h-index and its alternative: A Review

11/08/2018 ∙ by Anand Bihari, et al. ∙ NIT Patna 0

In recent years, several Scientometrics and Bibliometrics indicators were proposed to evaluate the scientific impact of individuals, institutions, colleges, universities and research teams. The h-index gives a major breakthrough in the research community to evaluate the scientific impact of an individual. It got a lot of attention due to its simplicity and several other indicators were proposed to extend the properties of h-index as well as to overcome shortcomings of h-index. In this literature review, we have discussed the advantages and limitations of almost all Scientometrics as well as Bibliometrics indicators which have been categorized into seven categories :(i) Complement of h-index, (ii) Based on total number of authors, (iii) Based on publication age, (iv) Combination of two indices, (v) Based on excess citation count, (vi) Based on total publication count, (vii) Based on other variants. The main objective of this article is to study all those indicators which have been proposed to evaluate the scientific impact of an individual researcher or a group of researchers.

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1 Introduction

Research is a cyclic process to spread knowledge and innovation in the society for its betterment. The one of the most important product of research is the research articles, that express the knowledge and working process of the proposed model. A scholar first studies the previously deployed methodology. He then looks for ways in which the acceptability of existing methods can be improved or, proposes a new methodology that overcome the limitation(s) of existing ones. His final outcome is published in the form of a research article. While writing the research article, the scholar refers published research articles. The acceptability of the new technique or research article is measured in terms of citation count (that is, how many research articles refer it) and the impact of the scholar is measured in terms of total citation count earned by all published articles, in which the scholar is present as a author or as a co-author.

During the last few decades, it seems that the evaluation of the scientific impact of a scholar or a group of scholars was a significant assignment. In many cases, it is compulsory to know the scientific impact of a scholar, for example, at the time of hiring a new faculty member, promotion of faculty members, continuation of research grants, etc. To do this, mostly they used either the total number of publications, total citation count, average citation count or citation count per publication. It is not uncommon to find a scholar who has published high number of articles, but has less scientific impact (i.e, citation count) than a scholar who has published less number of articles but has a greater scientific impact (citation count). Similarly, a scholar can gain more number of citations with the help of a few articles, however another scholar can gain similar number of citations in a distributed manner. In such a case both scholars are treated equally. As similar to the total publication and citation count, the average number of citation counts may be reflected by the single highly influenced article. However, the total publication count, the total citation count and the average citation count may not be a good measure, because they do not consider the overall impact and the productivity of scholars, and hence, also fail in comparing the scholars’ scientific impact. Thus, we can not use the total publication count, total citation count and average citation count as a measure to assess the scientific impact of scholars. We need a more fine grained approach that consider more number of attributes, because nowadays the research assignment, project grant, faculty promotion, award distribution etc. related decisions are made based on the individual scientific excellence or the performance of the individual in the group of scholars. Several publication indexing databases such as Google Scholar (walters2007google), ISI Web of Science, Scopus, DBLP, PubMed, SciELO, CiteSeer, CiteSeerX and many more are available to manage the research publications of scholars. To validate the properties of proposed indicators, scholars use these publication databases (bar2008informetrics).

The h-index was proposed by hirsch2005index, gives a major breakthrough in the scientific community to assess the scientific impact of scholars and gained a lot of attention due to its simplicity. The h-index covers the productivity and the impact of scholars and is better than the total number of publications and mean number of citation counts lehmann2006measures; hirsch2007does. But the h-index does not consider the impact of excess citation count and leaves a huge amount of citation count unaccounted. It also fails in comparing the scientific impact of scholars have similar index value. Based on the limitations of the h-index, several other indicators were proposed to overcome the shortcoming of h-index and enhance the scientific evaluation process with more variability. The h-index shows the popularity of a scholar, but it does not mean that the scholar is more prestigious. Because the h-index does not exclude the self citation count and does not differentiate the source of the citation. The citation counts from different type of articles such as Patents, Journal articles (reputed or non-reputed), conference proceedings and book chapters are treated equally. However, the citation sources and the citation count have an impact on the quality of citation, which is generally not considered ding2011popular.

Nowadays, the increase in collaborations between scholars affect the scientific productivity and the impact of research publications, because, generally, the research projects are too large and interdisciplinary. In this case, if we consider the h-index as an assessment tool, then every scholar in multi-authored articles gets full credit of its citation count. However the contribution of all scholars are not equal, so we can not distribute citation equally to all scholars. In this context, several indices were designed to give credit to all scholars and evaluate the scientific impact of scholars based on their credit. cole1973social considers only the first author, multiple1981price gives equal fractional credit to all scholars, i.e. 1/k, where k is the total number of scholars. hodge1981publication, sekercioglu2008quantifying and hagen2010harmonic gives credit to all scholars as per their proportional rank. van1997fractional and trueba2004robust used the arithmetic counting for credit allocation, egghe2000methods used the geometric series for credit allocation and several other methods are used to distribute shared-credit between all scholars. Most of the indices used the mathematical equation to share the credit among scholars, but we cannot express the contribution of scholars in the form such a mathematical expression.

In scientific assessment of scholars, the h-index considers only the few highly cited articles. However the articles having at least one citation count have significance in scientific assessment of scholars. In this context garcia2009multidimensional designed a multidimensional h-index that covers all cited articles. Several other indices use different mechanism to consider the impact of all cited articles in scientific assessment of scholars.

It seems that any one indicator does not fulfil all the requirements of scientific assessment process of a scholar. Hence, a combination of two or more different types of indicators is required. The combination of the properties of two or more different measures is a good step to evaluate the scientific impact of individuals martin1996use; van2003holy. In this context hg-index alonso2010hg, -index cabrerizo2010q2 and the -index liu2012modifying) were proposed.

One of the main issues related to most of the proposed indices is that the proposed indices do not consider the publication consistency issue. In some cases, it seems that the some of the scholars have publications only during their doctoral research, post-doctoral research and when working on the research projects. However, some of the scholars publish articles continuously. This issue do not affect the h-index, but has significance in scientific assessment. In this context, the career year h-indices were designed by mahbuba2012diffusion.. The main objective of this article is to make an extensive literature review on h-index and its variants to focus on following points:

  • The definition of h-index along with its advantages and limitation.

  • Literature review on the variants of h-index which are based on different parameters such as total number of citation count, total publication age, total number of collaborators, normalize citation count based on number of co-authors, total number of citers (citing authors) and many more.

In this article, section 2 presents the definition of the h-index along with its advantages and limitations. In section 3 contains the definition of the variants of the h-index along with its advantages and limitations. In section 4, we draw the conclusion of this review work.

2 h-index

To apprise the scientific impact of scholars, several publication based indicators such as the total number of publications, total citation count, and the average number of citations per paper can be used. However, these indicators have limitations. For example, when considering the total number of publications, a scholar can publish a number of papers, but still have a low scientific attraction in the research community. When considering the total citation count, only a few articles with very high citation counts can hide the very low citation count values of the vast majority of the published articles. When considering the average number of citations per paper, it does not capture the importance of the high impact articles. Based on the limitations of these indicators, Hirsch hirsch2005index proposed a new indicator called h-index. Formally, the h-index is defined as:

“The h-index of a scholar is h if h of his/her research articles have at least h citation count each and rest of the articles may have h or less citation count.”

