Theory and practise of the g-index

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1 Theoy and pactise of the g-index by L. Egghe (*), Univesiteit Hasselt (UHasselt), Campus Diepenbeek, Agoalaan, B-3590 Diepenbeek, Belgium Univesiteit Antwepen (UA), Campus Die Eiken, Univesiteitsplein, B-60 Wilijk, Belgium ABSTRACT The g-index is intoduced as an impovement of the h-index of Hisch to measue the global citation pefomance of a set of aticles. If this set is anked in deceasing ode of the numbe of citations that they eceived, the g-index is the (unique) lagest numbe such that the top g aticles eceived (togethe) at least g set of aticles and we have that g h. citations. We pove the unique existence of g fo any The geneal Lotkaian theoy of the g-index is pesented and we show that g= T (*) Pemanent addess. Key wods and phases: g-index, h-index, Lotka, citation pefomance, Pice medallist. Acknowledgement: The autho is gateful to Ds. M. Goovaets fo the pepaation of the citation data of the Pice medallists (Januay 006).

2 whee > is the Lotkaian exponent and whee T denotes the total numbe of souces. We then pesent the g-index of the (still active) Pice medallists fo thei complete caees up to 97 and compae it with the h-index. It is shown that the g-index inheits all the good popeties of the h-index and, in addition, bette takes into account the citation scoes of the top aticles. This yields a bette distinction between and ode of the scientists fom the point of view of visibility. I. Intoduction Recently the physicist Hisch (see Hisch (005)) intoduced the so-called h-index see also Ball (005), Baun, Glänzel and Schubet (005), Glänzel (006a,b), Egghe and Rousseau (006). Fo any geneal set of papes one can aange these papes in deceasing ode of the numbe of citations they eceived. The h-index is then the lagest ank h = such that the pape on this ank (and hence also all papes on ank,,h) has h o moe citations. Hence the papes on anks h+, h+, have not moe than h citations. Although intoduced by a physicist, this new science indicato has been well-eceived in scientometics (infometics). In the above mentioned efeences it was agued that the h- index is a simple single numbe incopoating publication as well as citation data (hence compising quantitative as well as qualitative o visibility aspects) and hence has an advantage ove numbes such as numbe of significant papes (which is abitay) o numbe of citations to each of the (say) q most cited papes (which again is not a single numbe). The h- index is also obust in the sense that it is insensitive to a set of uncited (o lowly cited) papes but also it is insensitive to one o seveal outstandingly highly cited papes. This last aspect can be consideed as a dawback of the h-index. Let us discuss this point futhe. Highly cited papes ae, of couse, impotant fo the detemination of the value h of the h- index. But once a pape is selected to belong to the top h papes, this pape is not used any moe in the detemination of h, as a vaiable ove time. Indeed, once a pape is selected to the top goup, the h-index calculated in subsequent yeas is not at all influenced by this pape s

3 3 eceived citations futhe on: even if the pape doubles o tiples its numbe of citations (o even moe) the subsequent h-indexes ae not influenced by this. We think it is an advantage of the h-index not to take into account the tail papes (with low numbe of citations) but it should (being a measue of oveall citation pefomance) take into account the citation evolution of the most cited papes! In ode to ovecome this disadvantage, whilst keeping the advantages of the h-index, we make the following emak: by definition of the h-index, the papes on ank,,h each have at least h citations, hence these h papes togethe have at least citations. But it could well be (see examples futhe on) that the fist h+ papes have togethe ( h+ ) o moe citations (hee we use the fact that, most pobably, the top papes have much moe than h citations) and the same might be tue fo anks h+ (the top ( h+ ) papes having togethe at least h ( h+ ) citations) o even highe. Theefoe, in the Lette Egghe (006a) we intoduced a simple vaiant of the h-index: the g- index. Definition I.: A set of papes has a g-index g if g is the highest ank such that the top g papes have, togethe, at least g citations. This also means that the top g+ papes have less than ( g+ ) papes. The following poposition (also emaked in Egghe (006a)) is tivial. Poposition I.: In all cases one has that g h () Poof: Since h satisfies the equiement that the top h papes have at least h papes and since g is the lagest numbe with this popety, it is clea that g h.

