- cationofthatpaper,degree,closenessandbetweennesscentrality havebeenregardedasprototypicalmeasuresthatcapturemost importantaspectsofcentrality.Theonlyothermeasureaswell- knownastheseiseigenvectorcentrality(Bonacich,1972),along withitsvariants(Bonacich,1987;BrinandPage,1998).

Inthispaper,wefocusonclosenessandbetweenness,which arebasedonanunderlyingconceptofsomethingflowingthrough anetworkalongoptimalpaths.Consistentwiththeimageryusedin Freeman’sseminalpaper,weassumethetiesinournetworkscanbe viewedascommunicationchannels,althoughitshouldbeclearthat ourresultsapplytoanykindofnetworkforwhichflows,geodesics, closeness,andbetweennesshavemeaningfulinterpretations.

Betweennessisgenerallyemployedwiththeunderstandingthat itcaptures thepotentialfor controlofcommunicationbetween

夽ResearchpartiallysupportedbyDFGundergrantBr2158/6-1.

∗Correspondingauthor.Tel.:+497531884433;fax:+497531883577.

E-mailaddresses:Ulrik.Brandes@uni-konstanz.de(U.Brandes), sborgatti@uky.edu(S.P.Borgatti),lin@aris.ss.uci.edu(L.C.Freeman).

actors.For closeness,Freeman(1979)actually outlinestwo dif- ferentpossibleinterpretations:eitherasindependencefromsuch controlbyothers(closenessasindependence)orasameasureof accessorefficiency(closenessasefficiency).Herewefocusonthe interpretationofindependenceasitisreferredtoinmanyempir- icalstudiessuchasBrass(1984),Rowley(1997),andPowelletal.

(1996).

Freeman(1980)showsthattheinterpretivedualityofclose- nessandbetweennessasmeasuresofindependenceandcontrolis quantitativelyjustified.Ithasbeenwidelyoverlooked,though,that thisjustificationisestablishedviaasharedunderlyingdependency relation.Instead,itisoftenstatedthatthemeasuresarerelated becausebotharedefinedintermsofgeodesics.Wewillarguethat thisviewisrathermisleading,andthatcloseness-as-independence andcloseness-as-efficiencyareactuallytwodifferentconceptsthat happentoagreeonnon-valuednetworks.Thecommongeneraliza- tionofclosenesstovaluednetworksisinlinewiththeefficiency interpretation only. We therefore propose new generalizations ofclosenesstodirected,disconnected,andvaluednetworksthat maintainthe independenceinterpretation and thus theduality with(commongeneralizationsof)betweenness.

We start bydefining necessaryterminology and introducing thebasicconceptof adependencycubein Section 2.Therela- tionsbetweendependenciesandthedualindicesofclosenessand betweennessarederivedinSection3,leadingtoourre-definitionof closeness-as-independenceinSection 2.1.InSections5and6,we

Konstanzer Online-Publikations-System (KOPS) URL: http://nbn-resolving.de/urn:nbn:de:bsz:352-0-311031 Erschienen in: Social Networks ; 44 (2016). - S. 153-159

https://dx.doi.org/10.1016/j.socnet.2015.08.003

showhowthisgeneralizestodirectedandvaluednetworkswhile maintainingthedualitywithbetweenness.WeconcludeinSection 7.

2. Preliminaries

Weassumethatnetworksarerepresentedasgraphsanduse standardterminologysuchasfoundinBollobás(1998)orDiestel (2010).

An(undirected) graphG=(V,E)consistsofa setVofvertices (alsocallednodes)representingactorsandasetE⊆

V

2

of(undi- rected)edges(alsocalledlinks)representingtiesbetweenactors.

Anedgeisthusanunorderedpairofverticesrepresentingasym- metricrelationship.Ifthereexistsanedgee={u,v_}∈E,wesaythat uandVareadjacentandthatuandVareincidenttoe.Wewilluse n=|V|forthenumberofverticesandm=|E|forthenumberofedges ofagraph.