The graphical representation of citation distribution of a scholar is shown in Fig.1.

Figure 1: Citation distribution of scholars

The graph based definition of h-index is that the h-index is the size of largest square fitted under the curve (i.e. citation count), that is the total number of publications which is under the largest square. The area above the largest square fitted under the curve represents the excess citation count of core articles and the area to the right of largest square fitted under the curve represents the total citation count of tail articles, which are not used in h-index computation.

For calculation of h-index, first the publications are arranged in the descending order of their citation count. The publications are then assigned a rank based on their order in the sorted list. The h-index is the maximum rank a publication where the citation count is equal to or greater than the rank of publication. This index almost covers the productivity and impact of scholars.

2.1 Advantages and limitations

The advantage of h-index are the following:

  1. It is too simple to compute hirsch2005index and does not require any data processing franceschini2010analysis.

  2. It produces a single number that combines both the quality and the quantity of the scholars’ publications hirsch2005index.

  3. It performs better than the total number of articles, total citation count, average citation count, citation per articles, number of highly cited articles, journal citation score & field citation score in scientific assessment of scholars hirsch2005index; van2006comparison.

  4. It can be easily obtained from any publication indexing databases such as Google Scholar, Web of Science etc. hirsch2005index; alonso2009h; hirsch2014meaning.

  5. It is closely related to the total number of publications that have significant influence liu2009properties.

  6. A small error in the citation distribution does not result in a huge change in the index value rousseau2007influence; vanclay2007robustness.

  7. A single highly cited article does not affect the index value egghe2009mathematical. Further, small changes in the citation count of articles do not affect the index value much hirsch2014meaning.

  8. It is also useful in the assessment of the impact of a journal mingers2012using.

On the other hand, it suffers with some of the shortcomings, which had been addressed by Hirsch himself and other scholars:

  1. The citation practice of articles is different in different fields, so it is not useful in comparing the scientific impact of scholars of different fields hirsch2005index; schreiber2007self; liu2009properties; waltman2011inconsistency; waltman2012inconsistency.

  2. It is also not suitable in comparing the scientific impact of scholars’ having different research careerskelly2006h.

  3. Once an article is selected for h-core, further citation will not be important in scientific assessment egghe2006theory; egghe2006improvement.

  4. It is very difficult to collect complete citation information of scholars.

  5. It also considers the self citation count in h-index computation.van2006comparison.

  6. It produces a single natural number, that affects the discriminative power of h-index tol2008rational; liu2014empirical when comparing two scholars with the same h-index value.

  7. The index value never decreases jin2007r.

  8. Generally, a research article is written by a group of scholars who rarely contribute equally. However the h-index gives full credit of citations to all scholars burrell2007should. So it is not fair to evaluate the scientific impact of scholars.

  9. It does not give any extra credit to highly cited articles van2008generalizing; zhang2009index; Bihari2017.

  10. It completely ignores the impact of h-tail articles, whereas some of the h-tail articles’ citation counts are equal or very close to the h-index value garcia2009multidimensional; Bihari2017.

  11. The publications inconsistency also does not affect the index value mahbuba2012diffusion; Bihari2018.

To overcome the above mentioned limitations, a lot of research has been done by the scholars to provide an efficient alternative to assess the impact of scholars. Some scholars used the properties of h-index and give an effective alternative measure. The alternative measures of h-index are categorized in the following seven categories:

  1. Complement of h-index (Section 2.2).

  2. Based on publication age (Section 2.3).

  3. Based on total number of authors (Section 2.4).

  4. Combination of two indices (Section 2.5).

  5. Based on excess citation count (Section 2.6).

  6. Based on the total number of publications (Section 2.7).

  7. Other types of indices (Section 2.8).

    1. Based on the core tail ratio (Section 2.8.1).

    2. Based on improving h-index to higher values (Section 2.8.2).

    3. Based on variants of citation process (Section 2.8.3).

    4. Miscellaneous indices (Section 2.8.4).

2.2 Complement of h-index

As discussed in the above section, the h-index does not give any extra credit to highly cited articles. To overcome this shortcoming of the h-index, several research complemented the h-index and gave alternatives to assess the scientific impact of scholars with the consideration of the impact of the highly cited h-core articles, the total number of h-core articles, the publication age and the total number of publications. The publication age is the number of years since the first publication of an author.

The citation count of an article shows the scientific impact of that article and the highly cited articles play an important role in the scientific assessment of scholars. However, the h-index considers only the h number of citation count from the h-core articles, whereas many h-core articles have more than h-citation count. To overcome this big-hit problem of h-index, the g-index has been proposed by egghe2006theory. The g-index gives more credit to highly cited articles as well as includes maximum number of articles citation in scientific assessment. Formally, it is defined as:

“The g-index of a scholar is the largest number g such that the top g articles have at least or more citations together.”

The main objective of this index is to overcome the big hit problem of h-index, that is, “once an article is selected in h-core, further citation is not countable in scientific assessment” bornmann2008there. The main difference between the h-index and the g-index is that the latter is based on the cumulative citation count, whereas the former is based on individual citation count of articles. The small number of highly cited articles give the highest g-index than the greater number of averagely cited papers alonso2010hg; this is also one of the main limitations of the g-index egghe2009econometric. To overcome the limitations of g-index, kosmulski2006new proposed a new index called h(2)-index, defined as:

“The h(2)-index of scholars is the highest natural number such that the h(2) most cited articles received at least citation each.”

For example, if the h(2)-index of a scholar is 10, it means scholar’s 10 publications have a minimum of 100 citations each. This index is almost similar to the g-index, the only difference being the method of calculation. The g-index is based on cumulative sum of citation count, whereas the h(2)-index is based on individual citation count. This index considers only the few highly cited articles and penalizes all those scholars who published articles having average citation counts. To overcome the limitations of h(2)-index, wu2008w proposed a new variant of h-index named w-index and it is defined as:

“The w-index of scholars is w such that the top w articles have at least 10w citations each.”

For example, if the w-index of a scholar is 5, it means that the scholar’s five articles have a minimum of 50 citations each. If the h(2) index of a scholar is 5, it means scholar’s five articles have at least 25 citation each. The w-index is more or less similar to the h(2)-index, but it is less strict than the h(2)-index. It requires only 10 times citation count increment to increase the index value to next. It penalizes all those young scholars who have just started working or those who do not have enough publications.

tol2009h mentioned that the h-index used only highly cited publications, the g-index overcome this shortcoming of the h-index, but it is sensitive to non cited articles. Based on this limitation, the author proposed two different indices called f-index and t-index based on harmonic and geometric average. The f-index of an author is defined as:

(1)

where is the citation count of article.

