11institutetext: Anand Bihari 22institutetext: Department of Computer Science
& Engineering
National Institute of Technology Patna, Bihar, India 22email:
<EMAIL_ADDRESS><EMAIL_ADDRESS>33institutetext: Sudhakar Tripathi
44institutetext: Department of Information & Technology
R. E. C. Ambedkar Nagar, Uttar Pradesh India 44email<EMAIL_ADDRESS>55institutetext: Akshay Deepak 66institutetext: Department of Computer Science
& Engineering
National Institute of Technology Patna, Bihar, India 66email:
<EMAIL_ADDRESS>
# h-index and its alternative: A Review

Anand Bihari  Sudhakar Tripathi  Akshay Deepak

###### Abstract

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
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
advantages and limitations of almost all Scientometrics as well as
Bibliometrics indicators which have been categorised in eight 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
and (viii) Based on account of self citation count. The main objective of this
article is to study all those indicators which has been proposed to evaluate
the scientific impact of individual researcher or a group of researchers.

###### Keywords:

h-index Scientometrics Indices Bibliometric Indices

††journal:

##  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 (Walters, (2007)), 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 (Bar-Ilan,
(2008)).

The h-index was proposed by Hirsch, (2005), 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 Lehmann et al., (2006); Hirsch, (2007). 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 Ding and Cronin, (2011).

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. Cole and Cole, (1973) considers only the first author,
de Solla Price and J., (1981) gives equal fractional credit to all scholars,
i.e. 1/k, where k is the total number of scholars. Hodge et al., (1981),
Sekercioglu, (2008) and Hagen, (2010) gives credit to all scholars as per
their proportional rank. Van Hooydonk, (1997) and Trueba and Guerrero, (2004)
used the arithmetic counting for credit allocation, Egghe et al., (2000) 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
García-Pérez, (2009) 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 Martin, (1996); Van Leeuwen et al., (2003).
In this context hg-index Alonso et al., (2010), $q^{2}$-index Cabrerizo et
al., (2010) and the $h_{mc}$-index Liu and Fang, (2012)) 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 Mahbuba and Rousseau, (2012).. 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 Hirsch, (2005) 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. $h^{2}$ 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. 1.

It is too simple to compute Hirsch, (2005) and does not require any data
processing Franceschini and Maisano, 2010b .

  2. 2.

It produces a single number that combines both the quality and the quantity of
the scholars’ publications Hirsch, (2005).

  3. 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 Hirsch, (2005); Van Raan, (2006).

  4. 4.

It can be easily obtained from any publication indexing databases such as
Google Scholar, Web of Science etc. Hirsch, (2005); Alonso et al., (2009);
Hirsch and Buela-Casal, (2014).

  5. 5.

It is closely related to the total number of publications that have
significant influence Liu and Rousseau, (2009).

  6. 6.

A small error in the citation distribution does not result in a huge change in
the index value Rousseau, (2007); Vanclay, (2007).

  7. 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 Hirsch and Buela-Casal, (2014).

  8. 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. 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
Hirsch, (2005); Schreiber, (2007); Liu and Rousseau, (2009); Waltman and van
Eck, (2011); Waltman and Van Eck, (2012).

  2. 2.

It is also not suitable in comparing the scientific impact of scholars’ having
different research careersKelly and Jennions, (2006).

  3. 3.

Once an article is selected for h-core, further citation will not be important
in scientific assessment Egghe, 2006b ; Egghe, 2006a .

  4. 4.

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

  5. 5.

It also considers the self citation count in h-index computation.Van Raan,
(2006).

  6. 6.

It produces a single natural number, that affects the discriminative power of
h-index Tol, (2008); Liu and Yang, (2014) when comparing two scholars with the
same h-index value.

  7. 7.

The index value never decreases Jin et al., (2007).

  8. 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 Burrell, 2007c . So it is not fair to evaluate the scientific impact
of scholars.

  9. 9.

It does not give any extra credit to highly cited articles Van Eck and
Waltman, (2008); Zhang, 2009a ; Bihari and Tripathi, (2017).

  10. 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
García-Pérez, (2009); Bihari and Tripathi, (2017).

  11. 11.

The publications inconsistency also does not affect the index value Mahbuba
and Rousseau, (2012); Bihari and Tripathi, (2018).

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. 1.

Complement of h-index (Section 2.2).

  2. 2.

Based on publication age (Section 2.3).

  3. 3.

Based on total number of authors (Section 2.4).

  4. 4.

Combination of two indices (Section 2.5).

  5. 5.

Based on excess citation count (Section 2.6).

  6. 6.

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

  7. 7.

Other types of indices (Section 2.8).

    1. (a)

Based on the core tail ratio (Section 2.8.1).

    2. (b)

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

    3. (c)

Based on variants of citation process (Section 2.8.3).

    4. (d)

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 Egghe, 2006b . 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 $g^{2}$ 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” Bornmann et al., (2008). 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 Alonso et
al., (2010); this is also one of the main limitations of the g-index
egghe2009econometric. To overcome the limitations of g-index, Kosmulski,
(2006) 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 $(h(2))^{2}$ 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, Wu, (2008) 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.

