Mathematical Theory of the h- and g-index in Case of Fractional Counting of Authorship

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1 Matematical Teory of te - and -Index in Case of Fractional Countin of Autorsip Leo Ee Universiteit Hasselt (UHasselt), Campus Diepenbeek, Aoralaan, B-3590 Diepenbeek, Belium and Universiteit Antwerpen (UA), Campus Drie Eiken, Universiteitsplein, B-260 Wilrijk, Belium leoee@uasseltbe Tis article studies te -index (Hirsc index) and te -index of autors, in case one counts autorsip of te cited articles in a fractional way Tere are two ways to do tis: One counts te citations to tese papers in a fractional way or one counts te ranks of te papers in a fractional way as credit for an autor In bot cases, we define te fractional - and -indexes, and we present inequalities (bot upper and lower bounds) between tese fractional - and -indexes and teir correspondin unweited values (also involvin, of course, te coautorsip distribution) Werever applicable, examples and counterexamples are provided In a concrete example (te publication citation list of te present autor), we make explicit calculations of tese fractional - and -indexes and sow tat tey are not very different from te unweited ones Introduction In Auust 2005, JE Hirsc proposed is famous -index: A scientist as as -index if of is or er papers ave at least citations eac and te oter papers ave no more tan citations eac (Hirsc, 2005; and in te arxiv repository of September 29, 2005: arxiv:pysics/ v5) A simpler, but equivalent, formulation of te definition of te -index is as follows: Rank te papers of a scientist in decreasin order of te number of citations tey ave received Ten tis scientist as -index if r = is te iest rank suc tat te first papers ave at least citations Since its definition (in pysics) in Auust 2005, te -index as attracted lots of attention Te rowt of te number of papers on te -index and related indexes is spectacular, and it is nowadays virtuallmpossible to present a complete reference list We limit ourselves to some early Received April 3, 2007; revised July 7, 2007; accepted February 5, 2008 Tis paper corrected te oriinal formulation (arxiv:pysics/ v3) in te sense tat no more replaced te word fewer (in arxiv, 7 Auust 2005) wic was an erroneous formulation ( could not exist in tis early definition) 2008 ASIS&T Publised online 2 May 2008 in Wiley InterScience (wwwintersciencewileycom) DOI: 0002/asi20845 reactions on te -index (Ball, 2005; Bornmann & Daniel, 2005; Braun, Glänzel, & Scubert, 2005; Ee & Rousseau, 2006; Glänzel, 2006a,b; Popov, 2005; Rousseau, 2007; van Raan, 2006) It is seldom seen tat a new indicator as suc an impact on te scientific minds: Te -index is already accepted at tis early stae as an indicator, presented in Scopus and te Web of Science, wenever a rankin is retrieved accordin to received citations Te -index is also applied to journals, see Braun et al (2005), or to topics (replacin scientist ), see Banks (2006) Te advantaes of te -index are described in te above publications: It is a simple sinle number incorporatin publication as well as citation data (ence comprisin quantitative as well as qualitative or visibility aspects) Anoter advantae of te -index is tat it is insensitive to a set of uncited (or lowly cited) papers (wic every scientist as due to several reasons, e, te publication of requested local articles for wic even a reputed scientist cannot be blamed) In Ee (2006a; see also 2006b,c), a disadvantae of te -index is revealed: Te -index is also insensitive to one or several outstandinly ily cited papers Indeed, altou ily cited papers are important for te determination of te value of te -index, once suc a ily cited paper is selected to belon to te top papers, its actual number of citations (at any time) is not used anymore Indeed, once a paper is selected to te top roup, te -index calculated in subsequent years is not influenced by tis paper s received citations furter on, even if tis paper doubles or triples its number of citations In Ee (2006a,b,c), we introduced te -index in order to overcome te above-mentioned disadvantae of te -index wile keepin its advantaes Notice tat by te very definition of te -index, te papers on rank,, eac ave at least citations, and ence tese papers ave, toeter, at least 2 citations But is not necessarily te larest rank wit tis property Terefore, te -index is defined as te larest rank suc tat te first papers ave, toeter, at least 2 citations It is obvious tat in all cases Bvin JOURNAL OF THE AMERICAN SOCIETY FOR INFORMATION SCIENCE AND TECHNOLOGY, 59(0):608 66, 2008