The f-index never goes beyond the total number of publications. This index gives higher weight to the least cited articles as compared to the highly cited articles. Based on this limitation, the author proposed a new index called t-index, defined as:

(2)

This index uses many properties of f-index. Both are comparatively difficult to calculate and their values lie between h and g-index. .

woeginger2008axiomatic proposed the other variants of h-index called w-index. This is almost similar to h-index.

“The w-index of a scholar is the largest value w for which their w articles have at least 1, 2, 3,…..w citation count.”
Mathematically, it is defined as:

(3)

where is the citation count of article and w is the maximum number of publications.

The main difference between h-index and the w-index is that the h-index describes the largest square area under the citation curve, whereas the w-index describe the largest isosceles right angle triangle under the citation curve.

The h-index can be used in comparing scientific impact of scholars xu2015new. However, the use of h-index in when comparing two scholars can be controversial, because the total number of core element and the citation count of core elements are often not same for two different scholars having a common h-index value. To overcome this limitation of h-index, xu2015new proposed a new variant of h-index called Gh-index. The Gh-index of a scholar , denoted , is defined as:

(4)

where is the total number of publications of scholar and is the h-index of the scholar.

Above mentioned indices consider only the highly cited articles, however, it seems that the number of highly cited articles and the citation count of the highly cited articles can be different for scholars with a common h-index value. The h-index fails in comparing the scientific impact of scholars in such a case. To overcome this limitation, jin2006h proposed a new index called A-index. Formally, it is defined as:

“The A-index of a scholar is the average number of h-core articles citation count.”

Mathematically, it defined as:

(5)

where A is the A-index of the scholar, h represents the h-index and is the citation count of the article.

In the case of high h-index with lower citation count of h-core articles, the A-index may be penalized due to division by h-index burrell2007h. In such a case, the one or two highly influential articles reflect the overall index value. So, instead of the average number of citation count, the square root of the sum of h-core articles is more pertinent. Based on this assumption, jin2007r proposed another index called R-index, which is further discussed in glanzel2007r. Formally, it is defined as:

“The R-index of scholars is the square root of the sum of h-core articles citation count.”

Mathematically, the R-index is defined as:

(6)

where h represents the h-index and is the citation count of the article.

However, the R-index penalizes all those scholars who have long h-cores. A bigger h-core may penalize the R-index and result in the possibility of getting a lower index value than the scholars having relatively smaller h-cores. To overcome this shortcoming, panaretos2009assessing proposed a new index called -index. Formally, it is defined as:

“The -index of scholars is the square root of the sum of square roots of the citation counts of the h-core articles .”

Mathematically, the -index is defined as:

(7)

where is the citation count of article.

But this index also suffers with the variability in the citation count of the h-core. If h-core articles have less variation, it results in high -index value. To handle variability in h-core citation count, the coefficient of variation (CV) can be use. Based on CV, the author proposed another indicator called . The -index is simply the subtraction of the CV score of the h-core articles from the -index. In this regard,bornmann2007b mentioned that it is very difficult to set the cutoff value for every author and none of the indicators addressed this issue. To overcome this limitation, bornmann2007b used the field specific reference standard for setting the cutoff value with the help of ESI (Essential Science Indicator) from Thomson Reuters and proposed a new index called b-index. Formally, it is defined as:

“The b-index of a scholar is b such that at least b articles belongs to the top 10% of the publication in a specific field.”

Mathematically, the b-index is defined as:

(8)

where,
is the b-index of scholar,
represents publication year,
represents the total publication count,
represents the citation count of article in year , and
is the 10% article published in year .

The main objective of this index is to identify the field specific prominent actor, because no one is good in all fields. It shows the field specific interest of scholars. But the main limitation of this index is the calculation process. It is difficult to calculate compared to the h-index.

The average and the square root of the citation count of the h-core articles has been used in scientific assessment of scholars in A-index and R-index respectively. Generally, it seems that the citation distribution of scholars’ article is skewed, therefore, instead of the average or the square root of the citation count, the median number of citation is much better to assess the scientific impact of scholars. In this way,

bornmann2008there proposed a new index called m-index. Formally, it is defined as:

“The m-index of a scholar is the median number of citation count of the h-core articles.”

The -index glanzel2010hirsch has also been discussed in the same way as the m-index.

egghe2008h state that the publication count and the citation count vary from scholar to scholar. The variation between the publication count and the citation count captures the sensitivity of performance of scholars. Based on the sensitivity of the performance changes, they proposed the weighted h-index egghe2008h. This index is almost similar to the r-index, but the difference is in the definition of h-core articles. In this index, instead of ranking an article based on its position in the sorted list, the authors used weighted ranking mechanism to rank the articles. The weighted rank of an article is defined as:

(9)

where is the h-index and is the citation count of the article.

The weighted rank of the article is the ratio of cumulative sum of the citation count of the top articles and the original h-index. Then, the weighted h-index of a scholar is the square root of the sum of the citation counts of the weighted core articles. Mathematically, it is defined as:

(10)

where, is the citation count of the article and is the largest rank among all publications such that the weighted rank .

In research community, the h-index is used in comparing the scientific impact of scholars. However, the h-index does not consider the research career of scholars. Hence the question arises: “How can we compare the scientific impact of scholars having same h-index but different research careers?” vaidya2005v argued that it is not fair to compare a scholar who gives 100% of his/her time with another scholar who gives only 40%, though both have the same h-index value. To overcome this limitation the v-index was introduced. The v-index of a scholar is the h-index value adjusted with the publication age. Mathematically, the v-index is defined as:

(11)

where:
is the h-index,
is the current year, and
is the year of first publication.

The presumption of h-index is that its value is almost equal to the career length. Based on this concept, burrell2007hirsch proposed a new index called m-quotient that can be used in comparing scientific impact of scholars with different publication age.

“The m-quotient of scholars is defined as: m=h/.”
where, represents the h-index and is the publication age. This index is almost similar to the v-index.

As discussed earlier, the h-index can be used in comparing the scientific impact of scholars. If we compare the scientific impact of two different scholars who do not have an equal number of articles, then it is not fair to use h-index as a comparing tool. To overcome this limitation of h-index, several new indicators complimented the h-index in the context of the total number of publications.

sidiropoulos2007generalized addressed the above mentioned limitation of h-index and proposed a new index called Normalized h-index. Formally, it is defined as:

(12)

where, represents the h-index and is the total number of articles.

Another similar approach called called v-index was proposed by riikonen2008national. Formally, it is defined as:

“The v-index of a scholar is the ratio of the h-index and the total number of publications.”

In recent years, the collaboration among scholars plays an important role in completion of the research work. hirch2010index mentioned that an article with 20 citations and published by a group of scholars is played equally in the h-index with other articles with 20 citations but published by single authors. However, the significance of both articles is not the same. Based on this scenario, the author proposed a new index called -index.

“The -index of scholars is such that the top k articles have at least k citations each and the co-authors of each article also have an h-index value of at least k.”
This index is very difficult to calculate because it requires article citation count as well as co-author’s h-index. Further, this index also penalizes articles published with collaborative efforts. Suppose an article got a good number of citations but the co-authors do not have h-index values equal to the citation count, then that article does not contribute to the -core. Whereas, an article that has less citation count, but the co-authors have h-index greater or equal to the citation count, belongs to the

-core. Another limitation of this index is that it drastically decreases after some time. If an author has collaborated with same age authors, then there is a probability that their

-index will decrease in future, whereas, had the author collaborated with younger scholars, their -index would have increased.