4 4 An example shows the easy calculation of the h-index and the g-index. The data ae the autho s own citation data deived fom the Web of Knowledge (WoK). It must be undelined, howeve, that the eal citation data can be much highe due to seveal easons: - only souce jounals, selected by Thomson ISI ae used, - unclea citations (even to souce jounals, e.g. to appea etc.) ae not counted in the WoK. In the table below TC stands fo the total numbe of citations fo each pape on ank and ΣTC stands fo the cumulative numbe of citations to the papes on ank,..., (fo each ). The bold face typed numbes give the explanation fo the h-index and the g-index g= 9. Indeed h= 3 is the highest ank such that all papes on ank,...,h have at least 3 citations (and hence the papes on ank 4 o highe have not moe than 3 citations). Also g= 9 is the highest ank such that the top 9 papes have at least 38> 36); on ank 0 we have 39 < 0 = 400 citations. h= 3 9 = 36 =,,... citations (hee Table. Ranking of the papes of L. Egghe accoding to thei numbe of citations eceived (souce: WoK). TC Σ TC

5 5 In the last section of this aticle we will compae the h- and g-indexes of the (active) Pice medallists (updated calculations of the h-index as in Glänzel and Pesson (005) and new calculations of the g-index) showing the advantage of the g-index above the h-index but in the next section we will give the mathematical theoy of the g-index based on Lotka s law C () = () f j j j, C> 0, > (it will tun out that, if we let j to be abitay lage which we assume hee fo the sake of simplicity we need to take total numbe of souces (= papes hee)) > ). In case of () we will show that (T = g= T (3), hence by Glänzel (006b) o Egghe and Rousseau (006), since one showed thee that T h=, we have g= h> h (4) Also the elation of g with the total numbe A of items (= citations hee) is given. Befoe this theoy is developed we will, fistly, show the geneal existence theoem fo the g-index: fo any set of papes we always have that the g-index exists and is unique. Note, cf. Baun, Glänzel and Schubet (005), Egghe (006a), Egghe and Rousseau (006), that any set of papes can be taken hee, e.g. the papes of a scientist but also a yea s poduction (aticles) in a jounal can be used.

6 6 II. Mathematical theoy of the g-index Fist we will give a mathematically exact definition of the g-index in continuous vaiables. II. Mathematical definition of the g-index Let f() j ( j ) denote the geneal size-fequency function of the system (which can be moe geneal than the papes-citation elation: we can wok in geneal infomation poduction pocesses (IPPs) whee we have souces that poduce items cf. Egghe and Rousseau (990), Egghe (005)). We do not suppose f to be Lotkaian at this moment. Let g ()( [ 0,T] ) denote the geneal ank-fequency function (the function g ( ) should not be confused with the g- index; we keep the f() j and g () notation since this has been done in all pevious aticles and books on this topic thoughout the text it will be clea whethe we deal with the function g () o with the g-index g). The geneal (defining) elation between the functions f() j and g () is as follows: g j f j' dj j () ( ) (5) = = ' Indeed, if = g () j (the invese of the function g) ( ) then g () = j and thee ae souces with an item density value lage than o equal to j. Denote () = g'd' ( ) (6) G 0 the cumulative numbe of items in the souces up to ank (i.e. the top souces). Definition: The ank is the g-index: = g of this system if is the highest value such that () G (7)

7 7 Note that this is the exact fomulation of the g-index as poposed in Section I in pactical systems. II. Existence theoem fo the g-index Theoem II.. Evey geneal system has a unique g-index. Poof: Define, fo all ] 0,T] G () () = (8) H and we define H0 ( ) = limh ( ) = g0 ( ). We fist pove that H stictly deceases on [ 0,T]. Indeed 0 > () G() g H' () = < 0 since () () ( ) g G g ' d ' < = 0 since the function g is stictly deceasing (by (5)) fo all values of ] 0,T]. Since H0 ( ) = limh ( ) we hence have that H stictly deceases on [ 0,T ]. If HT ( ) Tthen GT ( ) T 0 > and since this is the lagest possible value, we have the unique g-index g= T. Suppose now that HT ( ) < T. Define F () = H () (9) () G F () =

