Characterisation of the χ -index and the rec -index
aa r X i v : . [ c s . D L ] J un Characterisation of the χ -index and the rec -index Mark Levene [email protected] , Trevor Fenner [email protected] , andJudit Bar-Ilan [email protected] Department of Computer Science and Information Systems,University of London, London WC1E 7HX, U.K. Department of Information Science, Bar-Ilan University, Ramat Gan, Israel
Abstract
Axiomatic characterisation of a bibliometric index provides insight into the propertiesthat the index satisfies and facilitates the comparison of different indices. A geometricgeneralisation of the h -index, called the χ -index, has recently been proposed to addresssome of the problems with the h -index, in particular, the fact that it is not scale invariant,i.e., multiplying the number of citations of each publication by a positive constant maychange the relative ranking of two researchers. While the square of the h -index is the areaof the largest square under the citation curve of a researcher, the square of the χ -index,which we call the rec -index (or rectangle -index), is the area of the largest rectangle underthe citation curve. Our main contribution here is to provide a characterisation of the rec -index via three properties: monotonicity , uniform citation and uniform equivalence .Monotonicity is a natural property that we would expect any bibliometric index to satisfy,while the other two properties constrain the value of the rec -index to be the area of thelargest rectangle under the citation curve. The rec -index also allows us to distinguishbetween influential researchers who have relatively few, but highly-cited, publicationsand prolific researchers who have many, but less-cited, publications. Keywords: h -index, χ -index, rec -index, bibliometric index, core publications, quantity versusquality, axiomatic characterisation Axiomatic characterisation of a bibliometric index [Mar09, BM14] provides insight into theproperties that the index satisfies and facilitates the comparison of different indices. (Ax-iomatic characterisation is used in a number of areas for the same purpose.)Since Garfield’s foundational work in bibliometrics [Gar79], a plethora of bibliometricindices have been suggested [RB15, TB16], from a simple count of the total number of ci-tations to the more sophisticated h -index [Hir05]. These have often been motivated by theongoing debate between quantity (as measured by the number of publications) and quality(as measured by the number of citations to those publications). A review of many of these[WSL14], where a comparison of 108 bibliometric indicators was presented, concluded that,in order to gauge the overall impact of a researcher, several indicators should be used. Inparticular, many variants of the h -index have been proposed and, in [BMHD11], it was shownthat generally there is high correlation between the h -index and 37 of its variants. Moreover,1 critical view of the h -index and its variants, which was provided in [Sch18], argued that itis not a good indicator of recent impact and suggested instead a timed h -index over a slidingtime window. Another criticism of the h -index and its variants is that it treats all citationsin the same way, ignoring the relevance and impact of citations. This issue can be addressedby Markovian methods, such as PageRank [Lev10], giving rise to indices based on authoror publication-level citation networks, such as the Eigenfactor score [WJD + h -index,called the χ -index [FHLB18], which has recently been proposed to address some of the prob-lems with the h -index, in particular the fact that it is not scale invariant [PR16].The h -index is determined by the largest square that fits under the citation curve ofa researcher when plotting the number of citations to individual publications in decreasingorder. On the other hand, the χ -index is determined by the largest area rectangle that fitsunder the citation curve. The rec -index (or rectangle -index) is defined to be the area of thisrectangle and is the square of the χ -index. The rec -index is thus a member of the class ofgeometric indices that approximate the area under the citation curve of a researcher. Suchindices can address problems attached to, for example, the citation count, which takes intoaccount all citations. The h -index penalises both highly-cited publications and publicationswith only a few citations. On the other hand, the χ -index is more balanced than the h -indexin this respect, as it allows us to cater for both influential researchers with a few very highly-cited publications and prolific researchers who may have many publications but relatively fewcitations per publication.Our main contribution here is to provide a first characterisation of the rec -index viathree properties: monotonicity , uniform citation and uniform equivalence . Monotonicity is anatural property that we would expect any bibliometric index to satisfy, while the other twoproperties relate to the rectangle under the citation curve that determines the index. Uniformcitation specifies that when the shape of the citation curve is uniform, i.e. rectangular, thenthe value of the index is the total citation count of all the publications. Complementingthis property, uniform equivalence specifies that when the shape of the citation curve is notuniform, i.e. not rectangular, the value of the index is equal to that of some uniform citationcurve that can obtained by omitting some number of citations.We note that many different properties may be used to characterise a bibliometric index,and there is no general agreement on which are the most compelling. For example, there are anumber of distinct characterisations of the h -index, such as those presented in [Woe08b] and2Que11a]. However, if a number of properties are proved to characterise a given index, theacceptance of any other index would necessitate the violation of at least one of the propertiesused to characterise the given index.The rest of the paper is organised as follows. In Section 2, we introduce the rec -index.In Section 3, we define and discuss properties of the rec -index and other indices. Then, inSection 4, we present an axiomatic characterisation of the rec -index. Finally, in Section 5,we give our concluding remarks. rec -index and related bibliometric indices We assume that a researcher publishes n publications, where n ≥
0, which are represented bya citation vector of positive integers, x = h x , x , . . . , x n i , where x i is the number of citationsto publication i , sorted in descending order, i.e. x i ≥ x j for 1 ≤ i < j ≤ n . (We note that onecould consider only a subset of a researcher’s publications in the citation vector, for example,by only allowing journal publications, or publications in high-impact venues.)The citation curve is the curve arising from plotting the number of citations against theranking of the publications as a histogram specified by the citation vector.A bibliometric index is a function f that maps citation vectors to the set of non-negativereal numbers. As in [Woe08b], we assume the baseline condition that, for the empty citationvector x = hi , we have f ( x ) = 0.In this paper, we concentrate on characterising the rec -index directly, following the ap-proach adopted in [Woe08b, Woe08a, Que11a, Que11b], rather than characterising the bib-liometric ranking induced by the index, as was done in [Mar09, BM14]. This stems from ourparticular interest in the properties of geometric indices. It is evident that any two indices,such as the χ -index and the rec -index, that are monotonic transformations of each other areequivalent with respect to the induced rankings.The citation count index for a citation vector x = h x , x , . . . , x n i is the L1 norm of x ,denoted by k x k and defined by k x k = n X i =1 x i . (1)We say that a citation vector x = h x , x , . . . , x n i is dominated by a citation vector y = h y , y , . . . , y m i , written as x ⊑ y , if n ≤ m and x i ≤ y i for all i , 1 ≤ i ≤ n .It is worth recalling some known bibliometric indices [WSL14, TB16]: the citation count,as defined in (1); the publication count n ; the maximum citation index x ; the Euclideanindex E ( x ) [PR16], which is the Euclidean norm of x , i.e., E ( x ) = qP ni =1 x i ; the h -index[Hir05]; and the g -index [Egg06], which is a variant of the h -index giving extra weight tohighly-cited publications.The h -index [Hir05], in particular, has gained popularity due to its relative simplicity,ease of calculation, and its ingenious method of combining the quality and quantity of aresearcher’s outputs. It is defined as the maximum number h of the researcher’s publicationssuch that each has at least h citations, i.e. for a citation vector, x = h x , x , . . . , x n i the h -index is the largest h for which x h ≥ h . 