Table 1 shows the summary of the indices which complimented the h-index.

Index Definition Publication
g-index “The g-index of a scholar is the largest number g such that the top g articles have at least or more citations together.” egghe2006improvement
h(2)-index “The h(2)-index of scholars is the highest natural number such that the h(2) most cited articles received at least citation each.” kosmulski2006new
w-index “The w-index of scholars is w such that the top w articles have at least 10w citations each.” wu2008w
f-index , where is the citation count of article. tol2009h
t-index tol2009h
Woeginger w-index “The w-index of a scholar is the largest value w for which their w articles have at least 1, 2, 3,…..w citation count.” woeginger2008axiomatic
Gh-index xu2015new
A-index “The A-index of a scholar is the average number of h-core articles citation count.” A= ,where A is the A-index of the scholar, h represents the h-index and is the citation count of the article. jin2006h
R-index “The R-index of scholars is the square root of the sum of h-core articles citation count.” , where h represents the h-index and is the citation count of the article. jin2007r
-index “The -index of scholars is the square root of the sum of square roots of the citation counts of the h-core articles .” , where is the citation count of article. panaretos2009assessing
b-index “The b-index of a scholar is b such that at least b articles belongs to the top 10% of the publication in a specific field.” bornmann2007b
m-index “The m-index of a scholar is the median number of citation count of the h-core articles.” bornmann2008there
Weighted h-index , where, is the citation count of the article and is the largest rank among all publications such that the weighted rank . egghe2008h
v-index “ The v-index of a scholar is the h-index value adjusted with the publication age.” , where h is the h-index, is the current year, is the first publication year. vaidya2005v
m-quotient “The m-quotient of scholars is defined as: m=h/.” where, represents the h-index and is the publication age. burrell2007hirsch
Normalized h-index , where, represents the h-index and is the total number of articles. sidiropoulos2007generalized
Riikonen v-index “The v-index of a scholar is the ratio of the h-index and the total number of publications.” riikonen2008national
-index “The -index of scholars is such that the top k articles have at least k citations each and the co-authors of each article also have an h-index value of at least k.” hirch2010index
Table 1: Summary of Complement of h-index

2.3 Indices based on publication career

Generally, all citation based metrics are solely based on the number of publications and their citation counts. Whenever we compare the productivity of scholars, only the total number of publications and their citation count is considered. But the productivity of scholars are different at different stages of their careers. For example, a scholar who is retired or not active in research, but his/her articles are getting regular citations, is considered prominent. Whereas, a young scholar, who has published quite a few papers, but has a smaller number of citations (due to a small career) is not considered as prominent. Based on the above discussion, it is clear that the publication career of the scholars play an important role in the scientific assessment of scholars as well as in comparing the scientific impact of scholars.

On these lines, jin2007ar; jin2007r mentioned that if two scholars have an equal citation count of h-core articles with different publication careers, then the h-index of both scholars is the same. But this is not fair because their publication careers are different. To overcome this limitation of h-index, the AR-index has been proposed, which considers the total career of scholars. Formally, the AR-index of a scholar is defined as:

(13)

where and is the citation count and the age of the article respectively.

The main objective of this index is to give equal weightage to all the publications that are either published earlier or recently, and are useful in comparing the scientific impact of scholars having different lengths of publication career. But this index penalizes all those articles that were published earlier. The index value may decrease over time, but helps in estimating the recent scientific impact instead of the total scientific impact.

Sidiropoulos et. al. (2007)sidiropoulos2007generalized proposed the Contemporary h-index to give more credit to the citation of newer articles than the older ones. Formally, the Contemporary h-index is defined as:

“The contemporary h-index of a scholar is such that their articles have at least score each.”

The score of the article is defined as:

(14)

where,
is the current year,
is the publication year,
is the citation count of the article, and
are the coefficients set by the user.

If is 1, then the score of an article is the total citation count divided by the age of the article. It produces very small values that help to derive a new meaningful variant. The main objective of this index is to give more credit to recent articles rather than the older ones. However, the Contemporary h-index penalizes all those old articles that are continually earning citation till date. To overcome this limitation, sidiropoulos2007generalized proposed the Trend h-index to measure the current impact of scholars by the recent citation count of all articles. It is defined as:

“The trend h-index of a scholar is the largest number such that for his/her articles, each have a score .”

The score of the article is defined as:

(15)

where the symbols have the same meaning as defined in Eq.14.

This index requires year wise citations of all publications, which is one of the main limitations of the Trend h-index and makes its computation much more complex than the h-index. Another issue related to this index is the choice of and parameters. It is very difficult to assign a reasonable value for these parameters.

It is very difficult to differentiate between two scholars having equal h-index as well as equal citation counts in h-core articles. However, it seems that the h-index of some scholars remains unchanged for some time, while other scholar’s citation as well as h-index rises. In this context, the dynamic h-type-index () was designed by rousseau2008proposal. This covers the size of core articles and how that size changes with the time. Formally, it is defined as:

(16)

where,
R(y) is the R-index of the scholar at the career year, and
is the recent increment in the h-index for the year.

In this case the value of y is set by the user and it is very difficult to set a reasonable time window for the scientific assessment of a scholar. Instead of considering the total career of a scholar or any fixed time window, a decade based assessment is more precise. Based on this concept kosmulski2009new proposed the h-index per decade :

“The hpd-index of a scholar is hpd such that the hpd articles have at least hpd citation per decade () each.”

The adjusted citation per decade is defined as:

(17)

where is the decade year and it must be greater than the publication year.

Instead of the total time window or decade based scientific assessment, fiala2014current considers only recent three year’s citation time window. Pan, R. K. and Fortunato, S. (2014) pan2014author and Fortunato, S. (2014) fortunato2014author used the last five year citation time window for the scientific assessment of scholars. Instead of fixed time-stamp, a variable time-stamp is much better schreiber2015restricting. By using this concept schreiber2015restricting considered a variable timestamp , that is either whole publication career or a reasonable time window, and proposed the timed h-index (). Formally, it is defined as:

“The timed h-index of a scholar is the largest integer k such that the k articles have at least k citation count each during the defined citation time window.”
The decade year or fixed time window do not consider the overall impact of a scholar. It seems that some of the scholars published articles throughout the career, while some scholars published articles during their PhD career or published occasionally. In both cases, the evaluation of scientific impact of scholars is based on the citation count earned by such articles, but their contribution is very different. In this context, mahbuba2012diffusion; mahbuba2013year

proposed a set of indicators based on yearly impact of scholars. The year based indices are classified into four categories, where source is year and the items are (i) the total number of publications in a particular year, (ii) the total number of citations earned by all publications published in a particular year, (iii) the total number of citations earned in a particular year from all publications that are published in any year and (iv) diffusion of citation count based on the age of the publications.