2 practical examples (on te - and -index of te Price medalists), we sow te advantae of te -index over te -index, and we present (in Ee and Rousseau, 2006 and Ee, 2006a) formulae for te - and -index in case we ave a Lotkaian paper citation information production process (see Ee, 2005 on Lotkaian IPPs) In informetrics, lots of attention as been iven to coautored papers, ie, papers written by several autors Te natural question to be posed is as follows: How are te credits of eac autor counted, e, does every autor in a tree-autored paper et a credit of (total countin) or does every autor et a credit of 3 (fractional countin)? (see Ee, 2005, Ee, Rousseau, and Van Hooydonk, 2000) In eneral, fractional countin is preferred since tis does not increase te total weit of a sinle paper Te problem raised in tis article (mentioned to me by R Rousseau in an oral communication) is ow to calculate - and -indexes for autors wen we use a fractional creditin system Rousseau suests ivin an autor of an m-autored paper only a credit of c m if te paper received c citations Tis will be studied in tis article and will be called fractional countin on citations It is clear tat also anoter fractional creditin system is possible: fractional countin on papers, ie, for eac autor in an m-autored paper, te paper occupies only a fractional rank of m Tis fractional creditin system will also be studied in tis article in te framework of te - and -index In te next section, we define exactly bot fractional creditin systems, ive te definitions of te fractional - and -indexes in eac case, and present concrete examples In te Matematical Teory of te Indexes f, f, F, and F section, we present inequalities (bot upper and lower bounds) between te fractional -index and te unweited ( classical ) -index, also incorporatin of course te coautorsip distribution, ie, te function = #autors in te paper on rank i Te same is done for te -index, and tese inequalities are proved in bot fractional creditin systems Examples and, if applicable, counterexamples are presented We also sow tat te proved inequalities are optimal (ie, tat tey cannot be improved) In Te Indexes,, f, f, F and F for L Ee section, we calculate te non-fractional - and -indexes for te present autor and compare tese values wit te fractional - and -indexes (aain in bot fractional creditin systems) Te article closes by makin concludin remarks, includin te formulation of some open problems and topics for furter researc Te - and -Index for Fractional Autorsip Counts Introduction Te calculation of te classical - and -index of an autor (or a topic, ), denoted and respectivels based on a ranked list as in Table : Te papers of te autor are ranked in decreasin order of te number of citations tey ave received TABLE General form of a ranked list of papers in decreasin order of te number of received citations r (rank paper) # (number of citations) y 2 y 2 3 y 3 T TABLE 2 Visualization of te calculation of te -index r # y 2 y 2 y y + y + T TABLE 3 Visualization of te calculation of te -index r r 2 # y T # y T y y 2 4 y 2 y + y 2 4 ( ) 2 y ( ) 2 2 y 2 + ( + ) 2 + y + <( + ) 2 T T 2 y T T < T 2 Here T denotes te total number of papers (under consideration) Note tat (y, y 2,,y T ) is decreasin Ten te -index is defined as te larest rank r = suc tat y Tis definition is visualized as in Table 2 (wic we will also use furter on): r = is te larest rank suc tat y since y + < + Te -index is defined as te larest rank r = suc tat 2 It is obvious tat in all cases Here, we assume tat T In te case of T T 2, we can define = T, or better (see Ee, 2006a), we can add, in Table 2, fictitious articles wit zero citations: We add enou of tese articles so tat T (and even < T) were we denote by T te new number of articles (includin te fictitious ones) So, wit tis possible extension, te calculation of te -index is visualized as in Table 3: r = is te larest rank suc tat 2 because + <( + ) 2 JOURNAL OF THE AMERICAN SOCIETY FOR INFORMATION SCIENCE AND TECHNOLOGY Auust DOI: 0002/asi

