A novel triangle mapping technique to study the h-index based citation distribution.

0.7 MB·English

by  Zhang, Chun-Ting

A novel triangle mapping technique to
study the h-index based citation
SUBJECTAREAS: distribution
COMPUTATIONAL
BIOLOGYAND
BIOINFORMATICS
Chun-TingZhang
BIOPHYSICS
SYSTEMSBIOLOGY
DepartmentofPhysics,TianjinUniversity,Tianjin300072,China.
BIOTECHNOLOGY
Theh-indexhasreceivedwideattentioninrecentyears.Theareaunderthecitationfunctionisdividedby
Received theh-indexintothreeparts,representingh-squared,excessandh-tailcitations.Theh-indexbyitselfdoes
16November2012 notcarryinformationforexcessandh-tailcitations,whichcanplayanevenmoredominantrolethanh-
indexindeterminingthecitationcurve,andthereforeitisnecessarytoexaminetherelationsamongthem.A
Accepted trianglemappingtechniqueisproposedheretomapthethreepercentagesofthesecitationsontoapoint
3December2012 withinaregulartriangle.Byviewingthedistributionofmappingpoints,shapesofthecitationfunctionscan
bestudiedinaperceivableform.Asanexample,thedistributionofthemappingpointsfor100mostprolific
Published
economistsisstudiedbythistechnique.
3January2013
Theh-index,proposedbyHirsch1forevaluatingtheacademicimpactofindividualresearchers,hasreceived
wideattentioninrecentyears.Thecitationsreceivedbyallpapersofagivenresearchercanbecharacterized
Correspondenceand
byacitationdistributionfunction,wherethey-axiscorrespondstothecitationsreceivedbyapaper,whereas
requestsformaterials
thex-axisrepresentsthepaperrankarrangedindescendingorderofcitations(Fig.1).Thedistributionofcitation
shouldbeaddressedto
versepaperrankiscalledthecitationdistributionfunctionorcurve,denotedbyC(x),inwhichthepaperxreceives
C.T.Z.(ctzhang@tju. C(x)citations.Theh-indexwassimplydefinedasC(h)~h1.Theareaunderthecitationdistributioncurveis
edu.cn) dividedbytheh-indexintotwoparts:thoseoftheh-core2andtheh-tail3.Theformerisfurtherdividedinto
anothertwoparts:thoseofexcesscitations4andh-squaredcitations1.Asaconsequence,thetotalcitationsare
divided into three different parts: h-squared, excess and h-tail citations (Fig. 1). Indeed, the h-index lacks
information for the excess and the h-tail citations, keeping only the citations related to the h-index (h2).
Theoretically,onlywhenh2 isdominantamongthethreeparts,theh-indexcanproperlyreflecttheacademic
performanceofthescientistunderstudy,otherwise,theh-indexleadstobiasedevaluation.Thequestionthat
whethertheh-indexdominatesthecitationsornotdependsontheshapeofcitationdistributionfunction.
AspointedoutbyBornmannandco-workers5,foranisohindexgroup(scientistshavingthesameh-index),
theirassociatedcitationdistributionfunctionsmaydisplayquitedifferentshapes.Therefore,tostudyhowto
applyh-indexfairly,itisnecessarytostudytheshapeofthecitationdistributionfunctions,andthecurrentstudy
aimstoaddressthisquestionbyusingatrianglemappingtechnique.Oneoftheadvantagesofcitationtriangle
method is that the comparison of different shapes of the citation distribution functions can be performed
intuitively.Byviewingthedistributionofmappingpointswithinthetriangle,theshapesofthecitationdistri-
butionfunctionscanbestudiedwithaperceivablemanner.Basedonthismethod,weareabletostudythedegree
withwhichtheh-indexisapplicableproperly.Itishopedthatthetechniquepresentedhereisusefulforusingthe
h-indextoevaluateacademicperformanceinamoreunbiasedway.
