Dyamic h-idex: he Hirsch idex i fucio of ime by L. Egghe Uiversiei Hassel (UHassel), Campus Diepebeek, Agoralaa, B-3590 Diepebeek, Belgium ad Uiversiei Awerpe (UA), Campus Drie Eike, Uiversieisplei, B-260 Wilrijk, Belgium leo.egghe@uhassel.be ABSTRACT Whe we have a group of papers ad whe we fix he prese ime we ca deermie he uique umber h beig he umber of papers ha received h or more ciaios while he oher papers received a umber of ciaios which is o larger ha h. I his paper we deermie he ime depedece of he h-idex. This is impora o describe he expeced career evoluio of a scieis s work or of a joural s producio i a fixed year. We use he earlier esablished cumulaive h ciaio disribuio. We show ha ( ) ) h = - a T Permae address Key words ad phrases: Hirsch idex, h-idex, ime depede, Loka, agig rae.
2 where a is he agig rae, is he expoe of Loka s law of he sysem ad T is he oal umber of papers i he group. For = + we refid he seady sae (saic) formula which we proved i a previous paper. h= T Fucioal properies of he above formula are proved. Amogs several resuls we show (for, a, T fixed) ha h is a cocavely icreasig fucio of ime, asympoically bouded by T. I. Iroducio Le us have ay group of papers: his ca be he lis of publicaios of a scieis, he se of aricles i a year of publicaio i a joural, a bibliography ad so o. The h-idex is his umber such ha we have exacly h papers wih h or more ciaios while he oher papers have o more ha h ciaios. This h-idex has bee iroduced i Hirsch (2005) (see also Ball (2005)). Iroduced i physics, he h-idex was well-received by iformericias see Borma ad Daiel (2005) ad Brau, Gläzel ad Schuber (2005). The h-idex has he followig advaages (see Hirsch (2005), Brau, Gläzel ad Schuber (2005)): - I is a sigle umber icorporaig boh publicaio ad ciaio scores ad hece has a advaage over hese sigle measures ad over measures such as umber of sigifica papers (which is arbirary (which is o so for he h-idex)) or umber of ciaios o each of he q mos cied papers (which agai is o a sigle umber). - I is robus i he sese ha i is isesiive o a accideal se of ucied (or lowly cied) papers ad also o oe or several ousadigly highly cied papers. - I combies he feaures quaiy (umber of papers) wih qualiy (or visibiliy) i he sese of ciaio raes. The prese auhor shares he opiio of Brau, Gläzel ad Schuber ha he h-idex will be he opic of may iformerics aricles i he (ear) fuure ad hece, i deserves a cocise mahemaical sudy. I Egghe ad Rousseau (2006) we already showed ha ay sysem has a uique h-idex (here he h-idex oio was exeded o geeral iformaio producio processes, where sources (e.g. aricles) produce iems (e.g. ciaios)). Furher we showed
3 ha, whe we have a Lokaia sysem, i.e. a sysem such ha he law of Loka is valid (see Egghe ad Rousseau (990) or Egghe (2005)) C ( ) = () f j j C> 0, >, j³, ad where we have T sources (e.g. aricles) i oal, he h-idex equals T h= (2) Properies of his fucioal relaio are also proved i Egghe ad Rousseau (2006). I is evide ha i is a ieresig aspec o kow he evoluio of he h-idex over ime for a se of papers. This is sudied i his paper. I Egghe ad Rao (200) we deermied he cumulaive h ciaio disribuio, i.e. he cumulaive disribuio ( ) of he imes a which he papers (i such a geeral se of papers) receive heir h ciaio, i oher words: he cumulaive fracio of papers (amogs he ever cied papers) ha have received ciaios a ime. The fidig of his paper is ha such a disribuio ca be used o calculae, for every, he (hece -depede) h-idex. I he ex secio we will elaborae he model, where we will prove ha he ime depede h-idex equals ( ) ) h = - a T (3) where T deoes he oal umber of ever cied aricles, is Loka s expoe (he sysem aricles-ciaios supposed o be Lokaia) ad where a is he agig rae of he ciaios. We he prove he followig properies of (3): - The h-idex as fucio of (a,, T cosa) is a cocavely icreasig fucio wih horizoal asympoe a heigh T. Hece for goig o, we refid he ime idepede resul (2) proved i Egghe ad Rousseau (2006).
4 - The h-idex as fucio of (a, T, cosa) ad a (, T, cosa) is decreasig ad (of course) icreasig i T (a,, cosa). II. The h-idex as fucio of ime Le us recall a resul, proved i Egghe ad Rao (200) o he cumulaive h ciaio disribuio, i.e. he cumulaive disribuio over ime a which a paper will receive is h ciaio ( =,2,3,... ). Here he fracios are calculaed wih respec o he populaio of eveually cied papers. We have he followig resul. Theorem II. Egghe ad Rao (200)) (i) Le C ( ) deoe he cumulaive ciaio disribuio, i.e. he cumulaive disribuio of he fracios of ciaios a ime (i.e. ime afer publicaio). The he cumulaive h ciaio disribuio ( ) (wih respec o all papers ha are ever cied) is equal o ( ) ( ) æcö = ç çè ø (4) where > deoes Loka s expoe i he law of Loka () ha describes he aricle-ciaios relaioship. (ii) I case of expoeial agig, wih agig rae a, 0< a <, we have ha C( ) = - a ad hece ( ) æ- a ö = ç çè ø (5) From his, he -depede h-idex ca be deermied.