Apathfromasenders∈Vtoareceiverr∈V,or(s,r)-pathforshort, isanalternatingsequenceofverticesandedgesthatstartswiths, endswithr,andinwhicheveryvertexisincidenttoboththeedges thatcomebeforeandafteritinthesequence.Agraphisconnected, ifeverypairofverticesislinkedbyapath.

Inthis and the followingsection, allgraphs areassumed to beundirectedandconnected.Thedefinitionswillbeextendedto directedandvalued graphsinSections 5and6,where wealso considerdisconnectedgraphs.

2.1. Distanceandclosenesscentrality

Closenesscentrality,asthenamesuggests,isanindexdefinedin termsofadistance.Letthelengthofan(s,r)-pathbethenumberof edgescontainedinit.Wedefinethe(shortest−path)distance,dist(s, r),ofs,r∈Vastheminimumlengthofany(s,r)-path.Recallthat weconsideronlyconnectedgraphsfornowandobservethatdist(s, s)=0foralls∈V.

ThedistancematrixD=(dist(s,r))_s,r∈Vofanundirectedgraphis symmetric,sothatthetotaldistance,dist(v),ofavertexv_∈V is obtainedaseithertherowandcolumnsums

dist(v)=

r∈V

dist(v,r)=

s∈V

dist(s,v).

Thelargertheassociateddistancesum,thefartheravertexisfrom theothers,whichiswhyavertexisconsideredmorecentral,in terms ofcloseness, if itsassociated value is smaller(Sabidussi, 1966).

Becauseofthisreversalinranking,closenesscentralityofavertex s∈Visusuallydefinedastheinverseofthetotal(or,equivalently, average)distance(Bavelas,1950;Beauchamp,1965),

cC(s)=

r∈V

dist(s,r)

−1

= dist(s)⁻¹,

butsometimesalsobysubtractionfromanupperboundonthe maximumdistance(ValenteandForeman,1998).

2.2. Dependencyandbetweennesscentrality

Betweenness centrality is based on the idea that brokering positionsbetweenothersprovidetheopportunitytointerceptor influencetheircommunication.Again,theassumptionisthatcom- municationishappeningalongshortestpaths.

Denoteby(s,r)thenumberofshortest(s,r)-paths,andlet(s, r|b)bethenumberofshortest(s,r)-pathspassingthroughsome brokeringvertexb∈V\{s,r}.Forconsistency,let(s,s)=1,and(s,

r|b)=0ifb∈{s,r}.Ifallshortestpathsareequallylikelytobechosen, theratioı(s,b,r)=^(s,r|b)_(s,r) givestheprobabilitythatbisinvolved intheindirectcommunicationofswithr.Thetermı(s,b,r)iswell- definedbecause(s,r)>0(fornow,weassumeconnectedgraphs) andreferredtoasthedependencyofasendersandareceiverron abrokerb.Fromthebroker’sperspectiveitrepresentsthedegree ofcontrolthatbhasoverthecommunicationfromstor.

Betweennesscentralityisdefinedasthetotaldependencyofcom- municatingpairsonabrokerb∈V,

cB(b)=

s,r∈V

ı(s,b,r),

andthuscorrespondstob’soverallpotentialforcontrol.

Inthenextsectionwerecallandextendalargelyunknownresult ofFreeman(1980)showingthatthedependenciesgiverisetoa dyadicrelationthatrelatesclosenessandbetweennessquantita- tively.

3. Dyadicdependenciesandduality

Thedependenciesdefinedaboveformathree-waytensor,i.e.,a generalizedmatrix=(ı(s,b,r))_s,b,r∈V,thedependencycube.Ithas firstbeenconsideredexplicitlybyBorgattiandBonacich(1989), whoreferredtoitasthegeodesiccube.Thecubeassumestherole ofarepositoryofelementaryinformationaboutallcommunication triplesconsistingofasender,areceiver,andapotentialbrokerin between.Ifalln³entriesarerequired,astraightforwardalgorithm ofBatagelj(1994)canusedtodeterminethemintimeO(n³).

Theabovedefinitionofbetweennesscorrespondstoasumma- tionoverthe(s,r)-planeinthedependencycube,andanumberof otherinterestingquantitiesandinsightscanbeobtainedbysum- mingoverothersubsetsofelementsof.Thesearedetailednext andsummarizedinFig.1.