8 8 Since H0 ( ) > 0(since the function g stictly deceases (by (5)) and by (8)) and HT ( ) < Twe have F0 ( ) > 0and FT ( ) < 0. Hence, since F is continuous, thee is a value such that F () = 0. By (8) and (9) we hence have the existence of a value such that () G = Note that this satisfies (7) and that it is the highest possible value that satisfies (7): indeed, H stictly deceases, so, fo evey value ' > we have ( ) < H ( ) H' By (8): ( ) G () G' ' < = ( ) G ' < ' < ' contadicting (7). Hence this unique value is the g-index: = g. Note that, except if ( ) GT T, we can pove that the g-index always satisfies (7) with an equality sign instead of. Now we will give fomulae fo the g-index in tems of paametes that appea in Lotkaian infometics. II.3 Fomulae fo the g-index in Lotkaian systems If GT ( ) T then we know fom the poof of Theoem II.. that the g-index satisfies (7) with an equality sign: ( ) Gg= g (0)

9 9 Othewise (if GT ( ) > T) we take g= T. We have the following theoem. Theoem II.3.: Given the law of Lotka C () = () f j j j, C> 0, >, we have that the g-index equals g= T () if T T and g= T if T > T. Hee T denotes the total numbe of souces. Poof: Fist poof: The fist (cf. (5)) g () j f( j' ) dj j = = ' C j = (3) souces yield a total numbe of items (since > ) j'f ( j' ) dj' j (4) j C =

10 0 (cf. also Egghe (005), Chapte II). So, by (0) we have = g if C j g = (5) and if this g satisfies g T (othewise take g= T). By (3): ( ) j= C (6) (6) (fo = g) in (5) yields C g = C g= T C using that T = as follows fom (3) by taking j=. This value is taken as the g- index if it is T and we take g = T if it is stictly lage than T. Second poof: Now we wok diectly with fomula (0). Note that Lotka s law () is equivalent with Zipf s law B () = (7) g β

11 B, β > 0, ] 0,T] whee we have the elations C B = (8) β = (9) (cf. Egghe (005), Execise II...6 but see also the Appendix in Egghe and Rousseau (006) whee a poof is given.). Note that by (9) > is equivalent with 0< β <. If that is the case, (0) gives 0 g B d = g β hence, since 0< β < B g g β β = Hence B g = β β+ (0) Now (8) and (9) in (0) again yield fomula (). Coollay II.3.: If g is the g-index and h is the h-index of a Lotkaian system with exponent >, then g= h ()

12 (if this value is T ; othewise g= T). Poof: This follows eadily fom () and the fact that poved, appoximatively, in Glänzel (006b)). T h=, see Egghe and Rousseau (006) (also Taking j= C in (3) and (4) we see that the total numbe of souces T equals C the total numbe of items A equals, hence and that A μ = = () T equals the aveage numbe of items pe souce (cf. also Egghe (005), Chapte II). Hence we have the following coollay Coollay II.3.3: If μ is as above we have in case of () g= μ h (3) The g-index in function of and A is as in Coollay II.3.4. Coollay II.3.4: We have g= A (4) (if this value is T ).

13 3 Poof: This follows eadily fom () and (). We can also detemine the item density j fo which we have = g. In pactical cases this means the numbe of items in the souce at ank g. Note that this is h fo the h-index = h, by definition of the Hisch index. Fo the g-index we have: if the value in () is > T we have j= g( T) = and if the value in () is T we substitute = g in (7), using (8) and (9) and the fact that yielding C T =, j T = (6) We immediately see that j< h which is logical since g > h and the item density is h in case = h. Fomula (6) pesents a concete fomula fo the item density cut-off place. Note that, although can be any value >, we do not have lim j 0. Indeed since the = > validity of () is limited to g T we have, by () that T T fom which it follows that A μ = = T T (7) This implies in (6) that

14 4 T j= A, by (7). The case T > T (8) (hence whee we take g = T) occus in the following case: fom (8) it follows that > T (9) By () we have A μ = > T T hence A> T (30) Equivalently, (9) gives the condition in : T < (3) T Note that T > T