Results
Wehereproposeanoveltrianglemappingtechniquetostudytherelationsamongh-squared,excessandh-tail
citations.Foraregulartriangle,thesumofthedistancesfromanyinteriorpointtothethreesidesisequaltoa
constant,theheightofthetriangle.Notethatthesumofthepercentagesforh2,e2andt2isalsoaconstant,which
equalsto1.Basedonthischaracteristic,percentagesforthese3kindsofcitationsaremappedontoapointina
regulartriangle(Fig.2A).RefertotheMethodsectionfordetails.
Firstofall,letusconsidertwoconcreteexamples.AccordingtothecitationinformationprovidedbyDodson6,
C ~1700,h2~625(h~25),e2~477(e~21:84),andwefindH~0:37,E~0:28andT~0:35.Therefore,the
total
mappingpointcorrespondingtoDodsonissituatedattheregionNo.4,wheretheh-indexisapplicable(Fig.2B).
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tox~{0:456,y~{0:183.Itsmappingpointissituatedatthesmall
triangleNo.5,ane-indexabsolutelydominatedregions.Accordingto
Bornmannetal5andColeandCole8,ScientistAiscalledperfectionist-
typescientist,whohasratherfewbutveryhighlycitedpublications.
ForscientistB,E50.39,H50.48andT50.13,correspondingto
x~{0:150, y~0:147. Its mapping point is situated at the small
triangleNo.2,aboundaryregionbetweenh-indexande-indexdomi-
nated regions. According to references5,8, ScientistB is called a pro-
lific-type scientist, who publishes a large number of high-impact
papers. For scientist C, E 5 0.10, H 5 0.33 and T 5 0.57, corres-
pondingtox~0:271,y~{0:003.Itsmappingpointissituatedatthe
smalltriangleNo.8,at-indexdominatedregion.ScientistCiscalleda
mass producer5,8, who publishes a larger number of papers that are
lowlycited.Itcanbeseenfromtheaboveanalysis,thelocationsofthe
mapping points carry the information of the types of scientists.
Therefore,thetrianglemappingtechniqueisparticularlyusefulwhen
theacademicimpactofalargenumberofscientistsisstudied.Inthat
case,clusteringanalysiscanbeperformedbasedonthemappingpoint
Figure1|Thecitationdistributioncurve. They-axiscorrespondstothe locations,andthereforescientistscanbeclassifiedaccordingtotheir
citationsreceivedbyapaper,whereasthex-axisrepresentsthepaperrank academicperformance.
arrangedindescendingorderofcitations.Theareaunderthecitation Recently, Baum introduced a new parameter, called Excess-Tail
distributioncurveisdividedbytheh-indexintothreeparts:h2,e2(excess) Ratio9,denotedbyR,whereR~E=T~e2=t2.Baumfoundthatfor
andt2(h-tail). mostcaseshestudied,Rv1,evenRvv1:Onlyforfewcases,Rw1.
The shapes of citation distribution functions for Rw1 are peaked,
ThesecondexampleisforthechemistBerniAlder,whereh2~2500
(h~50), e2~12996 (e~114) and C ~184004, and we find
total
H~0:14,E~0:71andT~0:15.Therefore,themappingpointcor-
respondingtoAlderissituatedattheregionNo.5,wherethee-index
isabsolutelydominant(Fig.2B).ThisexampleshowsthatAlder’sh-
indexseverelyunder-estimateshisacademicimpact,andinthiscase,
thee-indexshouldbeusedtogetherwiththeh-indexforafairevalu-
ation4.