3 TABLE 4 Illustration of te calculation of and TABLE 6 Simple example wit co-autorsip information r r 2 # # r # Autors # Citations > < TABLE 5 r Fractional citation counts # (fraction of citations) y /ϕ() 2 y 2 /ϕ(2) 3 y 3 /ϕ(3) T y T /ϕ(t) Let us illustrate te calculation of te - and -index on a very simple example (Table 4) We immediately see tat = 3 and tat = 4 Suppose now tat we also ave te information on te number of coautors of eac paper For paper i =,,T, we will, enerally, denote by te number of coautors of paper i How can we use tese numbers in order to calculate fractional - and -indexes, ie, - and -indexes in wic we consider an autor of paper i (ence, were tere are coautors in total) to be credited wit a value for eac i? For more on fractional (and oter) creditin systems, we refer te reader to Ee, Rousseau, and Van Hooydonk (2000) or to Ee (2005) Tere are two different ways to do tis: fractional count of te citations or fractional count of te papers Tis will be done in te next two subsections Fractional - and -Indexes Usin Fractional Citation Counts By fractional citation counts, we consider te followin variant of Table (see Table 5; ie, all citation counts are divided by, te number of autors of paper i) Wit tis operation, note tat Table 5 does not necessarily ive te papers in decreasin order of fractional citation counts So, we ave to rearrane Table 5 in order to ave a table in wic te are decreasin We suppose tis as been executed We now define te fractional -index, denoted by f,as te larest rank r = f suc tat y f ϕ(y f ) f () We define te fractional -index, denoted by f as te larest rank r = f suc tat f 2 f (2) TABLE 7 Calculation of f and f r # r 2 # > < So, tese definitions are exactly te same as in te nonweited case but we use te new Table 5 We illustrate tis on te simple example in Table 4 were we add coautorsip information (te values ) See Table 6 Paper as 0 citations and two coautors, so an autor of suc a paper receives a fractional citation count of 5 Te second paper as only tis autor and ence keeps its five citations Te same autor in te tird paper receives a score because tere are tree citations and tree autors Hence, tis score is lower tan in te fourt paper; tis paper keeps its two citations because tis autor is te only one Finally, for te same reason, te fift and sixt papers keep teir one citation For decreasin order, we ave to rearrane te papers as in Table 7 (Papers 3 and 4 are intercaned) It is now clear tat f = 2 and f = 3 Fractional - and -Indexes Usin Fractional Paper Counts In tis fractional countin metod, we leave te citation scores uncaned but cane te paper scores into te fractional countin metod Tis is a very classical way of ivin paper scores to an autor in a multiautored paper: An autor of a paper tat as m autors in total receives a fractional score of m Tese new paper counts replace te entire rankins r =, 2,,T, ie, is replaced by ϕ(), 2 is replaced by ϕ() + ϕ(2) and so on Table 8 makes tis very clear, bein te variant of Table in case of fractional paper counts Note tat wit tis operation, te order of te papers is not caned because te values of te citation scores is not caned (of course, te rank values are caned!) We now define te fractional -index, denoted by F,as te larest rank r = F suc tat F = k y k (3) 60 JOURNAL OF THE AMERICAN SOCIETY FOR INFORMATION SCIENCE AND TECHNOLOGY Auust 2008 DOI: 0002/asi