5 Theorem II.2: The -depede h-idex equals ( ) = ( ) = ( ) (6) h h,,t C T for ³ 0 ad where T deoes he oal umber of ever cied aricles uder sudy. I he special case of (ii) i he above heorem we have ( ) ( ) ( ) h = h,,t,a = - a T (7) Proof: For every, ( ) is he fracio (wih respec o he ever cied papers) of aricles which have or more ciaios a ime. The defiiio of he h-idex gives ha h= for his such ha ( ) T =. (8) Ideed, ( T ) is he umber of papers wih ciaios a ime or before, hece he umber of papers wih or more ciaios (ad auomaically he oher papers have less ha ciaios: his is, i he coiuous seig, equal o o more ha ciaios ). So equaio (8) is he defiig relaio for he (-depede) h-idex: h=. Usig (4) we hece have or h ( ) ( ) æcö = ç T = h çè h ø ( ( ) ) h = C T I case C( ) = - a (expoeial agig rae) we evidely have (7).
6 The ex corollary was proved i Egghe ad Rousseau (2006) (see also Gläzel (2006) for a approximaio i he discree case), hece i also belogs o he prese ime-depede heory as a limiig case. Corollary II.3: If we le we have, i all cases: T h= (9) Resul (9) was proved i Egghe ad Rousseau (2006) wihou supposig ay agig disribuio: here we oly used ha he disribuio of papers wih a cerai umber of ciaios follows Loka s law wih expoe >. I his model, he miimum umber of ciaios is, hece we were oly dealig wih papers ha are eveually cied. This is also he case i he prese model. The cumulaive disribuio of he ime of ciaios is give by C. ( ) Supposig ha limc( ) = auomaically implies ha we oly cosider papers ha are, eveually, cied. So i is logical ha he resul (9) is foud here agai. I is, everheless, remarkable ha he saic (ime-idepede) Loka resul is foud here as a limiig resul of our dyamic (ime-depede) heory for. We have he followig furher corollaries of Theorem II.2. Corollary II.4: The fucio h( ) i Theorem II.2 i he variable (while all he oher variables are kep cosa) is a cocavely sricly icreasig fucio ha rages bewee 0 ad T (see Fig. ).
7 h T / 0 Fig.. Graph of he ime-depede h-idex. Proof: I is readily see ha h' ( ) > 0 ad ( ) h'' < 0 for all 0 ³. Furhermore ( ) ( ) =. Fially h( 0) = 0 ad lim h( ) = T, he highes value. lim h' 0 h' 0 = + ad The resul i Corollary II.4 is logical bu shows ha, whe ime passes, i becomes more ad more difficul o icrease he h-idex of he se of papers uder sudy, e.g. for a se of joural aricles i a cerai year or for a scieis s publicaios. Noe ha i his las case we eglec he fac ha publicaios have differe publicaio imes. We leave i as a ope problem o exed his heory o his case bu he prese heory is a good approximaio for ha, ceraily for high as discussed above. Corollary II.5: (i) For a, T, fixed we have ha h is a decreasig fucio of (ii) For, T, fixed we have ha h is a decreasig fucio of a. (iii) For, a, fixed we (evidely) have ha h is a icreasig fucio of T. Proof: This follows readily by calculaig he correspodig derivaives.
8 Refereces P. Ball (2005). Idex aims for fair rakig of scieiss. Naure 436, 900. L. Borma ad H.-D. Daiel (2005). Does he h-idex for rakig of scieiss really work? Scieomerics 65(3), 39-392. T. Brau, W. Gläzel ad A. Schuber (2005). A Hirsch-ype idex for jourals. The Scieis 9(22), 8-0. L. Egghe (2005). Power Laws i he Iformaio Producio Process: Lokaia Iformerics. Elsevier, Oxford (UK). L. Egghe ad I.K. Ravichadra Rao (200). Theory of firs-ciaio disribuios ad applicaios. Mahemaical ad Compuer Modellig 34, 8-90. L. Egghe ad R. Rousseau (990). Iroducio o Iformerics. Quaiaive Mehods i Library, Documeaio ad Iformaio Sciece. Elsevier, Amserdam, he Neherlads. L. Egghe ad R. Rousseau (2006). A iformeric model for he h-idex. Prepri. W. Gläzel (2006). O he H-idex a mahemaical approach o a ew measure of publicaio aciviy ad ciaio impac. Scieomerics 67(2), o appear. J.E. Hirsch (2005). A idex o quaify a idividual s scieific research oupu. hp://xxx.arxiv.org/abs/physics/0508025.