Inwhatfollows,letusapplythecitationtrianglemethodtostudy
thecasesofcitationsofthe100mostprolificeconomists7.Thedata
usedtoderivethecorrespondingh-index,e-indexandt-indexwere
kindly provided by Dr. Tol. As a consequence, we calculated the
coordinatesx andy ofeachmapping pointcorresponding toeach
economist.Thedistributionofthe100pointsisshowedinFig.3A.As
wecanseethatonlytwopointsaresituatedattheregionNo.3,i.e.,an
h-indexdominantregion.Meanwhile,only11points(11%)aresitu-
atedabovethehorizontallineH~1=3ory~0,wheretheh-index
can be properly applicable. Accordingly, for the remaining cases
(89%), where Hv1=3 or yv0, the h-index should be used jointly
withthee-index,eventhet-index.Theaverageh-indexande-index
overthe100pointsare19and28.14,respectively,correspondingto
theaverage H and E being 0.26 and 0.48 (averagex~{0:13, y~
{0:07).Overall,tohaveafairandaccurateevaluation,theh-index
shouldbeusedtogetherwiththee-indexeventhet-indexformostof
the100mostprolificeconomists.
Theh-indexcapturesonlytheinformationofthecitationfunction
partially.However,theabovedistributionofthe100mappingpoints
withinthetriangleprovidesmoreinformationabouttheshapesofthe
corresponding citation functions. For example, the mapping points
withinthesmalltriangleNo.5indicatethattheircitationdistribution
functions are peaked on the beginning part. On the contrary, the
mapping points within the small triangle No. 9 indicate that their
citationdistributionfunctionsareflatwithalongtail.Inbothcases,
theh-indexseemsnotappropriateincapturingthemaininformation Figure2|Thecitationtrianglemethodinstudyingh-indexbased
ofcitationfunction.Tocomplementtheh-index,Bormannandco- citations. (A)AregulartriangleABCwithitsheightbeingequalto1and
workers5 introduced three parameters: the h2upper, h2center and centersituatedatO.ADescartescoordinatesystemx{yissetupwithits
h2lower,whichcorrespondtoE,HandT,respectively,inthispaper. originatO.Thethreesidesofthetrianglearedenotedbyh,eandt,andthe
Inotherwords,thetrianglemappingtechniqueprovidesanintuitive distancesofaninteriorpointP(x,y)tothemareequaltoH,EandT,
representationoftheh2upper,h2centerandh2lower.Bornmannand respectively.Therefore,thepointP(x,y)isthemappingpointforthethree
co-workers5studiedtheshapesofthecitationdistributionfunctionsof realnumbersH,EandT.(B)Theregulartriangleisdividedintonine
threescientists,A,BandC,belongingtoanisohindexgroupwithh5 smallerregulartriangles.TheintervalsofH,EandTforeachofthe9
14.ForscientistA,E50.82,H50.15andT50.03,corresponding smallertrianglesareshowninTable1.
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Figure3|Distributionsofmappingpointsinthecitationtriangle. (A)Thedistributionofthe100mappingpointsforeachofthe100mostprolific
economists. Notethatonly11points(11%)aresituatedattheregionswheretheh-indexcanbeapplicable(Hw1=3),indicatingthattheh-index
shouldbeusedjointlywiththee-index,eventhet-index,fortheremaining89economists.(B)Anexampletodemonstratethatthepowerparameter
lisoneofthekeyfactors,whichdeterminesthepositionofthemappingpoint.GivenC ~512andN~100,startingfromtheregionNo.3(theh-index
1
dominatedregion)withl~0:4,themappingpointmovestotheregionNo.6(thee-indexdominatedregion)withl~1:6.Interestingly,thetrack
ofthemappingpointsformsaclockwiserotatingcurve.
whereasforRv1theshapesofthecitationfunctionsareflatwitha asimplemathematicalmodelforthecitationdistributioncurveC(x)4
longtail.Therefore,theExcess-Tailratioisanappropriateparameter C
CðxÞ~ 1,C ~C(1)w0, x§1, lw0: ð1Þ
tocapturetheoverallshapesofthecitationfunctions.Accordingto xl 1
eq.(12),Rw1orRv1correspondstoxv0,orxw0,respectively.