3o motivate the χ -index, consider the following three citation vectors, the citation curvesof which are depicted in Figure 1: (i) h i , i.e. 1 publication with 100 citations, (ii) h , , · · · , i , i.e. 10 publications with 10 citations each, and (iii) h , , · · · , i , i.e. 100publications with 1 citation each. (Note that the diagram in Figure 1 is not drawn to scale.)Figure 1: Example of the geometric interpretation of the h and χ indices.The χ -index is defined as the square root of the maximum area rectangle that can fitunder the citation curve, while the h -index is the square root of the maximum area squarethat can fit under the citation curve.Formally, we first define the rec -index (or rectangle -index) of a researcher with citationvector x by rec ( x ) = max i ix i . (2)The χ -index [FHLB18] is then defined by χ ( x ) = p rec ( x ).Returning to our example shown in Figure 1, we see that all three researchers have a χ -index of 10, while researcher (ii) has an h -index of 10, but researchers (i) and (iii) bothhave an h -index of only 1. The h -index may be seen as balancing quality, on the one hand, by4avouring publications with a higher number of citations and quantity, on the other hand, bytaking into account all publications with a sufficient number of citations. However, such anapproach disadvantages a researcher, such as (i), with a few very highly-cited publications,who may have carried out some influential seminal research, and it also disadvantages aprolific researcher, such as (iii), who may have many publications but fewer citations perpublication. Now, if we let k denote a value of i that maximises ix i in (2), the rec -indexcan distinguish between more influential researchers for which x k > k , such as (i), and more prolific researchers for which k > x k , such as (iii). In this sense the χ -index avoids the debateof number of citations versus number of publications by awarding all three researchers thesame χ -index of 10.The rec -index is a member of the class of (two-dimensional) geometric indices, as is thesquare of the h -index, and also the half the square of the w -index [Woe08b], which is the areaof the maximal isosceles right-angled triangle under the citation curve; formally, the w -indexis the largest integer w such that the citation vector x contains w distinct publications with atleast 1 , , . . . , w citations, respectively. In a more general setting, the dimension of an indexcan be formally defined [Pra17], and is related to the requirement of dimensional homogeneity from physics, that we may only compare quantities that have the same units.Geometric indices are actually quite natural, as their goal is to consider the area underthe citation curve in order to encapsulate the essential citations for a set of core publicationsthat in some sense represent the output of a researcher. We note that the rec -index, thesquare of the h -index, the publication count, and the maximum citation index all include thesame number of citations for each core publication.The citation count, which includes all publications with at least one citation in the core, isof course a reasonable bibliometric index. However, it is often argued that the citation countis problematic; in particular, it may be inflated by a small number of publication having alarge number of citations or it may be sensitive to a long tail of publications each having onlyfew citations. The axiomatic characterisation we describe here can be viewed as contributingto this debate by discussing several characteristics of geometric indices. In this section we define a variety of properties of bibliometric indices, almost all of which aresatisfied by the rec -index, and then, in Section 4, we show that monotonicity combined with uniform citation and uniform equivalence characterise the rec -index.We assume throughout this section that f ( · ) is the index under consideration. The firstproperty, monotonicity, is a natural requirement for any bibliometric index, stating thatadding citations to the citation vector should not decrease the value of the index: Monotonicity (M):
If citation vector x is dominated by citation vector y , i.e. x ⊑ y ,then f ( x ) ≤ f ( y ).It is easy to verify that all the indices we consider are monotonic. We note that the citationcount satisfies the stronger property of strict monotonicity ( SM) , in which f ( x ) < f ( y ) when x = y . It also satisfies several other desirable properties, such as rank independence (addinga new publication with a given number of citations to two citation vectors does not changetheir relative ranking) and rank scale invariance (multiplying the number of citations of each5ublication by a positive constant does not change the relative ranking of two citation vectors)[PR16]. Neither the χ -index nor the h -index are rank independent. However, the χ -index isrank scale invariant, but the h -index is not. Moreover, the rec -index satisfies the followingstronger property of (linear) scale invariance: Scale invariance (SI): f ( C x ) = Cf ( x ), for any positive constant C .It is easy to see that scale invariance implies rank scale invariance.A natural form of symmetry can be attained via the conjugate partition of a citation vector x = h x , x , . . . , x n i , which is the publication vector p = h p , p , . . . , p m i , where m = x and p i is the number of publications with at least i citations [Woe08b]. Geometrically, the publicationvector is obtained by reflecting the geometric representation of the citation vector along themain diagonal. This is shown in Figure 2 for a citation vector h , , , i , on the left, andits conjugate partition publication vector h , , , , , i , on the right. This motivates thefollowing property.Figure 2: The citation vector is shown on the left and its conjugate partition, the correspond-ing publication vector, is shown on the right. Self-conjugacy (SC):