Tol, (2009) 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:

$f=\max_{f}\frac{1}{\frac{1}{f}\sum_{p=1}^{f}\frac{1}{cit_{p}}}\geq f$ (1)

where $cit_{p}$ is the citation count of $p^{th}$ 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:

$t=\max_{t}exp[\frac{1}{t}\sum_{k=1}^{t}ln(cit_{k})]\geq
t\leftrightarrow\max_{t}\prod_{k=1}^{t}cit_{k}^{\frac{1}{t}}\geq t$ (2)

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

Woeginger, (2008) 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:

$w=\underset{w}{max}(cit_{p}\geq
w-p+1),~{}~{}~{}~{}~{}~{}for~{}all~{}p~{}\leq~{}w$ (3)

where $cit_{p}$ is the citation count of $p^{th}$ 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 Xu et al.,
(2015). 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, Xu et al., (2015)
proposed a new variant of h-index called _Gh-index_. The Gh-index of a scholar
$a$, denoted $Gh^{a}$, is defined as:

$Gh^{a}=\sum_{p=1}^{m}sing(Cit(Pub_{p}^{a})-GH),~{}~{}where~{}sing(x)=\begin{cases}1,~{}~{}~{}~{}\textrm{$x\geq
0$},\\\ 0,~{}~{}~{}~{}\textrm{x $<$ 0}\end{cases}$ (4)

where $m$ is the total number of publications of scholar $a$ and $GH$ 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, Jin, (2006) 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:

$A=\frac{1}{h}\sum_{p=1}^{h}cit_{p}$ (5)

where A is the A-index of the scholar, h represents the h-index and $cit_{p}$
is the citation count of the $p^{th}$ 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 Burrell, 2007b . 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, Jin et
al., (2007) proposed another index called R-index, which is further discussed
in Glänzel, (2007). 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:

$R=\sqrt{\sum_{p=1}^{h}cit_{p}}$ (6)

where h represents the h-index and $cit_{p}$ is the citation count of the
$p^{th}$ 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, Panaretos and Malesios, (2009) proposed
a new index called $R_{m}$-index. Formally, it is defined as:

_“The $R_{m}$-index of scholars is the square root of the sum of square roots
of the citation counts of the h-core articles .” _

Mathematically, the $R_{m}$-index is defined as:

$R_{m}=\sqrt{\sum_{k=1}^{h}cit_{k}^{\frac{1}{2}}}$ (7)

where $cit_{k}$ is the citation count of $k^{th}$ 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
$R_{m}$-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 $R_{m-cv}$. The $R_{m-cv}$-index is simply the
subtraction of the CV score of the h-core articles from the $R_{m}$-index. In
this regard,Bornmann et al., (2007) 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, Bornmann et al., (2007) 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:

$b=\sum_{y=1}^{Y}\sum_{p=1}^{P}(cit_{py}\geq Pub_{10py})$ (8)

where,
$b$ is the b-index of scholar,
$Y$ represents publication year,
$P$ represents the total publication count,
$cit_{py}$ represents the citation count of $p^{th}$ article in year $y$, and
$Pub_{10py}$ is the 10% article published in year $y$.

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, Bornmann et al., (2008) 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 $\mu$-index Glänzel and Schubert, (2010) has also been discussed in the
same way as the _m-index_.

Egghe and Rousseau, (2008) 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 $(h_{w})$ Egghe and Rousseau, (2008). 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:

$R_{w}(k)=\frac{\sum_{p=1}^{k}cit_{p}}{h}$ (9)

where $h$ is the h-index and $cit_{p}$ is the citation count of the $p^{th}$
article.

The weighted rank of the $k^{th}$ article is the ratio of cumulative sum of
the citation count of the top $k$ 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:

$h_{w}=\sqrt{\sum_{k=1}^{R}cit_{k}}$ (10)

where, $cit_{k}$ is the citation count of the $k^{th}$ article and $R$ is the
largest rank among all publications such that the $k^{th}$ weighted rank $\leq
cit_{k}$.

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?” Vaidya,
(2005) 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:

$v=\frac{h}{p(y_{this}-y_{0})}$ (11)

where:
$h$ is the h-index,
$y_{this}$ is the current year, and
$y_{0}$ 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, Burrell, 2007a  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/ $pub_{age}$.”_
where, $h$ represents the h-index and $pub_{age}$ 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.

Sidiropoulos et al., (2007) addressed the above mentioned limitation of
h-index and proposed a new index called _Normalized h-index_. Formally, it is
defined as:

$normalized~{}h-index=\frac{h}{Pub_{count}}$ (12)

where, $h$ represents the h-index and $pub_{count}$ is the total number of
articles.

Another similar approach called called _v-index_ was proposed by Riikonen and
Vihinen, (2008). 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. Hirch, (2010) 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 $\overline{h}$-index.

_“The $\overline{h}$-index of scholars is $k$ 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
$\overline{h}$-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
$\overline{h}$-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 $\overline{h}$-index will
decrease in future, whereas, had the author collaborated with younger
scholars, their $\overline{h}$-index would have increased.