4 TABLE 8 r Fractional paper counts # Citations /ϕ() y /ϕ() + /ϕ(2) y 2 /ϕ() + /ϕ(2) + /ϕ(3) y 3 T / y T Teorem In all cases, f (6),, Proof: Let r =,,T denote te oriinal ranks in te unweited case: y y 2 y T and let π be tis permutation of {,,T}, wic ives te ranks in te fractional scorin system (fractional citation scores): TABLE 9 Calculation of F and F r # # >(38333) <(48333) 2 y π() ϕ(π()) y π(2) ϕ(π(2)) Consider te first values { yπ() ϕ(π()),, } y π() ϕ(π()) y π(t) ϕ(π(t)) Suppose π( j) {,, } for all j =,, Ten {y π(),,y π() }={y,,y } (7) (8) Similarly, we define te fractional -index, denoted by F,as te larest rank r = F suc tat ( k ) 2 k (4) were k F = (5) Note tat in tis aloritm, evidently te fractional - and -indexes, F and F, can be non-entire numbers, wic is not an objection in itself We illustrate tis on te same simple example in Table 6 Now, fractional countin on papers evidently leads us to Table 9 It is clear tat now F = 8333 and F = Tese simple definitions and examples illustrate te nature of te indexes f, f, F, and F In te next section, we present te matematical teory of tese indexes in relation wit teir non-weited variants and and, of course, also featurin te values bein te coautorsip distribution over te papers i =,,T Matematical Teory of te Indexes f, f, F and F In te sequel (as we did previously), we will denote by and te -index and -index, respectively, of te unweited system as iven by Table We start by studyin te fractional citation indexes f and f Matematical Teory of f and f Let us denote, for every positive number x R + : [x] = te reatest inteer tat is x We ave te followin eneral result for f : and, ence, because ϕ, f If tere is a j {,,} suc tat π(j) / {,,}, ten y π(j) ϕ(π(j)) y π(j) for all i =,,, because ϕ, π(j) / {,,} and te are decreasin In all tese cases, we ave tat y π(j) ϕ(π(j)) min,, ence, necessarily, f because j {,,} Tis concludes te proof tat f Let now x = (9),, For all i =,,[x] (because ϕ ), weave (definition of ) and ence ϕ(j) j=,, = x [x] i Hence, for all i =,,[x] we ave i (0) If tese [x] values constitute te [x] larest values, ten we ave proved tat f [x] because f is te larest inteer wit property Equation 0 If tese first [x] values do not constitute te [x] larest values, ten some of tese values y are replaced by larer numbers j ϕ(j), j {[x],,t}, say replacin a smaller value (i {,,[x]}) but for wic JOURNAL OF THE AMERICAN SOCIETY FOR INFORMATION SCIENCE AND TECHNOLOGY Auust DOI: 0002/asi

5 Equation 0 is valid Define te permutation j = π(i) in tis way, we ence ave ϕ(π(i)) i () Tis can be done for ever =,, [x], makin te ϕ(π(i)), i =,,[x] decreasin Because f is te larest inteer wit property Equation, we ence ave f [x] A similar result will be proved for f First, we need a Lemma Lemma 2: for every a N, a a a (2) Proof: By definition of te -index (unweited), we ave 2 (3) ( Because a = a ) and y,,y T decreases, we ave by Equation 3 tat a ( ) a 2 = a Teorem 3 In all cases f (4),, Proof: Te proof tat f can be read on te lines of te proof tat f in Teorem Denote by a = (5),, If a = 0, we remark tat Equation 4 (left-and inequality) is trivial Hence, we suppose a > 0 because a is an inteer, a Ten, a N, a (trivially), and ence we can apply Lemma 2 yieldin Hence, [,, [ ],, ],, (6),,,, because Hence, Equation 6 ives [,,,, ],, 2 (7) Let now π be tis permutation of {,,T} yieldin decreasin values, i =,,T Ten, obviously [,, ] ϕ(π(i)) ϕ (π(i)) [,, ],, 2 (8) by Equation 7 Because te ϕ(π(i)) are decreasin and f is te larest inteer wit tis property, we ave tat f,, completin te proof We continue wit te matematical study of te indexes F and F Matematical Teory of F and F Teorem 4 In all cases Proof: F (9) y (20) because ϕ and by definition of Because, in tis case of fractional paper counts, te values y,,y T are kept te same (see Table 0), te order of y,,y T is uncaned (decreasin order) Because we ave Equation 20 and F is te larest number suc tat F = k (see Equation 3), we ave tat k, ence F = k y k (2) 62 JOURNAL OF THE AMERICAN SOCIETY FOR INFORMATION SCIENCE AND TECHNOLOGY Auust 2008 DOI: 0002/asi