Firstobservethatanysummationofdependenciesı(s,b,r)over eitherthesenders,brokers,orreceiversyieldsavalued,asymmet- ricanddyadicrelation.Itrelateseitherbrokersandreceivers,or sendersandreceivers,orsendersandbrokersinasquarematrix andthusdefinesavaluednetwork.

Consider,forexample,thedependenciesı(s,b,·)ofsenderss onbrokersbobtainedfromsummationoverallreceivers.These canbeinterpretedasquantifyinghowlikelyitisthatbisinvolved inacommunicationoriginatingatsanddirectedatanyr,i.e.,to whichextentsdependsonbinsendingtotherestofthenetwork bytheefficientpaths.Theseone-sideddependencies¹thusforma newasymmetricandvaluedrelationbetweensendersandbrokers derivedfromtheoriginaladjacencyrelation.Since

cB(b)=

s,r∈V

ı(s,b,r)=

s∈V

ı(s,b,·),

betweennesscentralitycanalsobeinterpretedasindegreeinthe derivednetwork. Itthusquantifiestheextenttowhichsenders dependonb.It isinterestingtonotethat,for agivensenders, one-sideddependenciesı(s,b,·)canbecomputedbyaccumulating dependenciesonbrokersfartherawayfroms,sothatitiscompu- tationallymoreefficienttodeterminethemdirectlyratherthanby explicitlydeterminingallentriesofandsubsequentsummation (Brandes,2001).

Similarly, marginals ı(·, b, r) can be interpreted as the dependenciesofreceiversrongatekeepersbtoletincominginfor- mationthrough.Bysymmetry,betweennessintheoriginalgraph

1Freeman(1980)usesthetermpair-dependencieswhichweavoidasitisprone tomisinterpretationinourmoregeneralcontext.

/

,,) ...

... ..

/ ,, .. ,'

(a) 6(s,b,r)

=

a((s,rlb))

u s.r (b) 6(s, ·, r) = dist(s,

r) -1

(c) 6(s, b, ·)

(d)

6(·, b, r)

(e) 6(s, ·, ·) = dc(s) (f)

o(·, ·,

r)

=

c(:(r)

(g) o(-, b, ·)

=

c8(b) (h)

o(-, ·, ·)

= W(G)- R(G) Fig. 1. Marginals of the dependency cube.

corresponds to outdegree in the graph defined by the valued rela- tion 8(- , b, r).

A key observation is that the third matrix of marginals 8(s, ·.

r), the dependency of each pairs, ron the rest of the network, is almost identical to the matrix of shortest-path distances (recall that we are considering non-valued networks for now). This was already observed in F

reeman (1980), Borgatti and Bonacich (1989),

and independently in

Buche I (2009, Lemma 4.5.1 ). We include a

straightforward proof illustrated in F

ig. 2.

lemma 1.

In a connected grapl!, 8(s, ·•

r) ^Q

dist(s.

r)- 1

for all

s =/=rEV.

Proof.

At every distance i Q 1,

... ,

dist(s, r)

- 1

from s, each shortest path from s tor passes through exactly one broker b. so that for s

⁼¹⁼

r

1/4 1/4

' 2/4

~ ---·

...

K . ·--- _I

_{1/4 •} ^I _{__...} ^2/4

^{I " '} ~·---<cv

^4/4

: . . ...

2/4

I

^{2/4 1/4}

^I I

Fig. 2. The fractions "~;/~¹ of shortest (s. r)-paths passing through those vertices b that have the same distance from s add up to 1, so that 8(s. ·, r)= l:•ev8(s, b.

r)=dist(s. r)-1.

we have

8(s, ·, r)

= L::&(s, b, r)

beV

d!sr(s,r)-1

2::::

1=1

o-(s, rib) _ d' t( ) _1

( ) - zs s, r .

o- s, r

beV: dtsr(s,b )=f

=1 D

This observation leads to two interesting insights: the quan-

titative duality of closeness and betweenness, and the reason for the alternative interpretations of closeness-as-efficiency and closeness-as-independence.