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

Table 1: Summary of Complement of 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 $g^{2}$ or more citations together.”_ |  Egghe, 2006a
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 $(h(2))^{2}$ citation each.”_ |  Kosmulski, (2006)
w-index |  _“The w-index of scholars is w such that the top w articles have at least 10w citations each.”_ |  Wu, (2008)
f-index |  $f=\max_{f}\frac{1}{\frac{1}{f}\sum_{p=1}^{f}\frac{1}{cit_{p}}}\geq f$, where $cit_{p}$ is the citation count of $p^{th}$ article. |  Tol, (2009)
t-index |  $t=\max_{t}exp[\frac{1}{t}\sum_{k=1}^{t}ln(cit_{k})]\geq t\iff\max_{t}\prod_{k=1}^{t}cit_{k}^{\frac{1}{t}}\geq t$ |  Tol, (2009)
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.”_ |  Woeginger, (2008)
Gh-index |  $Gh^{a}=\sum_{p=1}^{m}sing(Cit(Pub_{p}^{a})-GH),~{}~{}where~{}sing(x)=\begin{cases}1,~{}~{}~{}~{}\textrm{$x\geq 0$},\\\ 0,~{}~{}~{}~{}\textrm{x $<$ 0}\end{cases}$ |  Xu et al., (2015)
A-index |  _“The A-index of a scholar is the average number of h-core articles citation count.”_ A= $A=\frac{1}{h}\sum_{p=1}^{h}cit_{p}$,where A is the A-index of the scholar, h represents the h-index and $cit_{p}$ is the citation count of the $p^{th}$ article. |  Jin, (2006)
R-index |  _“The R-index of scholars is the square root of the sum of h-core articles citation count.”_ $R=\sqrt{\sum_{p=1}^{h}cit_{p}}$, where h represents the h-index and $cit_{p}$ is the citation count of the $p^{th}$ article. |  Jin et al., (2007)
$R_{m}$-index |  _“The $R_{m}$-index of scholars is the square root of the sum of square roots of the citation counts of the h-core articles .” _ $R_{m}=\sqrt{\sum_{k=1}^{h}cit_{k}^{\frac{1}{2}}}$, where $cit_{k}$ is the citation count of $k^{th}$ article. |  Panaretos and Malesios, (2009)
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.”_ $b=\sum_{y=1}^{Y}\sum_{p=1}^{P}(cit_{py}\geq Pub_{10py})$ |  Bornmann et al., (2007)
m-index |  _“The m-index of a scholar is the median number of citation count of the h-core articles.”_ |  Bornmann et al., (2008)
Weighted h-index |  _$h_{w}=\sqrt{\sum_{k=1}^{R}cit_{k}}$ , where, $cit_{k}$ is the citation count of the $k^{th}$ article and $R$ is the largest rank among all publications such that the $k^{th}$ weighted rank $\leq cit_{k}$._ |  Egghe and Rousseau, (2008)
v-index |  _“ The v-index of a scholar is the h-index value adjusted with the publication age.”_ $v=\frac{h}{p(y_{this}-y_{0})}$, where h is the h-index, $y_{this}$ is the current year, $y_{0}$ is the first publication year. |  Vaidya, (2005)
m-quotient |  _“The m-quotient of scholars is defined as: m=h/ $pub_{age}$.”_ where, $h$ represents the h-index and $pub_{age}$ is the publication age. |  Burrell, 2007a
Normalized h-index |  $normalized~{}h-index=\frac{h}{Pub_{count}}$, where, $h$ represents the h-index and $Pub_{count}$ is the total number of articles. |  Sidiropoulos et al., (2007)
Riikonen v-index |  _“The v-index of a scholar is the ratio of the h-index and the total number of publications.”_ |  Riikonen and Vihinen, (2008)
$\overline{h}$-index |  _“The $\overline{h}$-index of scholars is $k$ 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.”_ |  Hirch, (2010)

###  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, Jin, (2007); Jin et al., (2007) 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:

$AR=\sqrt{\sum_{p=1}^{h}\frac{cit_{p}}{age_{p}}}$ (13)

where $cit_{p}$ and $age_{p}$ is the citation count and the age of the
$p^{th}$ 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)Sidiropoulos et al., (2007) 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 $h^{C}$ such that their $h^{C}$
articles have at least score $S^{C}\geq h^{C}$ each.”_

The score $S^{C}$ of the $k^{th}$ article is defined as:

$S^{C}_{k}=\gamma(year_{now}-year_{k}+1)^{-\delta}*|cit_{k}|$ (14)

where,
$year_{now}$ is the current year,
$year_{k}$ is the publication year,
$cit_{k}$ is the citation count of the $k^{th}$ article, and
$\gamma,\delta$ are the coefficients set by the user.

If $\delta$ 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, Sidiropoulos et al.,
(2007) 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 $h^{t}$ such that for
his/her $h^{t}$ articles, each have a score $S^{t}\geq h^{Ct}$.” _

The score of the $k^{th}$ article is defined as:

$S^{t}_{k}=\gamma\sum_{\forall c\in cit_{k}}(year_{now}-year_{c}+1)^{-\delta}$
(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
$\gamma$ and $\delta$ 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 ($h_{d}$) was designed by Rousseau and Ye, (2008). This covers
the size of core articles and how that size changes with the time. Formally,
it is defined as:

$h_{d}=R(y).V_{h}(y)$ (16)

where,
R(y) is the R-index of the scholar at the $y^{th}$ career year, and
$V_{h}(y)$ is the recent increment in the h-index for the $y^{th}$ 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 Kosmulski,
(2009) proposed the h-index per decade $(h_{pd})$:

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

The adjusted citation per decade is defined as:

$\displaystyle
ACPD=\frac{Citation~{}from~{}publication~{}year~{}to~{}year~{}y}{y-(publication~{}year-1)}$
(17)

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

Instead of the total time window or decade based scientific assessment, Fiala,
(2014) considers only recent three year’s citation time window. Pan, R. K. and
Fortunato, S. (2014) Pan and Fortunato, (2014) and Fortunato, S. (2014)
Fortunato, (2014) 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 Schreiber, 2015a . By using this concept Schreiber,
2015a  considered a variable timestamp $t$, that is either whole publication
career or a reasonable time window, and proposed the timed h-index
($h_{t}(y)$). 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, Mahbuba and Rousseau, (2012,
2013) 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 $h$, if $h$ of his/her citation year receives at
least sum of $h$ 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 $p$
such that the $p$ year’s articles have at least $p$ diffusion citation count
each.”_

The diffusion speed of the publication year $Y$ is the sum of the citation
counts of all such articles that are published in year $Y$ 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 Bihari and Tripathi, (2018) 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 Bihari and Tripathi, (2017). 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 $EM^{{}^{\prime}}$-index has been
designed by Bihari and Tripathi, (2018). The year based
$EM^{{}^{\prime}}$-index has been computed by using the methodology of
$EM^{{}^{\prime}}$-index, which is discussed in Bihari and Tripathi, (2017).