15 5 so that (3) can occu (with the condition > ). Remak II.3.5: It might seem stange that g= T is possible in this Lotkaian model. Note howeve that () implies that A T = A T So, fo evey T fixed, if we let A we have that (but > ) so that we ae within the limitations of ou theoy. In this case we have ( ) A= G T > T and hence g= T. In the next section we will apply the g-index to the publications and citations of the (still active) Pice medallists and compae these g-indexes with the h-indexes of these same data. III. Calculation and compaison of the h- and g- indexes of the (still active) Pice medallists In Glänzel and Pesson (005), the h-indexes fo the (still active) Pice medallists ae calculated. We could use these numbes and compae them with the hee defined g-index. Howeve fo this we need to extend the tables in Glänzel and Pesson (005) (since g h) and it is hadly impossible to do this since we should do this fo the maximal citing time August 005 (since then the tables in Glänzel and Pesson (005) wee poduced). So the easiest thing to do is to emake these tables fo the pesent time (Januay 006) and make them long enough so that, on the same tables, the h- as well as the g-index can be calculated.

16 6 We have opted not to limit the publication yea to 986 o highe (as was the case in Glänzel and Pesson (005)). Indeed, the h-indexes in Glänzel and Pesson (005) seemed a bit unnatual in seveal senses. E. Gafield did not have the highest h-index (which we, nomally, could expect) and H. Small scoed lowest of all the Pice medallists. The majo eason fo these obsevations is that, by limiting the publication yea to 986 o highe, one cuts away most publications (and pehaps the highest cited ones) of the elatively olde scientists. Since we want to make a compaison of scientists (and not to daw conclusions on infometics fields) we decided not to limit the publication yea (except to the evident limit 97 since befoe that date the ISI (now Thomson ISI) data do not exist. Fo the same eason we count all publications even if scientists have published in diffeent domains (e.g. T. Baun in chemisty and L. Egghe in mathematics). These publications wee not used in the Glänzel and Pesson study. Of couse, by not limiting the publication peiod and the publication field, one might ague that thee is a bias towads the olde scientists. This is tue but, with the h- and g-indexes, we want to indicate the oveall pefomance (visibility) of the scientists as they ae viewn today (in the sense of lifetime achievement ). We base ouselves on the Web of Knowledge (WoK) and hence we ae limited to the Thomson ISI data. This means that no citations to non-souce jounals o confeence poceedings aticles o books ae counted. In addition, no citations to incomplete efeences ae counted even if they ae to souce jounal aticles (e.g. a citation to JASIST, 00, to appea): these ae not collected in the WoK times cited data. So the actual h- and g-indexes can be somewhat highe but this effect plays fo evey scientist so that compaisons ae still possible and also these limitations do not jeopadise the possibility to compae the h- and g- index. The tables of citation data of the (still active) medallists ae found in the Appendix. The table stops one line below the g-index since this is all we need. The numbe denotes the ank of the publication and TC denotes the total numbe of citations to the pape on ank. The numbe ΣTC also the table of denotes the cumulative numbe of citations to the fist anked papes. Finally, values is pesented as well as the publication yea (PY) of the aticle on

17 7 ank. The h- and g-index detemination is highlighted in the tables in the Appendix. Table gives the esults in deceasing ode of h and g. Table. h- and g-indexes of Pice medallists in deceasing ode Name h-index Name g-index Gafield Nain Baun Van Raan Glänzel Moed Schubet Small Matin Egghe Ingwesen Leydesdoff Rousseau White Gafield Nain Small Baun Schubet Glänzel Matin Moed Van Raan Ingwesen White Egghe Leydesdoff Rousseau We leave the detailed (subjective) intepetation of Table to the eade but it is clea that the g-index column is moe in line with intuition and with the aw data in the Appendix than the h-index column. In othe wods, the g-index, as simple as the h-index (a single measue, containing publication and citation elements), contains moe compaative infomation fom the aw data than the h-index and esembles moe the oveall feeling of visibility o life time achievement. A possible inteesting measue is g, i.e. the elative incease of g with espect to h. The esult h is pesented in Table 3, in deceasing ode of g. Hee we see emakable ode changes with h espect to the h- o g-odeings.