(i) Career year h-index by publications: “ The career year h-index by publications of a scholar is h, if h of his/her publication year has at least h publications each.”
To compute the career year h-index by publications, first the total number of publications in each publication year are calculated. Then, they are arranged in descending order of their total publication count. Then, the career year h-index by publications is the maximum rank in which the publication count is equal to or greater than the year-rank. This year based index considers the year wise productivity of scholars. Suppose a scholar productivity is more than others but their scientific impact is very less, then we can not say that the scholar is more prominent than others.

(ii) Career year h-index by citations (Item: publication year citation): “ The career year h-index by citations of a scholar is h if h of his/her publication years have at least h citation each.”
This index considers the total number of citations earned from all articles that are published in a particular year. This shows the productivity year impact of a scholar. Suppose a scholar published more number of articles in their earlier stage of career or selective year, then the index value is very low than the scholars who published articles regularly. So, instead of publications year citation, the citation year citation is more pertinent in scientific assessment.

(iii) Career year h-index by citations : “ The career year h-index by citations of a scholar is , if of his/her citation year receives at least sum of citaton count each.”
To compute career year h-index by citations, we first calculate the total number of citations earned in every year from all publications that are published in any year. This index considers the year wise impact of scholars and produces a single number that is equal or less than the total research career. This index may be influenced by the older articles. A good number of earlier published articles affect the index value significantly. Instead of only year wise citation, the age of the publications may also play important role in the scientific assessment. To do this, the diffusion based h-index was designed.

(iv) Diffusion based h-index: “The diffusion based h-index of scholar is such that the year’s articles have at least diffusion citation count each.”

The diffusion speed of the publication year is the sum of the citation counts of all such articles that are published in year divided by the age of the publication. All year wise indices consider the year wise impact of scholars rather than the individual publication citation count. It requires year wise publication count, citation count of all articles which are published in respective years. Finally, we can conclude that the all year based can be used as an alternative in scientific assessment of scholars. The career year h-indices use the h-index methodology to compute the overall impact of scholar. As we know that the traditional h-index suffers with big-hit and ignorance of tail-citation issue and the career year h-indices do not account these issue.

To overcome the big-hit problems or consideration of excess value in scientific assessment of scholars, the year based EM-index has been proposed by Bihari2018 with three different parameters. The source of the year based EM-index is year and the items are (i) Total publication count in a particular year, (ii) Publication year citation count and (iii) citation year citation. By using these three different item values, the year based EM-index has been designed and named (i) Year based EM-index by publications, (ii) Year based EM-index by publication year citation and (iii) Year based EM-index by citations. The year based EM-index are computed by using the EM-index methodology which is discussed in Bihari2017. The year based EM-index produce a set of value along with the global index value. The elements of the year based EM-index help in comparing scientific impact of scholar have similar index value. However, the career year h-indices and the year based EM-index considers only the core item value and leave some important item’s value, that may have very near to the core item value. To incorporate the importance of such tail-item value with the consideration of excess citation value, the year based -index has been designed by Bihari2018. The year based -index has been computed by using the methodology of -index, which is discussed in Bihari2017.

From the above discussion, it can be concluded that the career of the publication can be used as an important factor in the scientific assessment of scholars and helps in comparing the impact of junior and senior scientists. The research done till now in comparison of scientific impact of scholars is not sufficient and do not make clarity, hence, need some effort in this context. Summary of the publication career based index is shown in table 2.

Index Definition Publication
AR-index “The AR-index of a scholar is the square root of the sum of the normalized citations of h-core articles.” , where and is the citation count and the age of the article respectively. jin2007r
Contemporary h-index “The contemporary h-index of a scholar is such that their articles have at least score each.” sidiropoulos2007generalized
Trendy h-index “The trend h-index of a scholar is the largest number such that for his/her articles, each have a score .” sidiropoulos2007generalized
Dynamic h-type index , where, R(y) is the R-index of the scholar at the career year, and is the recent increment in the h-index for the year. rousseau2008proposal
Table 2: Summary of Index based on age of publication
Index Definition Publication
hpd-index “The hpd-index of a scholar is hpd such that the hpd articles have at least hpd citation per decade () each .” , where is the decade year and it must be greater than the publication year. kosmulski2009new
Timed h-index “The timed h-index of a scholar is the largest integer k such that the k articles have at least k citation count each during the defined citation time window.”
schreiber2015restricting
Career year h-index by publications “ The career year h-index by publications of a scholar is h, if h of his/her publication year has at least h publications each.” mahbuba2012diffusion
Career year h-index by publication year citations “ The career year h-index by citations of a scholar is h if h of his/her publication years have at least h citation each.” mahbuba2012diffusion
Career year h-index by citations “ The career year h-index by citations of a scholar is , if of his/her citation year receives at least sum of citation count each.” mahbuba2012diffusion
Diffusion based h-index “The diffusion based h-index of scholar is such that the year’s articles have at least diffusion citation count each.” mahbuba2012diffusion
Table 2: Summary of Index based on total number of author Continue..

2.4 Indices based on total number of author

In the research community, most of the research work is done by the group of scholars and the evaluation of the scientific impact of a scholar is based on their articles’ citation count. In the scientific assessment process, all authors get full credit of articles citation count. But rarely they contribute equally. To overcome this shortcoming, cole1973social considers only the first author and completely ignore the co-authors. But, it is not fair in the case of multi-authored articles (wan2007pure). lindsey1980production gives full credit to every scholar. multiple1981price used the fractional allocation between all authors, i.e., , where is the total number of scholars. hodge1981publication, sekercioglu2008quantifying and hagen2010harmonic share the credit among all scholars in proportion to their rank.van1997fractional; trueba2004robust used the arithmetic counting for credit allocation between author and co-authors. egghe2000methods used the geometric series for credit allocation. Several other methods are used to distributed share credit between all scholars prathap2010there; abbasi2010evaluating; altmann2009evaluating; liu2012modifying; hu2009loads.

The citation count of an article should be distributed to all co-authors based on their role in the article batista2006possible. However, it is very difficult to know the role of each scholar in an article. In this way, simply divide the h-index by the average number of scholars in h-core articles and named the proposed indicator is -index. It is defined as:

“The -index of a scholar is the ratio of the h-index and the average number of scholars in the h-core articles.”
Mathematically, it is defined as:

(18)

where is the h-index and is the average number of scholars from h-core articles.

Further, imperial2007usefulness discusses the impact of -index. This index penalizes the collaborative effort of scholars because the h-index value is divided by the average number of scholars. If every h-core publication has only one author then the -index value is equal to h-index. A few high co-authored articles may affect the index value drastically. To overcome this limitation of -index, wan2007pure proposed a new index called pure h-index. It is almost similar to the -index. The only difference is the denominator. In the -index, the denominator is the average number of scholars in the h-core articles, while in the pure h-index, the denominator is the square root of the average number of scholars in the h-core articles. The pure h-index is defined as :

(19)

where is the h-index and is the average number of scholars in the h-core articles.