6 TABLE 0 Example of F > TABLE Calculation of F r # Autors # Citations # Cit r (fractional) # Cit > < TABLE 2 Calculation of F provin te left inequalitn Equation 9 By Equation 2 F y k (22) and because k, we ave, by definition of, y k (23) Equations 22 and 23 yield F, ence completin tis proof For F only one inequalits valid Teorem 5 In all cases Proof: ( F (24) ) 2 2 (25) because ϕ and by definition of Now F is te larest number k F = (26) for wic ( k ) 2 k (27) (see Equations 4 and 5) So, by Equations 25, 26, and 27, we ave F ϕ (i) Tables 4 and 9 already yielded an example of F < Let us now present an example of F > Example Tis is an example in wic all papers ave five autors From rank 5 on, all papers ave one citation and five autors It is clear tat = 3 and = 4 For te sake of completeness and illustration of te inequalities (Equation 9), we also calculate F (see Table ) r (fractional) # Cit >(72) <(74) 2 We clearly ave F = < and also F is satisfied because 06 = < F Now we calculate F usin Table 2 It now follows tat F = 72 > Note tat is satisfied because F = 08 < F Remark: Te above teory can also be applied to fractional country scores (instead of fractional autor scores) cf Ee et al (2000) Here, for all i is still valid but te values are not always entire (as in te case of number of autors) An example will illustrate tis: Suppose a paper i as five autors and tat country c appears tree times (ie, tree of tese five autors are of country c) Ten, te fractional score, used all over tis paper (in te fractional citation scores as well as in te fractional paper scores) equals = 3 5 JOURNAL OF THE AMERICAN SOCIETY FOR INFORMATION SCIENCE AND TECHNOLOGY Auust DOI: 0002/asi

7 TABLE 3 Fictitious example r # Autors # Citations ence = 5 3 All te proved results are valid for tese fractional - and -indexes of a country Te same can be said about researc roups etc Te Inequalities, Proved Above, Are Optimal It is easy to sow tat all inequalities proved above (ie, Equations 6, 4, 9, and 24) are optimal and ence cannot be improved Tis is clear for te rit-and sides of te inequalities by takin any example were all papers are written by one autor Ten f = F = and f = F = For te left-and sides of te inequalities, we present te example in Table 3 (fictitious example) Here = 5 for all i =, 2, 3, 4, 5 Clearly = 5 Now, we ave tat f = F = = =,,, sowin tat te left-and sides of Equations 6 and 9 can be reaced Also = 5 and it is easy to see tat f = F = = =,,, sowin tat te left-and sides of Equations 4 and 24 can be reaced So, tis example sows tat te proved inequalities cannot be improved and tey support te matematical teory We close tis paper by presentin and discussin te,, f, f, F, and F values of te present autor (on January 8, 2007) Tis will sow tat te extreme differences between te fractional and non-fractional - and -indices, obtained in te above example, are not true in practice TABLE ) Calculation of te - and -index of L Ee (as of January # Aut r r 2 # Cit # Cit > < Te second column presents te new rankins accordin to te fractional citation scores (tird column) Note tat te fourt column presents te cumulative fractional citation scores accordin to te second column We find f = 2 and f = 8 Finally we calculate te - and -index F and F for fractional paper counts Note tat now te order of te papers does not cane (but teir rank values do!) See Table 6 From Table 6, it is clear tat F = and tat F = Te Indexes,, f, f, F and F forlee A searc in te Web of Science on January 8, 2007, yielded Table 4 for te present autor from wic and (unweited) can be determined (we stop at y T = 8 because we will not need more articles as will become clear furter on) We added te number of autors of eac paper for furter use (first column) It is clear from Table 3 tat = 4 and = 20 for te present autor Te fractional citation scores are presented in Table 5, sorted on te new ranks Conclusions and Suestions for Furter Researc In tis article, te - and -indexes of autors are extended to teir fractional versions Tis is done in two different ways One metod considers fractional citation counts were for eac m-autored paper, a citation count of s divided by m Te correspondin - and -index is denoted by f and f Anoter metod leaves te citation counts uncaned but replaces te ranks by te fractional paper count: A paper wit m autors adds a fractional rank of m Te correspondin - and -index is denoted by F and F 64 JOURNAL OF THE AMERICAN SOCIETY FOR INFORMATION SCIENCE AND TECHNOLOGY Auust 2008 DOI: 0002/asi