ThetotalcitationsreceivedbyNpapers,C ,is
total
N
Discussion C ~ðC(x)dx~ C1 (cid:2)N1{l{1(cid:3): ð2Þ
Inwhatfollows,wewanttoexplorethekeyfactorsthatdeterminethe total 1{l
shapeofthecitationdistributionfunction.Aspreviously,weassume 1
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Table1|IntervalsofH,EandTforeachofthe9regions(smalltriangles)withinthecitationtriangle
No. H E T Featureremark
1 Hw2=3 EzTv1=3 EzTv1=3 Theh-indexabsolutelydominatedregion
2 Hw1=3 Ew1=3 Tv1=3 Boundarybetweenh-indexande-indexdominatedregions
3 Hw1=3 Ev1=3 Tv1=3 Theh-indexdominatedregion
4 Hw1=3 Ev1=3 Tw1=3 Boundarybetweenh-indexandt-indexdominatedregions
5 HzTv1=3 Ew2=3 HzTv1=3 Thee-indexabsolutelydominatedregion
6 Hv1=3 Ew1=3 Tv1=3 Thee-indexdominatedregion
7 Hv1=3 Ew1=3 Tw1=3 Boundarybetweene-indexandt-indexdominatedregions
8 Hv1=3 Ev1=3 Tw1=3 Thet-indexdominatedregion
9 HzEv1=3 HzEv1=3 Tw2=3 Thet-indexabsolutelydominatedregion
Basedoneq.(1),itwasshownthat4,10 receivedbythepapersintheh-coreareequaltoh2ze2.Theh-indexdividesthetotal
citationsofascientistintotwoparts:thefirstpartisoftheh-core,whereasthesecond
hlz1~C1: ð3Þ oneisoftheh-tail3.Forconvenience,wedefinethesquarerootofcitationsreceivedby
However,weshouldhavehvN,whichleadsto ablylpaallppearpseinrsthoefah-stcaiielnatsistht,eCtt-oitnald,eisx.cTohmepreofsoerde,otfhtehnreuempbaerrtso:fht2o,tae2lcaintadtito2n,si.ree.,ceived
lnC
lwl0~lnN1{1: ð4Þ Ctotal~h2ze2zt2, ð9Þ
Meanwhile,wehave4 whereh,eandtaretheh-,e-andt-index,respectively.Letting
e2~ 1 (cid:4)C {lClz21(cid:5): ð5Þ H~h2=Ctotal,E~e2=Ctotal,T~t2=Ctotal, ð10Þ
l{1 1 1
wehave
Usingeqs.(1)–(5),wefind HzEzT~1: ð11Þ
H~ h2 ~(1{l)|C111{zll,lwl ,l=1: ð6Þ Fsiodresanisyerqeuguallatrottrhiaenhgeleig,hthteosfutmheotfritahnegdleis.tCanocnessidfreormaraengyulianrtetrriioarngploeinAtBtCotwheiththirtese
C N1{l{1 0 heightequalto1(Fig.2A).LetthecenterofthetrianglebedenotedbyO,andanx2y
total
coordinatesystemissetupasshowninFig.2A.Basedoneq.(11)andthefeatureof
1{l theregulartriangle,thesetofthreerealnumbersH,EandTismappedontoapoint
e2 l|C1zl{1 P(x,y)withinthetriangle,asshowninFig.2A.Simplecalculationshowsthat
E~Ctotal~ N1{1l{1 ,lwl0,l=1: ð7Þ (x~(T{E)=pffi3ffiffi~(1{H{2E)=pffi3ffiffi,
ð12Þ
y~H{1=3:
Therefore,theconditionunderwhichtheh-indexcanbedominant
shouldsatisfyHw1=3,or Thetrianglecanbedividedinto9smallertriangles(regions)asshowninFig.2B.We
1{l denotethembyNo.1throughNo.9,respectively.Eachregionischaracterizedbya
(1{l)|C1zl 1 specialintervalofthethreerealnumbersH,EandT,respectively.Forexample,atthe
N1{l{11 w3: ð8Þ regionNo.1,Hw2=3andEzTv1=3,indicatingthath2isabsolutelydominantat
thisregionascomparedwithe2andt2.Similarly,attheregionNo.5,Ew2=3and
To have an intuitive picture, we consider some numerical HzTv1=3,indicatingthate2isabsolutelydominantascomparedwithh2andt2.At
examples as follows. Taking C ~512, N~100 and letting theregionNo.9,Tw2=3andHzEv1=3,indicatingthatt2isabsolutelydominant