Let p be the conjugate partition of the citation vector x , then f ( p ) = f ( x ).Clearly the rec and h indices, as well as the citation count, are all self-conjugate. On theother hand, we note that the publication count and the maximum citation index are conjugatesof each other. Self-conjugacy implies a balanced approach between influence (quality) andprolificity (quantity).Some indices tend to emphasise influence, for example, the maximum citation index, Eu-clidean index and g -index, whereas others, such as the publication count, emphasise prolificity.Should we wish to emphasise influence rather than prolificity, we may define a version of the rec -index, the rec I -index, in which we restrict the maximum in (2) to be over those i forwhich i ≤ x i . Conversely, should we wish to emphasise prolificity, we may instead define6he conjugate index, the rec P -index, by using the corresponding publication vector p andrestricting i so that i ≤ p i .Many people, probably the majority, tend to favour indices that emphasise influence. Itis therefore worth noting our findings in [FHLB18], where the citations of a large numberof researchers, from a Google Scholar data set made available by Radicchi and Castellano[RC13], were analysed and their rec -indices calculated. Table 11 in [FHLB18] shows that93% of the researchers for which the χ -index was significantly larger than the h -index weremore influential than prolific, i.e. for these researchers x k > k . This indicates that, in general,the rec -index satisfies the tendency to favour influence.Following the conclusion in [WSL14] that several bibliometric indicators should be usedto gauge the overall impact of a researcher, we suggest that both quality and quantity maybe assessed by using the pair of indices ( rec I , rec P ).We now concentrate on the rec -index. A typical citation curve, corresponding to thecitation vector h , , , i , is shown in Figure 3, with circles indicating where a new citationcan be added. We observe that each addition of a new citation completes a rectangle. Ifthe new citation is added to publication k , then the newly formed rectangle has width k and height x k + 1. For example, if we add a citation to the third publication, producing thecitation vector h , , , i , then the rec -index will increase from 9 to 12, but if we add it tothe fourth publication, producing the citation vector h , , , i , then the rec -index will notincrease.Figure 3: A citation vector with circles indicating where a new citation may be added.For any citation vector x = h x , x , . . . , x n i , we write x [ k ] for the citation vector obtainedfrom x by adding a single citation to publication k , where 1 ≤ k ≤ n + 1, thereby increasingits citation count from x k to x k + 1. If k = n + 1, we assume that x k = 0. We note that k must be the smallest index j for which x j = x k .By the definition of the rec -index, it is straightforward to see that rec ( x [ k ] ) ≥ k ( x k + 1),7nd thus rec ( x [ k ] ) = max ( rec ( x ) , k ( x k + 1)) . (3)The following property encapsulates this observation. Rectangle completion (RC):
For any citation vector x , f ( x [ k ] ) = max ( f ( x ) , k ( x k + 1)) . Noting that rectangle completion implies monotonicity , it is then straightforward to verifythat a bibliometric index f satisfies rectangle completion (together with the baseline condition)if and only if it is the rec -index. Obviously, as it essentially encapsulates the definition of the rec -index, rectangle completion is not particularly useful as a characterisation of the rec -index;however, it provides an alternative and constructive definition of the index.Consider the situation when rec ( x ) = kx k . We note that (i) if x k = k , the rec -index isequal to the square of the h -index, (ii) if x k > k , the researcher tends towards being moreinfluential, and (iii) if x k < k , the researcher tends towards being more prolific. Thus theshape of the maximum area rectangle will indicate an interpretation of the index value. Wealso note that, when the histogram of the citation curve is a rectangle, the distribution ofcitations is uniform . In this case the core includes all publications.More formally, we say that a citation vector u = h u , u , . . . , u n i is uniform if u = u = · · · = u n . It follows that rec ( x ) = k x k if and only if x is uniform. This observation suggeststhe following weaker form of this property. Uniform citation (UC):