18 8 Table 3. g -values of Pice medallists in deceasing ode h Name Gafield Small White Ingwesen Matin Schubet Baun Glänzel Moed Nain Egghe Leydesdoff Van Raan Rousseau g h

19 9 Appendix Tables of TC,, ΣTC, and PY fo each of the (still active) Pice medallists and detemination of the h- and g-index. Gafield E. TC Σ TC PY

20 Baun T. TC Σ TC PY

21 Small H. TC Σ TC PY

22 Van Raan A.F.J. TC Σ TC PY

23 3 Matin B. TC Σ TC PY Nain F. TC Σ TC PY

24 Schubet A. TC Σ TC PY

25 Glänzel W. TC Σ TC PY

26 Moed F.H. TC Σ TC PY Leydesdoff L. TC Σ TC PY

27 Egghe L. TC Σ TC PY Rousseau R. TC Σ TC PY

28 Ingwesen P. TC Σ TC PY

29 9 White H.D. TC Σ TC PY IV. Conclusions and open poblems In this pape we studied the g-index being an impovement of the h-index. The g-index g is the lagest ank (whee papes ae aanged in deceasing ode of the numbe of citations they eceived) such that the fist g papes have (togethe) at least g citations. We show that g h and that g always uniquely exists. We pesent fomulae fo g in Lotkaian infometics. We show that

30 30 g= T g= h if these values ae T ; othewise g= T. Hee is the Lotka exponent and T denotes the total numbe of souces (in the citation application this means the total numbe of eve cited papes). We then calculate the h- and g-indexes of the (still active) Pice medallists. Diffeent than in Glänzel and Pesson (005) we do not limit the publication peiod (except fo the fact that we do not use papes olde than published in 97 due to the fact that ISI has no data fo them) no do we limit the topic to infometics, hence the complete caees (up to 97) of the Pice medallists ae consideed. It is found that the anked g-index column esembles moe the oveall feeling of visibility o life time achievement than does the anked h-index column. We leave open the futhe exploation of the g-index, including the establishment of the g- index in function of time. In Egghe (006b) we wee able to do this fo the h-index based on the cumulative n nt citation distibution (see Egghe and Rao (00)) and in a fothcoming pape we will do the same fo the g-index based on a time-dependent Lotkaian theoy. We also leave open the constuction of othe h- o g-like indexes and the compaison of these new indexes with the h- and g-index. It would also be inteesting to wok out moe pactical cases (in othe fields) of h- and g-index compaisons. Such case studies can lean a lot on the advantages and/o disadvantages of the h-index and the g-index.

31 3 Refeences Ball P. (005). Index aims fo fai anking of scientists. Natue 436, 900. Baun T., Glänzel W. and Schubet A. (005). A Hisch-type index fo jounals. The Scientist 9(), 8. Egghe L. (005). Powe Laws in the Infomation Poduction Pocess: Lotkaian Infometics. Elsevie, Oxfod (UK). Egghe L. (006a). An impovement of the h-index: the g-index. Pepint. Egghe L. (006b). Dynamic h-index: the Hisch index in function of time. Pepint. Egghe L. and Rao I.K.R. (00). Theoy of fist-citation distibutions and applications. Mathematical and Compute Modelling 34(-), Egghe L. and Rousseau R. (990). Intoduction to Infometics. Quantitative Methods in Libay, Documentation and Infomation Science. Elsevie, Amstedam (the Nethelands). Egghe L. and Rousseau R. (006). An infometic model of the Hisch index. Pepint. Glänzel W. (006a). On the oppotunities and limitations of the H-index. Science Focus, to appea. Glänzel W. (006b). On the H-index A mathematical appoach to a new measue of publication activity and citation impact. Scientometics 67(), to appea. Glänzel W. and Pesson O. (005). H-index fo Pice medallist. ISSI Newslette (4), 5-8. Hisch J.E. (005). An index to quantify an individual s scientific eseach output. axiv:physics/

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