8 TABLE 5 Calculation of f and f r (old) r (new) Fract cit Fract cit >(8) <(9) TABLE 6 Calculation of F and F r (fract) # Cit # Cit >(22666) <(23666) 2 We present te matematical teory of f, f, F, and F in function of and (te non-fractional and index) and of te distribution of te number of autors in paper i =,,T (were tere are T papers) and i is determined by te number of citations, in decreasin order y,,y T We prove tat in all cases, f,, ([x] denotes, for every positive number x, te larest entire number tat is x) In a similar way but wit a different proof, we sow tat in all cases, f,, Furter, we sow tat in all cases, and F F wile we sow by example tat F < as well as F > are possible We also sow by example tat tese inequalities cannot be improved: All extreme values can be reaced We cannot perform better (by e, presentin functional equalities between tese fractional indexes and teir non-fractional counterparts and ) witout involvin te citation scores,i=,,t We leave it as an open problem to work out exact formulae for f, f, F and F in function of,, and (i =,,T), but tis is even an open problem for and in function of,i=,,t We remark tat te indexes f, f, F, and F and teir teories can also be applied to te case of fractional country scores, were te remain but can ave non-entire, rational values We close te paper by calculatin,, f, f, F, and F for tis autor and note tat te correspondin - and -values are not far away from eac oter, ereby sowin tat te extreme cases are not true in practice We encourae to apply tese indexes,, f, f, F, and F to autors (journals, countries, institutes, topics, ) of several fields and to draw conclusions concernin teir comparisons References Ball, P (2005) Index aims for fair rankin of scientists Nature 436, 900 Banks, MG (2006) An extension of te Hirsc index: Indexin scientific topics and compounds Scientometrics, 69(), 6 68 Bornmann, L, & Daniel, H-D (2005) Does te -index for rankin of scientists really work? Scientometrics, 65, Braun, T, Glänzel, W, & Scubert, A (2005) A Hirsc-type index for journals Te Scientist, 9(22) Ee, L (2005) Power laws in te information production process: Lotkaian informetrics Oxford, UK: Elsevier Ee, L (2006a) Teory and practice of te -index Scientometrics, 69(), 3 52 Ee, L (2006b) How to improve te -index Te Scientist, 20(3), 4 JOURNAL OF THE AMERICAN SOCIETY FOR INFORMATION SCIENCE AND TECHNOLOGY Auust DOI: 0002/asi

9 Ee, L (2006c) An improvement of te -index: Te -index ISSI Newsletter, 2(), 8 9 Ee, L, & Rousseau, R (2006) An informetric model for te Hirsc index Scientometrics, 69(), 2 29 Ee, L, Rousseau, R, & Van Hooydonk, G (2000) Metods for accreditin publications to autors or countries: Consequences for evaluation studies Journal of te American Society for Information Science, 5(2), Glänzel, W (2006a) On te H-index A matematical approac to a new measure of publication activity and citation impact Scientometrics, 67(2), Glänzel, W (2006b) On te opportunities and limitations of te H-index Science Focus, (), 0 (in Cinese) Hirsc, JE (2005) An index to quantify an individual s scientific researc output Proceedins of te National Academy of Sciences of te United States of America, 02, (Oriinal arxiv paper, arxiv:pysics/ v3, later corrected in arxiv:pysics/ v5) Popov, SB (2005) A parameter to quantify dynamics of a researcer s scientific activity arxiv:pysics/05083 Rousseau, R (2007) Te influence of missin publications on te Hirsc index Journal of Informetrics, (), 2 7 van Raan, AFJ (2006) Comparison of te Hirsc-index wit standard bibliometric indicators and wit peer judment for 47 cemistry researc roups Scientometrics, 67(), JOURNAL OF THE AMERICAN SOCIETY FOR INFORMATION SCIENCE AND TECHNOLOGY Auust 2008 DOI: 0002/asi