Further, the authors discussed the equivalent number of co-authors, proportional (arithmetic) and geometric assignment to share credit among all scholars in a multi-authored article. In case of the equivalent number of co-authors, the credit share of an individual in an article is 1/k (burrell1995fractional; van1997fractional), where k is the total number of authors in the article. In case of arithmetic (proportional) assignment (van1997fractional), the credit share of an individual(either author or co-author) in an article is defined as :

(20)

Where,
is the credit share of scholar A,
is the total number of scholars and
is the rank of the scholar in the publication.

In case of geometric assignment (egghe2000methods), the credit share of a scholar in an article is defined as:

(21)

where the symbols have their meaning as defined in Eq. 20

Based on these credit assignment schemes, the pure h-index of a scholar is defined as:

(22)

where h is the h-index and is the average credit share of h-core articles, defined as:

(23)

where the symbols have their meaning as defined in Eq. 20.

By using the same credit assignment scheme, the pure R-index of a scholar is defined as:

(24)

where,
E(author) is the average number of scholar in the h-core articles
C(a,p) is defined in Eq. 20.
h(pub) is the set of h-core articles.

It seems that a scholar may get high h-index, but he rarely contributes as a core author, while another scholar gets relatively lesser h-index, but mostly contributes as a core author. The h-index and the pure h-index does not account for this issue. To resolve this issue, chai2008adapted proposed a new index called adaptive pure h-index (. In order to determine the adaptive pure h-index, we first calculate the equivalent numbers of co-authors based on the pure h-index. Then, we compute the effective citation count. The effective citation count is the actual number of citation count divided by the square root of the equivalent number of authors. Then, we rank the publications in the descending order of their effective citation counts. The -index lies between and , where is the h-index based on the effective citation count () . Mathematically, the adaptive pure h-index () is defined as:

(25)

If and +1 are equal, then . If all articles are single authored, then = h-index. The adaptive pure h-index is also defined with arithmetic (proportional) and geometric credit assignment schemes.

Another similar approach the normalized -index was proposed by wohlin2009new. It evaluates the scientific impact of scholars using basic h-index with adjusted citation count. It is an extension of -index. The primary difference between -index, pure h-index, adaptive pure h-index and normalized -index is the distribution of citations among scholars. In this index, the citation distribution of an article to all scholars is the ratio of total citation count and the total number of scholars (, where is the citation count and Author_count(P) is the total number of scholars of article). Then the normalize -index of a scholar is the maximum rank in which the adjusted citation count is equal or greater than the rank value.

“The normalized -index of a scholar is k, if k of his/her articles have at least k normalized citation count each.”

egghe2008mathematical discussed the fractional credit allocation technique to share the credit among scholars in multi-authored articles and proposed the fractional h and g-index.

“The fractional h-index () of a scholar is , if of his/her articles have at least fractional citation count each.”
Mathematically, it is defined as:

(26)

Where,
is the fractional h-index of a scholar,
cit(K) is the citation count of the -article, and
Author(K) is the total number of authors in article.

The author applied the same technique to the g-index and proposed a new index called fractional g-index.

“The fractional g-index () of scholars is , if of his/her articles have at least cumulative fractional citation count each.”
Mathematically, it is defined as:

(27)

where the symbols have their meaning as defined in Eq. 26 Instead of fractional citation count, the fractional ranking of publications is used to design fractional h-index () (egghe2008mathematical). The effective fractional rank of the article is the cumulative sum of the effective rank of successive publications (). The effective rank of the article is defined as:

(28)

where, is the total number of successive articles and Author(p) is the total number of authors in the article.

Then the fractional h-index is defined as:

(29)

where, is the cumulative effective rank of article, is the citation count of article.
Another similar approach proposed by schreiber2008share called -index.

Instead of arithmetic and geometric distribution of citation counts,

hagen2008harmonic suggested the harmonic counting method. The harmonic share of the author in an article is defined as:

(30)

where, k is the rank of author in the authored article.

Based on harmonic credit allocation, the harmonic h-index is defined as:.

“The harmonic h-index of a scholar is , if of his/her articles have at least harmonic credits each.”

Similar approaches, weighted h-index and weighted citation h-cut, have been discussed by abbas2011weighted. These consider the total number of cited articles and the total number of co-authors. To share the citation credit among scholars, the positionally weighted and the equal weighted mechanisms are used. In the positionally weighted scheme, the first author gets more credit than the second one and the second author gets more credit than the third one and so on. Finally the summation of all weights is normalized to 1. The weight of the author in the authored article is defined as:

(31)

In equally weighted scheme all authors get equal weight; i.e. 1/m where m is the total number of authors.

“The weighted h-index of a scholar is the largest number k such that their k articles have at least k weighted citation aggregate each.”

Mathematically, the weighted h-index is defined as:

(32)

where,
P is the total credit score earned by positionally or equally weighted scheme,
is the citation count of the article and
is the weighted score of a scholar in the article.

The weighted citation H-cut of a scholar is the sum of the weighted citation count of the weighted h-core articles.

(33)

where the symbols have their meaning as defined in Eq. 32.

In the last 2 or 3 decades, the number of co-authors has continuously increased harsanyi1993multiple; kennedy2003multiple; greene2007demise. This plays a critical role in the distribution of citations among the scholars. If the number of co-authors is more, then the distribution based on the average number of authors, arithmetic counting, geometric counting and the harmonic count is not fair. To resolve this issue, zhang2009proposal design a new index called weighted h-index. In weighted h-index, the shared credit of the first and the corresponding authors are 1 and the author’s share credit is defined as:

(34)

where, m is the total number of authors and j is the rank of the author.
For example, let an article published by five authors. Let the first and the last author be the primary and the corresponding author respectively. Then both of the authors get full credit for publication. The rest of the authors earn credit based on their rank using the proportional counting method. By using the weighted citation count, the weighted h-index is defined as:

“The weighted h-index of a scholar is w, if w of his/her articles have at least w weighted citation count each.”

This index is almost similar to the -index, the only difference is the credit allocation of the first and the corresponding author. All of the credit allocation schemes consider only the mathematical equation to share the credit among scholars. However, the real scenario is different. The role of every scholar is different in different articles, hence, it is not fair to distribute the citation among scholars by using a mathematical equation. In this way, hu2010those categories the contribution of scholars in three different categories : (i) First author, (ii) Corresponding author and (iii) other author (whose contribution is not defined Based on this, shapiro1994contributions; hu2009loads) proposed a new index called -index (Majority based index). Based on the contribution of a scholar, the following four different measures can accumulate the performance of scholars: (i)Overall h-index, (ii)First author h-index, (iii)Corresponding author h-index and (iv)other contributor h-index.

The overall h-index is the original h-index, the first author h-index is the h-index computed from the citation count of all those articles in which the scholar was present as a first author. The corresponding author h-index is the h-index computed from the citation count of all those articles in which the author is present as a corresponding author. Finally, the other contributor h-index is the h-index computed form citation count of all those articles in which the scholar is present neither as the main author nor as the corresponding author. The relatively high value of the first author h-index indicates that the author mostly worked as a primary author. The relatively high value of the corresponding author h-index indicates that the author mostly worked as a corresponding author and the relatively high value of the other contributors’ h-index shows that the author mostly worked as a supportive author.