If the citation vector x is uniform then f ( x ) = k x k .This property makes the reasonable assertion that, if all publications have the same num-ber of citations, they should all be in the core and all citations to them included in theindex.As stated above rectangle completion is too contrived in the sense that it mimics thedefinition of the rec -index. So we now explore a way to replace it with uniform citation ,which gives a lower bound on the index, together with another property that gives an upperbound on the index. This additional property is: Uniform equivalence (UE):
For any citation vector x , there exists a uniform citationvector u dominated by x , i.e. u ⊑ x , for which f ( x ) = f ( u ).This property asserts that the same number of citations should be included for eachpublication in the core.The following proposition, which defines a property that is similar to Axiom D in [Woe08b],can easily be shown to follow from monotonicity , uniform citation and uniform equivalence . Proposition 3.1. If f satisfies properties M , UC and UE then it also satisfies the followingproperty: Citation increase (CI):
If we add a single citation to each publication in x , resultingin the citation vector y , then f ( y ) > f ( x ) . citation increase together with monotonicity and uniform citation do not imply uniform equivalence , as is the case for the bibliometric index, f ( x ) = rec ( x ) + k x k , (4)which satisfies citation increase but not uniform equivalence . rec -index The rankings induced by several bibliometric indices, including the h -index and g -index, werecharacterised in [Mar09, BM14], whereas, in [Woe08b, Woe08a] and [Que11a, Que11b], theauthors concentrated on characterising the h -index and g -index directly. These characteri-sations, as well as that presented here, are based on properties that address three types ofissues. First, the inclusion of fundamental properties like baseline and monotonicity shouldbe considered. The second issue is concerned with the conditions under which the value of theindex increases (see, for example, citation increase ). The third issue considers what changes tothe citation vector leave the index unchanged (for example, the h -index satisfies the propertyof independence of irrelevant citations [Que11b], cf. [Kon14], which essentially states thatadding a single citation to a core publication does not increase the index). Another categoryof properties that are important for characterising bibliometric indices are invariants, suchas scale invariance and self-conjugacy , which give transformations that change the index in apredictable manner or do not change the value at all.The main result in this section is Theorem 4.1, which provides an axiomatic characterisa-tion of the rec -index. We also show in Proposition 4.5 that the χ -index satisfies the desirableproperty that, subsequent to the addition of a single citation to the citation vector, the χ -index cannot increase by more than one. Theorem 4.1.
A bibliometric index f satisfies the three properties of monotonicity, uniformcitation and uniform equivalence if and only if it is the rec -index.Proof. It is clear from Section 3 that the rec -index satisfies these properties, so it remains toprove that they are sufficient.Let x be a citation vector and let u be a uniform citation vector such that k u k is maximalfor all u ⊑ x . Clearly, rec ( x ) = k u k . So, by monotonicity and uniform citation , we have f ( x ) ≥ f ( u ) = k u k . By uniform equivalence , we may let v be a uniform citation vector suchthat v ⊑ x and f ( x ) = f ( v ), and therefore f ( x ) = k v k by uniform citation . Since k u k ≥ k v k by the definition of u , it follows that f ( x ) = k u k = rec ( x ).The following two corollaries show that we may replace two of the properties in Theo-rem 4.1 by two simpler properties together with the intuitive property of scale invariance . Corollary 4.2.
The result of Theorem 4.1 holds if we replace monotonicity by the followingmore restrictive property.
Uniform monotonicity (UM): If x is uniform and x ⊑ y , then f ( x ) ≤ f ( y ) . orollary 4.3. The result of Theorem 4.1 holds if we replace uniform citation by the followingmore restrictive property together with scale invariance.
Uniform single citation (USC): If x = x = · · · = x n = 1 for the citation vector x ,then f ( x ) = k x k = n .Proof. Clearly
USC and SI imply UC .It is not difficult to demonstrate that the three properties of Theorem 4.1 characterisingthe rec -index are independent, i.e. omitting any one of them would render the theorem false. Proposition 4.4.
The three properties of Theorem 4.1 characterising the rec -index are in-dependent.Proof.