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.

Table 2: Summary of Index based on age of publication 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.”_ $AR=\sqrt{\sum_{p=1}^{h}\frac{cit_{p}}{age_{p}}}$, where $cit_{p}$ and $age_{p}$ is the citation count and the age of the $p^{th}$ article respectively. |  Jin et al., (2007)
Contemporary h-index |  _“The contemporary h-index of a scholar is $h^{C}$ such that their $h^{C}$ articles have at least score $S^{C}\geq h^{C}$ each.”_ |  Sidiropoulos et al., (2007)
Trendy h-index |  _“The trend h-index of a scholar is the largest number $h^{t}$ such that for his/her $h^{t}$ articles, each have a score $S^{t}\geq h^{Ct}$.”_ |  Sidiropoulos et al., (2007)
Dynamic h-type index |  $h_{d}=R(y).V_{h}(y)$, where, R(y) is the R-index of the scholar at the $y^{th}$ career year, and $V_{h}(y)$ is the recent increment in the h-index for the $y^{th}$ year. |  Rousseau and Ye, (2008)
Table 2: Summary of Index based on total number of author Continue.. 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 ( $ACPD\geq hpd$) each .”_ $ACPD=\frac{Citation~{}from~{}publication~{}year~{}to~{}year~{}y}{y-(publication~{}year-1)}$, where $y$ is the decade year and it must be greater than the publication year. |  Kosmulski, (2009)
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.”_ |
|  Schreiber, 2015a  |
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.”_ |  Mahbuba and Rousseau, (2012)
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.”_ |  Mahbuba and Rousseau, (2012)
Career year h-index by citations |  _“ The career year h-index by citations of a scholar is $h$, if $h$ of his/her citation year receives at least sum of $h$ citation count each.”_ |  Mahbuba and Rousseau, (2012)
Diffusion based h-index |  _“The diffusion based h-index of scholar is $p$ such that the $p$ year’s articles have at least $p$ diffusion citation count each.”_ |  Mahbuba and Rousseau, (2012)

###  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, Cole and Cole, (1973) considers only
the first author and completely ignore the co-authors. But, it is not fair in
the case of multi-authored articles (Wan et al., (2007)). Lindsey, (1980)
gives full credit to every scholar. de Solla Price and J., (1981) used the
fractional allocation between all authors, i.e., $1/k$, where $k$ is the total
number of scholars. Hodge et al., (1981), Sekercioglu, (2008) and Hagen,
(2010) share the credit among all scholars in proportion to their rank.Van
Hooydonk, (1997); Trueba and Guerrero, (2004) used the arithmetic counting for
credit allocation between author and co-authors. Egghe et al., (2000) used the
geometric series for credit allocation. Several other methods are used to
distributed share credit between all scholars Prathap, 2010b ; Abbasi et al.,
(2010); Altmann et al., (2009); Liu and Fang, (2012); Hu, (2009).

The citation count of an article should be distributed to all co-authors based
on their role in the article Batista et al., (2006). 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 $h_{i}$-index. It is defined as:

_“The $h_{i}$-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:

$h_{i}=\frac{h}{Avg_{A}}$ (18)

where $h$ is the h-index and $Avg_{A}$ is the average number of scholars from
h-core articles.

Further, Imperial and Rodríguez-Navarro, (2007) discusses the impact of
$h_{i}$-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 $h_{i}$-index value is
equal to h-index. A few high co-authored articles may affect the index value
drastically. To overcome this limitation of $h_{i}$-index, Wan et al., (2007)
proposed a new index called _pure h-index_. It is almost similar to the
$h_{i}$-index. The only difference is the denominator. In the $h_{i}$-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 :

$h_{p}(A)=\frac{h}{\sqrt{E(author)}}$ (19)

where $h$ is the h-index and $E(author)$ 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 (Burrell and
Rousseau, (1995); Van Hooydonk, (1997)), where k is the total number of
authors in the article. In case of arithmetic (proportional) assignment (Van
Hooydonk, (1997)), the credit share of an individual(either author or co-
author) in an article is defined as :

$C(A,P)=\frac{2(K+1-R)}{K(K+1)}$ (20)

Where,
$C(A,P)$ is the credit share of scholar A,
$K$ is the total number of scholars and
$R$ is the rank of the scholar in the $p^{th}$ publication.

In case of geometric assignment (Egghe et al., (2000)), the credit share of a
scholar in an article is defined as:

$\displaystyle C(A,P)=\frac{2^{(K-R)}}{2^{k}-1}$ (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:

$h_{p}(A)=\frac{h}{Avg_{Score}}$ (22)

where h is the h-index and $Avg_{Score}$ is the average credit share of h-core
articles, defined as:

$Avg_{Score}=\frac{\sum_{P=1}^{h}(C(A,P)}{h}$ (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:

$R_{p}(A)=\sqrt{\frac{\sum_{P\in h(pub)}C(A,P)}{E(author)}}$ (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, Chai et al., (2008) proposed a
new index called adaptive pure h-index ($h_{ap})$. 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
$h_{ap}$-index lies between $h_{eff}$ and $h_{eff}+1.$, where $h_{eff}$ is the
h-index based on the effective citation count ($cit_{eff}$) . Mathematically,
the adaptive pure h-index ($h_{ap}$) is defined as:

$h_{ap}=\frac{(h_{eff}+1)*cit_{eff}(h_{eff})-h_{eff}*cit_{eff}(h_{eff}+1)}{cit_{eff}(h_{eff})-cit_{eff}(h_{eff}+1)+1}$
(25)

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

Another similar approach the _normalized $h_{i}$-index_ was proposed by
Wohlin, (2009). It evaluates the scientific impact of scholars using basic
h-index with adjusted citation count. It is an extension of $h_{i}$-index. The
primary difference between $h_{i}$-index, pure h-index, adaptive pure h-index
and normalized $h_{i}$-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
($Cit_{P}/Author\\_count(P)$, where $Cit_{P}$ is the citation count and
Author_count(P) is the total number of scholars of $P^{th}$ article). Then the
normalize $h_{i}$-index of a scholar is the maximum rank in which the adjusted
citation count is equal or greater than the rank value.