l~0:4, 0:5, 0:6, 0:7, 0:8, 0:9, 1:1,11:2, 1:3, 1:4, 1:5, 1:6, respect- ascomparedwithh2ande2.Furthermore,attheregionNo.3,Hw1=3,Ev1=3,
Tv1=3,so,itiscalledanh-indexdominantregion;attheregionNo.6,Ew1=3,
ively, we calculate the values of H and E for each case. Using eq. Hv1=3,Tv1=3,so,itiscalledane-indexdominantregion;andattheregionNo.8,
(12),wefind12mappingpointsinthetriangle,asshowninFig.3B.It Tw1=3,Hv1=3,Ev1=3,so,itiscalledat-indexdominantregion.Finally,the
isinterestingtoseethatwiththeincreaseofthelvalue,thetrackof regionNo.2istheboundaryregionbetweentheh-indexande-indexdominant
themappingpointsformsaclockwiserotatingcurve.Thisexample regions,theregionNo.4istheboundaryregionbetweentheh-indexandt-index
dominantregions,andtheregionNo.7istheboundaryregionbetweenthee-index
showsthatthepowerparameterlisoneofthekeyfactorstodeter-
andt-indexdominantregions.Theabovedescriptionhassymmetryofaregular
minetheshapeofthecitationfunction.GivenC1 andN,thereisa triangle.ThetotaldescriptionissummarizedinTable1.
thresholdofl,whenlislessthanthisthreshold,theh-indexcanno ThethreerealnumbersH,EandTarethepercentagesofcitationsassociatedwith
longerbeproperlyapplicable.Infact,l??,h?1. theh-,e-andt-index,respectively.Ingeneral,Hshouldbegreaterthan1/3(oryw0),
wheretheh-indexisproperlyapplicable,otherwise,ifHv1=3(oryv0),theh-index
The main contribution of this paper is to propose the citation
under-evaluatestheacademicimpactoftheresearcherconcerned.Therefore,thefour
trianglemethod,bywhichtheshapesofcitationdistributionfunc- regionsNo.1,No.2,No.3andNo.4aretheregionswheretheh-indexcanbeproperly
tionscanbestudiedinaperceivableform.Basedonthedistribution applied(Hw1=3).TheregionsNo.2,No.5,No.6andNo.7aretheregionswherethee-
ofmappingpoints,applicabilityandlimitationoftheh-indexcanbe indexcanbeproperlyapplied(Ew1=3),whereasthoseofNo.4,No.7,No.8andNo.9
aretheregionswherethet-indexcanbeproperlyapplied(Tw1=3).Insummary,the
studied. Generally, the h-index is not properly applicable in the
h-indexcanonlybeproperlyappliedintheregionsNo.1,No.2,No.3andNo.4
e-index or t-index dominated regions. In those cases, the h-index (Hw1=3oryw0);andtheh-indexshouldbejointlyappliedtogetherwiththee-index
shouldbejointlyappliedtogetherwiththee-indexort-index.The ort-indexintheremainingregionsNo.5throughNo.9(Hv1=3oryv0).
proposedmappingtechniqueprovidesaplatformtostudytheaca-
demicimpact ofa group of scientists, because some mathematical
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h-indexbasedcitationdistribution.Sci.Rep.3,1023;DOI:10.1038/srep01023(2013).
Acknowledgements
IthankDr.F.GaoandDr.K.SongandforhelpsinpreparingFigures2–3.Iamgratefulto
Dr.RichardTolforkindlyprovidingthedatausedtocalculatethee-indexandt-indexfor
eachofthe100mostprolificeconomists.
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