Closeness and betweenness centrality are dual to each other conceptually. While one quantifies the independence from control of others, the other quantifies the potential control over (com- munication between) others.

The following lemma shows that the

relation is not only conceptual but holds quantitatively.

Corollary 2.

In a connected grapl!, l:bevSCs. b, ·) -

cc{s)-1 -

(n - 1

).

Proof. Using Lemma 1, we get

l:bev8(s,

b, ·)QS(s, ·, ·)-

l:rev8(s,

·,r)Q l:revldist(s, r)

-1]Qcc(s)-¹-

(n

-1). D

Closeness and betweenness are thus dual in the sense that they are obtained as (the inverse of) row and column sums (i.e., outde- gree and indegree) oftl1e dependency relation 8(s,

b, ·),

cs(b)

= L::&(s, b, ·)

and cc(s)-

¹ = (n -1

) +

L::&(s, b, · ).

Backed by formal arguments we can therefore state that between-

ness is in exactly the same sense a measure of control, or the dependency of others on an actor, as closeness is a measure of independence, or the lack of dependency on others.

As

exempli-

fied in Fig. 3, both are directly related via the dependency relation

8(s,

b, ·) =

Erev "~(D~l, for which inverse closeness corresponds to weighted outdegree (up to a constant) and betweenness to weighted indegree.

Moreover, as demonstrated by the examples in Fig. 4, the rank-

ings obtained from these dual notions may coincide but may also be quite different from each other.

While no deep mathematics are involved,and despite an explicit

derivation in

Freeman (1980), this relationship has been largely

overlooked. We deem it important, however, because it adds strong

support for the interpretation duality that empirical researchers

have been relying on, and even more so because it has important

consequences for the generalization of closeness to unconnected,

directed, and valued networks as discussed in the subsequent sec-

tions.

Fig.3.Inconnectedundirectedgraphs,inverseclosenessandbetweennessaretherowandcolumnsumsoftheasymmetricdependencyrelationı(s,b,·)=

rı(s,b,r).

4. Interpretationandadjustmentofclosenesscentrality For the interpretation of closeness centrality we focus on totaldistances,dist(s)=c_C(s)⁻¹,becausethesolepurposeoftaking inversesistoreversetheorderofvaluesandthiscouldbeachieved inanyofanumberofways.

Aninterpretationobtaineddirectlyfromitsdefinitionisoneof efficientlyreachingothers:

“...apointiscentraltothedegreethatthedistancesassoci- atedwithallitsgeodesicsareminimum.Shortdistancesmean fewermessagetransmissions,shortertimesandlowercosts.”

(Freeman,1979,p.225)

AsdiscussedinmoredetailinSection 6,thisistheinterpretation onwhichcommongeneralizationsofclosenesstovaluednetworks arebased.

Itappears,however,thattheinterpretationofclosenessasbeing dualtothepotentialforcontrolassociatedwithbetweenness,

“...apointisviewedascentraltotheextentthatitcanavoid thecontrolpotentialofothers.”(Freeman,1979,p.224) isthemoreprominent(see,e.g.,Brass,1984;Rowley,1997;Powell etal.,1996).Sincedyadicdependenciesadduptoshortestpath

distanceminusone,theyactuallycorrespondtothenumber of intermediariesonashortestpath.Thehigherthisnumber,themore dependentanactorisonothers.Fromthepointofviewoftheinde- pendenceinterpretation,therowsumsinthematrixofı(s,b,·)thus reflecttheintendedmeaningevenbetterthanthedistances.

Itwillproveusefulbeyondtheeliminationofconstantstodefine thefollowingvariantofclosenesscentrality.

Definition3. ForagraphG=(V,E),closeness-as-independenceis definedvia

c_C(s)⁻¹=ı(s,·,·)

foralls∈V.

Clearly,whencomparedtoc_C(s),thisdoesnotaffecttheranking ofverticesinaconnectedundirectedgraphbecauseallvaluesare shiftedequallyby(n−1),

c_C(s)⁻¹=ı(s,·,·)=

b∈V

ı(s,b,·)

=dist(s)−(n−1)=c_C(s)⁻¹−(n−1).