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

Egghe, (2008) 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 ( $h_{f}$) of a scholar is $h_{f}$, if $h_{f}$ of
his/her articles have at least $h_{f}$ fractional citation count each.”_
Mathematically, it is defined as:

$h_{f}=\underset{K}{max}(K\leq\frac{cit(K)}{Author(K)})$ (26)

Where,
$h_{f}$ is the fractional h-index of a scholar,
cit(K) is the citation count of the $k^{th}$-article, and
Author(K) is the total number of authors in $k^{th}$ article.

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

_“The fractional g-index ( $g_{f}$) of scholars is $g_{f}$, if $g_{f}$ of
his/her articles have at least $g_{f}^{2}$ cumulative fractional citation
count each.”_
Mathematically, it is defined as:

$g_{f}=\underset{p}{max}(\sum_{k=1}^{p}\frac{cit_{k}}{Author(k)}\geq p^{2})$
(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 ($h_{F}$) (Egghe, (2008)). The effective fractional
rank of the $p^{th}$ article is the cumulative sum of the effective rank of
successive publications ($R=1/(Author(p)$). The effective rank of the $n^{th}$
article is defined as:

$R_{eff}(n)=\sum_{p=1}^{n}\frac{1}{Author(p)}$ (28)

where, $n$ is the total number of successive articles and Author(p) is the
total number of authors in the $p^{th}$ article.

Then the fractional h-index is defined as:

$h_{F}=\underset{R_{eff}(p)}{max}(cit_{p}\geq R_{eff}(p)$ (29)

where, $R_{eff}(p)$ is the cumulative effective rank of $p^{th}$ article,
$cit_{p}$ is the citation count of $p^{th}$ article.
Another similar approach proposed by Schreiber, 2008b  called $h_{m}$-index.

Instead of arithmetic and geometric distribution of citation counts, Hagen,
(2008) suggested the harmonic counting method. The harmonic share of the
$k^{th}$ author in an article is defined as:

$HC(k)=\frac{(1/k)}{[1+(1/2)+.......+(1/m)]}$ (30)

where, k is the rank of author in the $m^{th}$ authored article.

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

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

Similar approaches, weighted h-index and weighted citation h-cut, have been
discussed by Abbas, (2011). 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 $k^{th}$ author in the $m^{th}$ authored article is defined as:

$wt_{k}=\frac{2(m-k+1)}{m(m+1)}$ (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:

$h_{W}|wt_{k}\in P=h_{W},if(\min_{k=1}^{h_{W}}\\{cit_{k}wt_{k}\\}\geq
h_{W},\max_{i\neq k}\\{cit_{k}wt_{k}\\}\leq h_{W})$ (32)

where,
P is the total credit score earned by positionally or equally weighted scheme,
$cit_{k}$ is the citation count of the $k^{th}$ article and
$wt_{k}$ is the weighted score of a scholar in the $k^{th}$ article.

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

$\xi_{w}=\sum_{k=1}^{|h_{w}|}cit_{k}wt_{k}$ (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 Harsanyi, (1993); Kennedy, (2003); Greene, (2007). 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, Zhang, 2009b  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 $j^{th}$ author’s share credit is
defined as:

$WC(j,m)=\frac{2(m-j+1)}{(m+1)(m-2)},m\geq 4,2\leq j\leq m-1$ (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 $h_{i}$-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, Hu et al., (2010) 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, Shapiro et al.,
(1994); Hu, (2009)) proposed a new index called $h_{maj}$-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, Bucur et al., (2015) 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:

$Hirsch(p,t)=(h(p),h(t))$ (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 Würtz and Schmidt, (2016). 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($h_{1}$), (ii) Second authorship($h_{2}$), (iii) Third
authorship($h_{3}$) and (iv) Last authorship h-index ($h_{last}$). The
relatively high value of $h_{1}$ indicates that the author mostly worked as a
primary author, the relatively high value of $h_{2}$,$h_{3}$ indicates that
the author mostly worked as a secondary author with major contribution, and
the relatively high value of $h_{last}$ indicates that the author mostly work
as a senior investigator or supervisor.

Instead of distribution of citation among scholars, Aziz et al. (2013)Aziz and
Rozing, (2013) 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 $k^{th}$ author in $m^{th}$ authored article is defined as:

$W(k,m)=\frac{1+|m+1-2k|}{1/2~{}m^{2}+m(1-D)}$ (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:

$p=1-\frac{ME}{Pub_{all}}$ (37)

where, $Pub_{all}$ is the total number of published articles and ME is the
monograph equivalent of an author, which is defined as:

$ME(A)=\sum_{p=1}^{Pub_{all}}W(A,p)$ (38)

where, $Pub_{all}$ is the total number of publications and W(A,p) is the
weight of the author A in the $p^{th}$ 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.

Prathap, 2010a  proposed _Fractional and Harmonic p-index_ to account for the
number of authors. In the fractional p-index ($p_{f}$), the fractional credit
of an author is the 1/(total number of authors) and is defined as:

$p_{f}=(Cit_{f}^{2}/P_{f})^{1/3}$ (39)

where $cit_{f}$ is the total fractional citation count and $P_{f}$ is the
cumulative sum of fractional rank of co-authors.