Instead of these four types of indicators, bucur2015updated categories authors into two categories: the primary author (main author and corresponding author) and non-primary author. With consideration of primary and non-primary author the Hirsch(p,t) was proposed. Formally, it is defined as:

(35)

where, h(p) represents the h-index computed from the citation count of all those articles in which the author was present as a main or a corresponding author, and h(t) represents the overall h-index.

Suppose a scholar published 3 articles with 2 citations each. Out of these three, 2 articles were published as the primary author (main and corresponding author). Then the h-index based on these two articles is 2 and the overall h-index is also 2. Hence, the Hirsch(p,t)=2,2.

Another similar approach has been discussed by wurtz2016stratified. The author mentioned that the first, second, third and last author’s contribution is equal. Based on this scenario, author proposed a new index called Stratified h-index, which combines following four types of indicators: (i) First authorship(), (ii) Second authorship(), (iii) Third authorship() and (iv) Last authorship h-index (). The relatively high value of indicates that the author mostly worked as a primary author, the relatively high value of , indicates that the author mostly worked as a secondary author with major contribution, and the relatively high value of indicates that the author mostly work as a senior investigator or supervisor.

Instead of distribution of citation among scholars, Aziz et al. (2013)aziz2013profit used the impact of collaborators in the scientific gain of scholars. To do this, the harmonic weighted scheme and the rank of the authors is used to estimate the scientific impact of scholars in an article. The weight of the author in authored article is defined as:

(36)

where D is 0, if the article authored by the even number of author and 1/2m, if the article is authored by an odd number of authors.

The sum of the weight of all articles is the number of monograph equivalent. The monograph equivalent is the total number of single authored articles. Then the profit (p)-index of a scholar is defined as:

(37)

where, is the total number of published articles and ME is the monograph equivalent of an author, which is defined as:

(38)

where, is the total number of publications and W(A,p) is the weight of the author A in the article.

The value of the profit h-index lies between 0 and 1, where 0 indicates that all articles are written by the primary author, i.e., the contribution of the co-authors is zero, or, the papers are singly authored.

prathap2010fractional proposed Fractional and Harmonic p-index to account for the number of authors. In the fractional p-index (), the fractional credit of an author is the 1/(total number of authors) and is defined as:

(39)

where is the total fractional citation count and is the cumulative sum of fractional rank of co-authors.

The total fractional citation count of an author is defined as:

(40)

where,
is the total number of articles,
is the rank of author in the article, and
is the citation count of the article.

In harmonic counting method, the weighted credit of the scholar in an authored article is defined as: (1/k)/(1+(1/2)+(1/3)+ …….+(1/m)) and the harmonic p-index of a scholar is defined as:

(41)

where, is the total harmonic credit of citation counts and is the cumulative sum of the harmonic rank.

Instead of arithmetical allocation of citation counts to an author, galam2011tailor used the Tailor Based Allocation (TBA) mechanism to share credit among scholars and proposed a new index called gh-index.

“The gh-index of a scholar is k, if k of his/her articles have at least k TBA based fractional citation count each.”

In this mechanism, the extra credit to the first and the to the last author was given. The tailor based credit allocation of first, last and the author is respectively defined as :

(42)
(43)
(44)

where, and is the total number of scholar in an article.

In case an article authored by two authors, the credit allocation has three choices: two to one third, three to one quarter and one to one half. In case of two to one third, the extra credit is given to the first and the last author is 2 and 1 respectively, in three to one quarter, the extra credit is 1 and 0 respectively, and in case of one to one half, the extra credit given is 0 and 1 respectively for the first and the last author respectively. In case of articles, authored by more than two authors, the decision of credit allocation depends on choice.

liu2011fairly presented a new mechanism to share credit among corresponding and non-corresponding authors. In this mechanism, the corresponding author gets more credit than the non-corresponding author. The credit of a scholar decreases, when the number of scholar increases. The following steps are used to share credit among scholars:

  1. First, the sequence of authors is rearranged in the following way: first author, corresponding author and the rest of the authors in the original sequence. For example: an article has been written by four scholars A1, A2, A3 and A4. If A4 is the corresponding author, then the sequence is like A1, A4, A2 and A3.

  2. The credit allocation to the author in an m-authored article is defined as:

    (45)

    where, is the integral constant greater than one.

    A smaller value give the credit balance among scholars rather than the maximum . Most Important Authors (MIA), i.e., the first author and the corresponding author, tie for the first rank. Their credit allocation is the average of and is defined as:

    (46)
  3. The normalized credit score of an individual scholar is defined as:

    (47)

    and

    (48)

    where is the sum of all credit given to every author.

  4. The citation allocation of the scholar in the article is defined as:

    (49)

Based on this credit allocation system two different indices were proposed called CCA h-index () and CCA g-index ()

“The -index of a scholar is , if of his/her articles have at least allocated citation count each.”

“The index of a scholar is , if of his/her articles have at least citation together.”

Generally, most of the indices give more credit to the first and the corresponding author, however, every author have their own importance in an article. The first and the corresponding author may have done most of the research work and guidance. The other authors may have analyzed the research work biswal2013absolute. So we cannot discard the importance of non-primary authors. To consider the importance of non-corresponding authors, the Absolute index(Ab-index) biswal2013absolute has been designed. In this index, the first author and the corresponding author get full credit and the rest of the authors get shares in decreasing arithmetic progression.

“The Ab-index of a scholar is the sum of the partial credit earned from all articles in which the scholar is present either as a first, corresponding or a co-author.”

Mathematically, the Ab-index of author is defined as:

(50)

where, is the partial credit of scholar in the article . The partial credit of the first and the corresponding author is equal, and is defined as:

(51)

where,
indicates the total partial credit of the first or the corresponding author in the article,
is the citation count of the article,
is the total number of authors and,
is the first author or/and the corresponding author.

In single authored articles, author gets full 100% credit. In case of one first author, one corresponding author and one other co-author, the credit share of the first and the corresponding author is 40% each, and 20% for the other author. In case of the first author also being the corresponding author, the credit share is 66.67% and the remaining author gets 33.33% credit share. In case of non-primary authors (neither the first nor the corresponding author), then the credit share of non-primary author is defined as:

(52)

where, , , and is the number of corresponding authors, first author, total number of authors and the citation counts of the article respectively.

Clearly, the computation of this index is a lot more complex than the h-index.

altmann2009evaluating proposed a new index called RP-index (Research Productivity). This index is based on normalized citation count (each citation count is divided by the age of the publication) and the contribution factor of the individual researcher in the group. It is defined as:

(53)

where, indicates the normalized citation score of author in article and is the contribution factor of author in article ; it lies between 0 & 1. If the contribution of all authors is equal, then is 1/(total number of authors).

“The RP-index of a scholar is RP, if RP of his/her articles have at least average RP normalized citation count each.”