Fig.4. Actorsthatothersdependonarenotnecessarilyindependent:darknessindicatesbetweennesswhereassizeindicatesindependence.In(b),theindependentactors inthemiddleareeasiertoavoidthantheouteroneswiththeexceptionofthependants.

Themodificationdoeshave,however,importantconsequencesfor networksof valued relations,and alsofor directed and discon- nectednetworksasshowninthenextsection.

5. Reachabilityanddisconnectednetworks

Closeness centrality is ill-defined on disconnected graphs because somedistances are undefined and it is not clear how to compare partial sums having different numbers of defined distances.Wewilladdressthisproblemtogetherwiththegeneral- izationoftheaboveresultstonetworksofasymmetricrelations.

AgraphG=(V,E)iscalleddirected,ordigraphforshort,ifits edgesaredefinedasorderedratherthanunorderedpairsofver- tices.Wewrite(v,w)∈E⊆V×Vorv→wtodistinguishdirected edgesfromundirectededges{v,w}∈

V

2

.

Apathiscalleddirected,ifeachofitsedgesisdirectedfromits precedingtoitssucceedingvertex.Ifthereexistsadirected(s,r)- path,rissaidtobereachablefroms.Reachabilityisthereflexiveand transitiveclosureofadjacencyandthusareflexiveandtransitive, butnotnecessarilysymmetric,relations→^*r.Wesaythatadigraph isstronglyconnected,ifeveryvertexisreachablefromeveryother vertex.

Thedefinition ofcloseness centrality generalizestostrongly connecteddigraphs,althoughasymmetrynowforcesustodecide whetherdistancesshouldbemeasuredfromortothefocalver- tex.Forconvenience,wewillonlyconsiderclosenesscentralityin termsofdistancesfromavertex,theothercaseissymmetric.

Fordigraphsthat arenotstrongly connected,thenumberof intermediariesanactordependsonismeaningfulonlyinrelation tothenumberofpossiblereceiversthatcanactuallybereached.

ForadigraphG=(V,E)letR(G)=|{(s,r)∈V×V:s=/ r,s→^*r}|bethe numberoforderednon-looppairsinthereachabilityrelation.For avertexv_∈V,wedefineR⁺(v)=|{r∈V : v_→^∗_r}|tobethenumber ofreachableverticesandR⁻(v)=|{s∈V : s→^∗v}|tobethenumber ofreachingvertices,sothatR(G)=

s∈VR⁺(s)=

r∈VR⁻(r).Further- more,welet

W(G)=

s,r∈V:s→^∗r

dist(s,r)

bethesumofalldefineddistances.ThissumisknownastheWiener index(Wiener,1947)andoftenusedasanetwork-levelcharacter- istic.Notethatcharacteristicpathlength(WattsandStrogatz,1998), oneofthedimensionstoassesswhetheranetworkisconsidereda smallworld,isdefinedastheaveragedistanceofanypairofver- tices,^W(G)_R(G),andthereforeyieldssimplyanormalizedversionofthe Wienerindex.

Thefollowingresultshowsthatthetotal(andthusalsoaverage) closeness-as-independenceandbetweennessareequal,andthat theycorrespondtotheWienerindexcorrectedforreachability.

Theorem4. ForadirectedgraphG=(V,E),

s∈V

c_C(s)⁻¹=W(G)−R(G)=

b∈V

c_B(b).

Proof. Again, we express all quantities in terms of three-way dependencies, and thus obtain

s∈Vc_C(s)⁻¹=

s∈V

b∈Vı(s,b,·)=

b∈V

s∈Vı(s,b,·)=

b∈VcB(b), i.e., bothtotals,forclosenessandbetweenness,areequaltothetotalof all dependencies

s,b,r∈Vı(s,b,r)=

s=/r∈V:s→^∗rdist(s,r)−1= W(G)−R(G).䊐

Closeness-as-independence and betweenness centrality are thuspartitionsofthesametotalvolume,whichissmallerformore

compactgraphs.However,theydividethisvolumeupindiffer- entwaysasdescribedintheprevioussection.Thiscommonscale mayalsobeofinterestinacomparativeanalysisofclosenessand betweenness interms of endogenous and exogenous centrality (EverettandBorgatti,2010).