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

$cit_{f}=\sum_{p=1}^{n}rank_{p}cit_{p}$ (40)

where,
$n$ is the total number of articles,
$rank_{p}$ is the rank of author in the $p^{th}$ article, and
$cit_{p}$ is the citation count of the $p^{th}$ article.

In harmonic counting method, the weighted credit of the $k^{th}$ scholar in an
$m$ 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:

$p_{h}=(Cit_{h}^{2}/P_{h})^{1/3}$ (41)

where, $cit_{h}$ is the total harmonic credit of citation counts and $P_{h}$
is the cumulative sum of the harmonic rank.

Instead of arithmetical allocation of citation counts to an author, Galam,
(2011) 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 $\alpha$ to the first and the $\beta$ to
the last author was given. The tailor based credit allocation of first, last
and the $t^{th}$ author is respectively defined as :

$g(1,n)=\frac{n+\alpha}{T_{n}}$ (42) $g(n,n)=\frac{n-1+\beta}{T_{n}}$ (43)
$g(t,n)=\frac{n-t}{T_{n}}$ (44)

where, $T_{n}=n(1+n)/2+\alpha+\beta$ and $n$ 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.

Liu and Fang, (2011) 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. 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. 2.

The credit allocation to the $k^{th}$ author in an m-authored article is
defined as:

$p(k,m)=m^{-\frac{1}{t}}k^{-(1-\frac{1}{t})}$ (45)

where, $t$ is the integral constant greater than one.

A smaller $t$ value give the credit balance among scholars rather than the
maximum $t$. 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 $p(k,m)$ and is defined as:

$p_{1-rank}(m)=\frac{\sum_{a=1}^{n_{(1-rank)}}m^{-\frac{1}{t}}k^{-(1-\frac{1}{t})}}{n_{(1-rank)}}$
(46)

  3. 3.

The normalized credit score of an individual scholar is defined as:

$p^{{}^{\prime}}(k,m)=\frac{p(k,m)}{S_{p}(m)}$ (47)

and

$p^{{}^{\prime}}_{1-rank}(m)=\frac{p_{(1-rank)}(m)}{S_{p}(m)}$ (48)

where $S_{p}(m)$ is the sum of all credit given to every author.

  4. 4.

The citation allocation of the $k^{th}$ scholar in the article $A$ is defined
as:

$CC_{(k,A)}=\begin{cases}cit_{A}\times p_{1-rank}(m),~{}~{}~{}~{}\textrm{For
first and corresponding author}\\\ cit_{A}\times
p^{{}^{\prime}}(k,m),~{}~{}~{}~{}\textrm{For others}\end{cases}$ (49)

Based on this credit allocation system two different indices were proposed
called CCA h-index ($h_{c}$) and CCA g-index ($g_{c}$)

_“The $h_{c}$-index of a scholar is $h_{c}$, if $h_{c}$ of his/her articles
have at least $h_{c}$ allocated citation count each.” _

_“The $g_{c}$ index of a scholar is $g_{c}$, if $g_{c}$ of his/her articles
have at least $g_{c}^{2}$ 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 Biswal, (2013). So we cannot discard the importance of non-primary
authors. To consider the importance of non-corresponding authors, the Absolute
index(Ab-index) Biswal, (2013) 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 $k^{th}$ author is defined as:

$Ab(k)=\sum_{A=1}^{m}PC_{A}(k)$ (50)

where, $PC_{A}(k)$ is the partial credit of scholar $k$ in the article $A$.
The partial credit of the first and the corresponding author is equal, and is
defined as:

$PC_{1_{st}/co}(p)=\frac{2*cit_{p}}{k+M}$ (51)

where,
$PC_{1_{st}/co}(p)$ indicates the total partial credit of the first or the
corresponding author in the $p^{th}$ article,
$cit_{p}$ is the citation count of the $p^{th}$ article,
$k$ is the total number of authors and,
$M$ 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 $r$ non-primary authors (neither the first nor the corresponding
author), then the credit share of $r^{th}$ non-primary author is defined as:

$PC_{r}=\frac{2*cit_{A}(k-r-CA+1)}{(k+f)(k-f+1)}$ (52)

where, $CA$, $f$, $k$ and $cit_{A}$ is the number of corresponding authors,
first author, total number of authors and the citation counts of the article
$A$ respectively.

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

Altmann et al., (2009) 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:

$NCT_{ak}=\frac{Cit_{k}}{age_{k}}*CF_{ak},0\leq CF_{ak}\leq 1$ (53)

where, $NCT_{ak}$ indicates the normalized citation score of author $a$ in
article $k$ and $CF_{ak}$ is the contribution factor of author $a$ in article
$k$; it lies between 0 & 1\. If the contribution of all authors is equal, then
$CF_{ak}$ 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:

$RP_{k}=max(RP|\frac{1}{RP}\sum_{p=1}^{RP}NCT_{pk}),RP\in P_{all}$ (54)

where, $P_{all}$ is the total number of publications of scholar. Altmann et
al., (2009) also gives a slightly modified definition of the RP-index as:

$\displaystyle
RP_{k}^{t}=\begin{cases}\frac{1}{RP_{k}}\sum_{t=1}^{RP_{k}}NCT_{tk},&\textrm{$if\frac{1}{RP_{k}}\sum_{t=1}^{RP_{k}}NCT_{tk}<RP_{k}+1$}\\\
RP_{k}+0.99,&\textrm{$if\frac{1}{RP_{k}}\sum_{t=1}^{RP_{k}}NCT_{tk}\geq
RP_{k}+1$}\end{cases}$ (55)

The contribution of collaborators also plays an important role in the
scientific assessment of a scholar. Based on this concept, Abbasi et al.,
(2009) and Abbasi et al., (2010) 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:

$RC_{k}=max(k|\frac{1}{n}\sum_{j=1}^{n}CCV_{j}\geq n),n\in M$ (56)

where, $M$ is the total number of collaborators and $CCV_{j}$ is the co-author
collaboration value of author $k$ with co-author $j$, and is defined as:

$\displaystyle
CCV_{j}=Total~{}number~{}of~{}collaborations~{}between~{}author(k,j)*RP_{j}^{{}^{\prime}}$
(57)

where $RP_{j}^{{}^{\prime}}$ is the RP-index of scholar j.