Mathematically, it is defined as:

(54)

where, is the total number of publications of scholar. altmann2009evaluating also gives a slightly modified definition of the RP-index as:

(55)

The contribution of collaborators also plays an important role in the scientific assessment of a scholar. Based on this concept, abbasi2009evaluating and abbasi2010evaluating proposed a new index called RC(Researcher Collaboration)-index and the CC
(Community Collaboration)-index.

“The RC-index of a scholar is the largest number R such that their R co-authors have at least R average co-author collaboration value each.”
Mathematically, the RC-index is defined as:

(56)

where, is the total number of collaborators and is the co-author collaboration value of author with co-author , and is defined as:

(57)

where is the RP-index of scholar j.

After a slight modification of the RC-index, the -index is defined as:

(58)

The values lies between and +1.

The Community collaboration index called CC-index is defined as:

(59)

After a slight modification of the CC-index, the -index is def]ined as:

(60)

In this section, we have discussed a number of indices. They consider the number of co-authors in the scientific assessment of scholars and the research articles. Several indices consider the total number of authors of h-core articles. Some of them are defined through somewhat complex mathematical equations, some consider the impact of only the first and the corresponding author, and so on. These all are only the assumption because a research article does not have any information related to author contributions. After a long journey of scientific assessment of scholars, the sharing of credits among scholars is still an open challenge. Summary of the indices based on total number of author shown in table 3

Index Definition Publication
-index “The -index of a scholar is the ratio of the h-index and the average number of scholars in the h-core articles.” , where is the h-index and is the average number of scholars from h-core articles. batista2006possible
Pure h-index , where is the h-index and is the average number of scholars in the h-core articles. wan2007pure
Pure R-index wan2007pure
Adaptive pure h-index() , where represents h-index based on effective citation count. chai2008adapted
Normalized -index “The normalized -index of a scholar is k, if k of his/her articles have at least k normalized citation count each.” wohlin2009new
Fractional h-index () “The fractional h-index () of a scholar is , if of his/her articles have at least fractional citation count each.” egghe2008mathematical
Fractional g-index () The fractional g-index () of an author is such that the top papers has at least cumulative fractional citation count each. egghe2008mathematical
Harmonic h-index “The harmonic h-index of a scholar is , if of his/her articles have at least harmonic credits each.” hagen2008harmonic
Weighted h-index “The weighted h-index of a scholar is the largest number k such that their k articles have at least k weighted citation aggregate each.” abbas2011weighted
Zhang Weighted h-index “The weighted h-index of a scholar is w, if w of his/her articles have at least w weighted citation count each.” zhang2009proposal
Hirsch(p,t) The Hirsch(p,t) is the combination of sum of h-index of author based on citation count of all those article where author present as a main & corresponding author and overall h-index of an author. (Hirsch(p,t)=h(p),h(t)) bucur2015updated
Profit h-index () , Where is the adjusted h-index of author and h is the actual h-index of the corresponding author. aziz2013profit
Fraction p-index () , where prathap2010fractional
Harmonic p-index() prathap2010fractional
gh-index “The gh-index of a scholar is k, if k of his/her articles have at least k TBA based fractional citation count each.” galam2011tailor
-index “The -index of a scholar is , if of his/her articles have at least allocated citation count each.” liu2011fairly
index “The index of a scholar is , if of his/her articles have at least citation together.” liu2011fairly
Ab-index “The Ab-index of a scholar is the sum of the partial credit earned from all articles in which the scholar is present either as a first, corresponding or a co-author.” (, where is the partial credit of author k in article A.) biswal2013absolute
RP-index (Research Productivity) “The RP-index of a scholar is RP, if RP of his/her articles have at least average RP normalized citation count each.” altmann2009evaluating
RC-index “The RC-index of a scholar is the largest number R such that their R co-authors have at least R average co-author collaboration value each.” abbasi2009evaluating
CC-index abbasi2010evaluating
Table 3: Summary of Index based on total number of author

2.5 Indices based on combination of two indices

To measure the scientific impact of scholars, several indices were proposed and every index measures the impact with different parameters. In order to find the global impact of a scholar, several parameters should be addressed that cover at least total productivity and the impact of scholars. Any single index covers only one parameter, so the combination of advantages of two or more indices is more precise martin1996use; van2003holy. In this context, the hg-index, -index and -index were proposed to assess the scientific impact of scholars.

The h-index considers only the highly cited articles and leaves a number of articles, however, the g-index tries to use a maximum number of articlesalonso2010hg. But both of the indices do not meet the sufficient requirement of the scientific assessment of scholars. So, the combination of these two indices called hg-index could be an effective alternative to assess the scientific impact of scholars. Formally, it is defined as:

“The hg-index of a scholar is the geometric mean of the h and g-index.”

Mathematically, the hg-index is defined as:

(61)

where, h and g represents the h and the g-index value of scholar.
The hg-index combines the advantages of h-index and g-index to produce a value near to the h-index than to g-index. The lower h-index penalizes the g-index and produces a lower index value.

cabrerizo2010q2 categorizes the indices in two categories: the first one is based on the productive core of a scholar like h-index(hirsch2005index), g-index(egghe2006theory), hg-index(alonso2010hg) & -index(kosmulski2006new), while the second one is based on the impact of core papers like a-index (jin2006h), m-index(bornmann2008there), AR-index(jin2007r) & -index(egghe2008h). But one measure alone, either based on the productive core or the impact of the productive core, is not suitable for the scientific assessment of scholars. To overcome this limitation of scientific assessment process, cabrerizo2010q2 proposed a new index called -index that considers both the number of productive core articles and the impact of core articles. Formally,

“The -index of scholars is the geometric mean of h-index and m-index.”

Mathematically, it is defined as:

(62)

where, the h and m is the h and m-index value of scholar.
The -index of a scholar is nearer to the h-index than the m-index. The low value of h-index penalizes the m-index and the resultant -index is relatively low.

Another index designed to combine the properties of -index (hagen2008harmonic) and the -index (liu2011fairly) is the -index (liu2012modifying). In -index, the citation count is distributed equally to all the scholars (, where is the citation count and is the total number of scholars). Instead of this, the author used combined credit allocation for most important author (MIA) (liu2011fairly and schreiber2008share) and the cumulative sum of the combined credit allocation, named effective paper count, is used for publication ranking. Mathematically, it is defined as:

(63)

where rk(p) is the rank of author in article and is the number of authors in paper .

“The -index of a scholar is k, if k of his/her articles have at least k effective article count each.”
Mathematically, it is defined as:

(64)

where, is the effective paper count of the article and is the citation count of the article. If the tail articles earn extra citation count in the near future and core articles citations remain unchanged, then -index will increase.

In this section, we have discussed all those indices that combine the advantages of two indices. From the above discussion, we can conclude that the combination of two indices could be an effective alternative to citation based indices, but still, each has limitations to emerge as a single index to completely asses the scientific impact of scholars. The summary of index based on the combination of two indices shown in table 4.

Index Definition Publication
hg-index “The hg-index of a scholar is the geometric mean of the h and g-index.” () alonso2010hg
-index “The -index of scholars is the geometric mean of h-index and m-index.” () cabrerizo2010q2