The following examples justify this claim.a) The index defined in (4) satisfies M and UC but not UE .b) The square of the h -index, the publication count n , the maximum citation index x , max ( n, x ) and min ( n, x ) all satisfy M and UE but not UC .c) The product nx n of the number of publications and the minimum number of citationssatisfies UC and UE but not M , nor UM .To summarise the properties that characterise the rec -index: UM or M implies thatadding new citations will not decrease the value of the index, while UC and UE provide,respectively, lower and upper bounds on its value. We recall that, since the rec -index is notstrictly monotonic, some citations may not contribute to the value of the index; however, allcore publications contribute an equal number of citations.We now present a construction that gives rise to a property equivalent to UE . Whileindices, such as the citation count (1), that include the full set of citations are strictly mono-tonic, indices, such as the rec -index, that include just a core set of publications typicallyincrease only after a batch of citations has been added. The construction we now presentcaptures this aspect of the rec -index.Consider a sequence of citation vectors S = x , x , . . . , x i , . . . , x s , (5)where x = hi and x s = x . When x i ⊑ x i +1 and k x i +1 k − k x i k = 1, for 1 ≤ i ≤ s −
1, we saythat S is a constructive sequence for x .We are interested in constructive sequences satisfying the property that, for each i , if f ( x i ) < f ( x i +1 ) then x i +1 is a uniform citation vector; we call such a constructive sequence f - incremental . This suggests the following property of a citation index. Uniform increment (UI):
For any citation vector x , there exists an f - incremental constructive sequence for x . 10t is not difficult to prove that UI implies UE . Moreover, the rec -index satisfies UI , andwe now present one method for constructing a rec - incremental sequence for a citation vector x .(i) Start from hi and construct a sequence of uniform citation vectors as follows.(ii) From the uniform citation vector u , add citations one-by-one to obtain a new uniformcitation vector, either by adding a new column (i.e. a new publication) or by adding anew row (cf. property CI ) to the rectangle corresponding to u .(iii) Repeat step (ii) until we obtain a citation vector u for which rec ( u ) = rec ( x ).(iv) Add the remaining citations one-by-one in any order until we obtain x .It is straightforward to show that it is always possible, in step (ii) above, to choose betweenadding a column or a row in such a way that the sequence is rec - incremental .It may be argued that it is natural that a one-dimensional bibliometric index should notincrease by more than one when a single citation is added to the citation vector. We nowprove that this holds for the χ -index. Proposition 4.5.
Let x be a citation vector and let x [ k ] be a citation vector obtained byadding a single citation to x . Then χ ( x [ k ] ) ≤ χ ( x ) + 1 .Proof. We recall that x [ k ] is obtained from x by adding a single citation to publication k ,thereby increasing its citation count to x k + 1. If rec ( x [ k ] ) = rec ( x ) the result holds trivially.So we may assume that rec ( x [ k ] ) = rec ( x ), in which case rec ( x [ k ] ) = k ( x k + 1) by (3).Now, since x [ k ] is a citation vector, we must have rec ( x ) ≥ max (cid:0) kx k , ( k − x k + 1) (cid:1) . Therefore, rec ( x ) ≥ rec ( x [ k ] ) − min ( k, x k + 1) ≥ rec ( x [ k ] ) − p k ( x k + 1) , and thus rec ( x [ k ] ) − rec ( x ) ≤ χ ( x [ k ] ) . It follows that χ ( x [ k ] ) − χ ( x ) ≤ χ ( x [ k ] ) χ ( x [ k ] ) + χ ( x ) ≤ . . Finally, we note that if it is required that the χ -index be an integer, then the ceilingfunction, which maps χ ( x ) to the least integer greater than or equal to χ ( x ), can be employed.11 Concluding remarks
Geometric indices, such as the rec -index, capture a core set of the publications that representa researcher’s total output. The χ -index (which is equal to the square root of the rec -index)can be viewed as a generalisation of the h -index, and has the advantage that it allows us todistinguish between more influential and more prolific researchers, depending on whether theheight of the largest area rectangle under the citation curve is greater than or less than itswidth, respectively.We presented several properties that are satisfied by the rec -index and proved, in Theo-rem 4.1, that the three properties of monotonicity , uniform citation and uniform equivalence characterise the rec -index. While monotonicity is a very natural property for any bibliometricindex, uniform citation and uniform equivalence are natural when it is required to include thesame number of citations for each publication in the core.Geometric indices, such as the rec -index, give better insight into the relationship betweeninfluence (quality) and prolificity (quantity) than indices, such as the h -index, that are moreconstrained in this respect. Acknowledgements
The authors would like to thank the reviewers for their constructive comments, which havehelped us to improve the paper.
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