After a slight modification of the RC-index, the $RC^{{}^{\prime}}$-index is
defined as:

$\left.\begin{aligned}
RC^{{}^{\prime}}_{k}=\begin{cases}\frac{1}{RC_{k}}\sum_{j=1}^{RC_{j}}CCV_{c},&\textrm{$if\frac{1}{RC_{k}}\sum_{j=1}^{RC_{j}}CCV_{c}<RC_{j}+1$}\\\
RC_{k}+0.99,&\textrm{$if\frac{1}{RC_{k}}\sum_{j=1}^{RC_{j}}CCV_{c}\geq
RC_{j}+1$}.\end{cases}\end{aligned}\right.$ (58)

The $RC^{{}^{\prime}}_{k}$ values lies between $RC_{k}$ and $RC_{k}$ +1.

The Community collaboration index called _CC-index_ is defined as:

$CC_{k}=max(k|\frac{1}{n}\sum_{j=1}^{n}RC_{j}\geq n),n\in M$ (59)

After a slight modification of the CC-index, the $CC^{{}^{\prime}}$-index is
def]ined as:

$\displaystyle
CC^{{}^{\prime}}_{k}=\begin{cases}\frac{1}{CC_{k}}\sum_{j=1}^{CC_{j}}RC_{c},&\textrm{$if\frac{1}{CC_{k}}\sum_{j=1}^{CC_{j}}RC_{c}<CC_{j}+1$}\\\
CC_{k}+0.99,&\textrm{$if\frac{1}{CC_{k}}\sum_{j=1}^{CC_{j}}RC_{c}\geq
CC_{j}+1$}.\end{cases}$ (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

Table 3: Summary of Index based on total number of author Index |  Definition |  Publication
---|---|---
$h_{i}$-index |  _“The $h_{i}$-index of a scholar is the ratio of the h-index and the average number of scholars in the h-core articles.”_ $h_{i}=\frac{h}{Avg_{A}}$, where $h$ is the h-index and $Avg_{A}$ is the average number of scholars from h-core articles. |  Batista et al., (2006)
Pure h-index |  _$h_{p}(A)=\frac{K}{\sqrt{E(author)}}$ , where $h$ is the h-index and $E(author)$ is the average number of scholars in the h-core articles._ |  Wan et al., (2007)
Pure R-index |  _$R_{p}(A)=\sqrt{\frac{\sum_{P\in h(pub)}C(A,P)}{E(author)}}$_ |  Wan et al., (2007)
Adaptive pure h-index($h_{ap}$) |  _$h_{ap}=\frac{(h_{eff}+1)*cit_{eff}(h_{eff})-h_{eff}*cit_{eff}(h_{eff}+1)}{cit_{eff}(h_{eff})-cit_{eff}(h_{eff}+1)+1}$ , where $h_{eff}$ represents h-index based on effective citation count._ |  Chai et al., (2008)
Normalized $h_{i}$-index |  _“The normalized $h_{i}$-index of a scholar is k, if k of his/her articles have at least k normalized citation count each.”_ |  Wohlin, (2009)
Fractional h-index ($h_{f}$) |  _“The fractional h-index ( $h_{f}$) of a scholar is $h_{f}$, if $h_{f}$ of his/her articles have at least $h_{f}$ fractional citation count each.”_ |  Egghe, (2008)
Fractional g-index ($g_{f}$) |  _The fractional g-index ( $g_{f}$) of an author is $g_{f}$ such that the top $g_{f}$ papers has at least $g_{f}^{2}$ cumulative fractional citation count each._ |  Egghe, (2008)
Harmonic h-index |  _“The harmonic h-index of a scholar is $H_{h}$, if $H_{h}$ of his/her articles have at least $H_{h}$ harmonic credits each.”_ |  Hagen, (2008)
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.”_ |  Abbas, (2011)
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.”_ |  Zhang, 2009b
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)) |  Bucur et al., (2015)
Profit h-index ($P_{h}$) |  $p_{h}=1-\frac{h_{a}}{h}$, Where $h_{a}$ is the adjusted h-index of author and h is the actual h-index of the corresponding author. |  Aziz and Rozing, (2013)
Fraction p-index ($p_{f}$) |  $p_{f}=(Cit_{f}^{2}/P_{f})^{1/3}$, where $cit_{f}=\sum_{p=1}^{n}rank_{p}cit_{p}$ |  Prathap, 2010a
Harmonic p-index($p_{h}$) |  $p_{h}=(Cit_{h}^{2}/P_{h})^{1/3}$ |  Prathap, 2010a
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.”_ |  Galam, (2011)
$h_{c}$-index |  _“The $h_{c}$-index of a scholar is $h_{c}$, if $h_{c}$ of his/her articles have at least $h_{c}$ allocated citation count each.”_ |  Liu and Fang, (2011)
$g_{c}$ index |  _“The $g_{c}$ index of a scholar is $g_{c}$, if $g_{c}$ of his/her articles have at least $g_{c}^{2}$ citation together.”_ |  Liu and Fang, (2011)
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.”_ ($Ab(k)=\sum_{A=1}^{m}PC_{A}(k)$, where $PC_{A}(k)$ is the partial credit of author k in article A.) |  Biswal, (2013)
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.”_ |  Altmann et al., (2009)
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.”_ |  Abbasi et al., (2009)
CC-index |  _$CC_{k}=max(k|\frac{1}{n}\sum_{j=1}^{n}RC_{j}\geq n),n\in M$_ |  Abbasi et al., (2010)