Fordisconnectedundirectedgraphsornon-stronglyconnected digraphs, in which the reachability relation is not complete, betweennessretainsitsinterpretationasthetotalpotentialforcon- trolofshortest-pathconnections.Fromthewould-bebroker’spoint ofview,itmaynotmakeadifferencewhethera givenpaircan connectviabetterpathsthatdonotinvolvethebroker,orcannot connectatall:eitherwaythebrokerwillnotbebrokeringbetween them.

Fromthepointofviewofanactoravoidingdependenceonbro- kers,however,itmayinfactmakeadifferencewhetherthesame numberofintermediariescontroltheconnectionstomanyorfew reachablereceivers.Intheabsenceofasubstantivejustificationfor combiningdependencywiththenumberofreachable receivers, closenesscentrality shouldthereforebetreated asabi-criterial index,i.e.,thetwovaluesforthetotalnumberofintermediaries andthenumberof reachablereceiversshouldnotbecombined intoasinglequantity.

6. Generalizeddistanceandvaluednetworks

Duringgeneralizationtovaluedgraphsthedifferencebetween closeness and our variant closeness-as-independence becomes most apparent. In fact, the duality with betweenness is maintained only by closeness-as-independence. This differ- ence highlights the fact that the interpretations of close- ness as either efficiency or independence are actually dis- tinct indices that happen to coincide on non-valued net- works.

LetavaluedgraphbedefinedasagraphG=(V,E;)withedge values:E→R.Suchvaluestypicallyhaveapositivesignandrep- resentadistanceorlagintheconnectionbetweenadjacentvertices, sothatthelengthofapathinavaluedgraphisgenerallydefined asthesumofthevaluesofitsedges(Flament,1963).Notethat thedefinitionofpathlengthasthenumberofedgesgiveninSec- tion 2.1isthespecialcaseinwhichalledgeshavealengthof1.

Distancesdist(s,r)arethendefinedasbefore,i.e.,astheminimum lengthofany(s,r)-path.Whileothervaluesarepossibleandother generalizationsofshortestorbestpathsexist(e.g.,YangandKnoke, 2001;Opsahletal.,2010),thisappearstobethemostfrequently employed.

From any generalization of path lengths to valued graphs we obtain straightforward generalizations of closeness and betweenness. Since the dependencies ı(s, b, r) are defined as the fraction of optimal (s, r)-paths passing through b, independent of the value associated with such paths, the interpretationofbetweennessispreservedinanysuchgeneral- ization.

Theinterpretationofclosenessasindicatingaccessefficiency isalsopreservedaslongasdistancesstillrepresentaneffortnec- essaryforthesendertoreachthereceiver.Thisisnotnecessarily true,however,fortheinterpretationofclosenessasindicatingthe independenceofanactorfromothers,becauseLemma1nolonger holds.ThisisillustratedinFig.5.

The rationale behind the latter interpretation of closeness wastheindependencefromintermediaries.Intermediaries,how- ever,featureinthestandarddefinitionofclosenessonlybecause their number corresponds to the number of edges in a path minus one. It is thus rather by coincidence than by design that distance and dependence almost agree in non-valued graphs.

Fig.5.Inthisvaluedgraph,dist(s,r)=6,andtherearethreeshortest(s,r)-pathwith 2,4,and4innerverticesforanaverageof3¹₃=ı(s,·,r)intermediariesthatsandr dependon.

Our variant closeness-as-independence index, on the other hand,preservesthedualitywithbetweennessestablishedabove forthecaseofnon-valuednetworks.Recallthatc_B(b)=

b∈Vı(s, b, ·)and c_C(s)⁻¹=

s∈Vı(s,b,·)innon-valuednetworks. Since theinterpretationsofı(s,b,r)andthusı(s,b,·)arenotaffected by generalization to valued networks, both betweenness and closeness-as-independencegeneralizestraightforwardlywiththe othergeneralizedquantities,andtheirdualityasindegreeandout- degreeismaintained.

Witha suitablydefined distance,closeness-as-independence canstillberegardedastheinverseofatotaldistanceasshown inthefollowingLemma1.

Lemma5. InavaluedgraphG=(V,E;),ı(s,·,r)equalstheaverage numberofinnerverticesinshortest(s,r)-pathsforanys,r∈V.

Proof. Foranypairofverticess,r∈V,ı(s,·,r)=

b∈V(s,r|b) (s,r) by definition.Sinceeveryinnervertexbcontributes1toeachshortest pathitiscontainedin,

b∈V(s,r|b)isthetotalnumberofoccur- rencesofinnerverticesinanyshortestpath,anddivisionbythe number(s,r)ofshortest(s,r)-pathyieldstheaverage.䊐

Therefore,sincec_C(s)=ı(s,·,·)=

r∈Vı(s,·,r)byDefinition 3,ouradjustmentofclosenesstomaintaintheindependenceinter- pretationcan alsobe seen asreplacing shortest-path distances dist(s,r)withtheaveragenumbersofintermediariesı(s,·,r)on shortestpaths.Notethatthisalsoholdsinnon-valuednetworks becauseallshortest(s,r)-pathshavethesamenumberdist(s,r)−1 ofinnervertices(Lemma1).

Like betweenness and closeness-as-efficiency, our variant closeness-as-independenceindexcan bedeterminedin O(nm+ n²logn)time because a single-source shortest-paths computa- tionfromasendersyieldsthedependenciesı(s,b,·)forallb∈V (Brandes,2001).

7. Discussion

Wehavepointedoutaformaldualitybetweenclosenessand betweennesscentralitythat,althoughknown,haslongbeenused onlyinconceptualterms.Thedualityisexpressedintermsofa derived relation,thedyadicdependencyof senders onbrokers.

Betweenness and closeness are in fact theweighted indegrees andoutdegreesinthenetworkofthisderivedrelation.Sincetotal betweennessandclosenessinagraphthusequalthetotalofthe dyadicdependencies,theyalsoequal thesumofdistances in a graphminusthenumberofreachablepairs.

Closenessandbetweennessyieldthesamerankingonpaths, stargraphs,cliques,andanumberofothergraphs.Itwillbeinter- estingtoinvestigatebyhowmuchtheycanactuallydiffer.Wegave anexample(apathofcliquesofvaryingsize)inwhichtherankings arealmostthereverseofeachother.

Theobserveddualitygeneralizestodirectedandnon-connected networks,nomatterwhetherclosenessisgeneralizedbyintroduc- ingafinitedistanceforunreachablepairsorbyconsideringtotal distanceandnumberofreachableverticesasatwo-dimensional index. By reversing edge directions, it is easily confirmed that the corresponding dependency of receivers on brokers corre- spondstoclosenessdefinedbydistancesto,ratherthanfrom,an actor.

Invaluednetworksitbecomesapparentthatthetwointerpre- tationsofclosenesscentralityasefficiencyandasindependence actually refer to two different concepts that happen to coin- cide (up to an additive constant) in non-valued networks.

Duality is maintained in valued networks only if the defini- tionof closeness isadapted. By replacing thesumof distances with a sum of dependencies, we effectively replace shortest- path distance with the expected number of intermediaries on a shortest path. Since this is in line with the original moti- vation for closeness centrality asan indicator of independence (Freeman, 1979), we consider it a strong argument for our newvariantincaseswhereclosenessisinterpretedasindepen- dence.

Finally, we note that the concept of dual centrality indices appliesmoregenerallytoallindicesthatareco-determinedbyan asymmetricrelationderivedfromtheoriginalnetwork.Closeness and betweennessarerow andcolumn sumsof dyadicdepend- encies,soindegreeandoutdegreeonotherrelationsareobvious extensions.Forcertainderivedrelations,however,wealsoexpect meaningful dualities to arise from left and right singular vec- tors.

Acknowledgements

ThisresearchwaspartiallysupportedbyDeutscheForschungs- gemeinschaft(DFG)undergrantBr2158/6-1.

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