###  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 Martin, (1996); Van Leeuwen et al., (2003). In
this context, the hg-index, $q^{2}$-index and $h_{mc}$-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 articlesAlonso
et al., (2010). 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:

$hg=\sqrt{h.g}$ (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.

Cabrerizo et al., (2010) categorizes the indices in two categories: the first
one is based on the productive core of a scholar like h-index(Hirsch, (2005)),
g-index(Egghe, 2006b ), hg-index(Alonso et al., (2010)) & $h^{(2)}$
-index(Kosmulski, (2006)), while the second one is based on the impact of core
papers like a-index (Jin, (2006)), m-index(Bornmann et al., (2008)), AR-
index(Jin et al., (2007)) & $h_{w}$-index(Egghe and Rousseau, (2008)). 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, Cabrerizo et al.,
(2010) proposed a new index called _$q^{2}$ -index_ that considers both the
number of productive core articles and the impact of core articles. Formally,

_“The $q^{2}$-index of scholars is the geometric mean of h-index and
m-index.”_

Mathematically, it is defined as:

$q^{2}=\sqrt{h.m}$ (62)

where, the h and m is the h and m-index value of scholar.
The $q^{2}$-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 $q^{2}$-index is
relatively low.

Another index designed to combine the properties of $h_{m}$-index (Hagen,
(2008)) and the $h_{c}$-index (Liu and Fang, (2011)) is the $h_{mc}$-index
(Liu and Fang, (2012)). In $h_{m}$-index, the citation count is distributed
equally to all the scholars ($cit/n$, where $cit$ is the citation count and
$n$ is the total number of scholars). Instead of this, the author used
combined credit allocation for most important author (MIA) (Liu and Fang,
(2011) and Schreiber, 2008b ) and the cumulative sum of the combined credit
allocation, named effective paper count, is used for publication ranking.
Mathematically, it is defined as:

$p_{eff}(k)=\sum_{p=1}^{k}p(rk(p),m(p)$ (63)

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

_“The $h_{mc}$-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:

$h_{mc}=\max_{k}(p_{eff}(k)\leq cit(k)$ (64)

where, $p_{eff}(k)$ is the effective paper count of the $k^{th}$ article and
$cit(k)$ is the citation count of the $k^{th}$ article. If the tail articles
earn extra citation count in the near future and core articles citations
remain unchanged, then $h_{mc}$-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.

Table 4: Summary of Index based on combination of two index Index |  Definition |  Publication
---|---|---
hg-index |  _“The hg-index of a scholar is the geometric mean of the h and g-index.”_ ($hg=\sqrt{h.g}$) |  Alonso et al., (2010)
$q^{2}$-index |  _“The $q^{2}$-index of scholars is the geometric mean of h-index and m-index.”_ ($q^{2}=\sqrt{h.m}$) |  Cabrerizo et al., (2010)
$h_{mc}$-index |  _“The $h_{mc}$-index of a scholar is k, if k of his/her articles have at least k effective article count each.”_ ($h_{mc}=\max_{k}(p_{eff}(k)\leq cit(k)$) |  Liu and Fang, (2012)

###  2.6 Indices based on excess citation count

The scientific impact of an article is measured in terms of citation count and
the scientific impact of a scholar is computed based on the citation count of
articles. An article that earned more citations than others is considered more
influential and a scholar having a higher index value than others is
considered a prominent actor. At some stage, further addition in citation
counts of an article does not contribute in improving the values of most of
the popular indices, but every citation count has its own importance in the
scientific assessment of scholars. Such citations are called excess citations.
To account for the importance of excess citation count, several indices were
designed.

Let scholar $X$ publish 10 articles with 100 citations each and scholar $Y$
also publish 10 articles, but with 10 citations each. In both cases, the
h-index value is equal, however, the scientific impact of scholar $X$ is more
than scholar $Y$. In another case, let scholar $X$ publish five articles with
100 citations each and scholar $Y$ also publish 10 articles, but with 8
citations each. The h-index of scholar $Y$ is more than scholar $X$, but the
total scientific impact of scholar $Y$ is significantly less than that of
scholar $X$. To overcome this limitation of h-index, Zhang, 2009a  proposed a
new index called _e-index_ which considers only the excess citation. The
e-index of a scholar is defined as:

$e^{2}=\sum_{p=1}^{h}(cit_{p}-h)=\sum_{p=1}^{h}cit_{p}-h^{2}$ (65)

where, $cit_{p}$ is the citation count of the $p^{th}$ article and $h$
represents the h-index. This index considers only the excess number of
citation counts and completely ignore the core citations. This is one of the
main limitations of e-index.

The _j-index_ proposed by Todeschini, (2011) uses the weighted citation
incremental approach. Here, the author suggested 12 arbitrary weighted
citation increments ($\Delta$wc) like 500, 250, 100, 50, 25, 10, 5, 4, 3, 2,
1.5 & 1.25 and the corresponding weight($wt$) as $1/R$, where $R$ is the rank
of weighted citation increment. Formally, the j-index is defined as:

$j=h+\frac{\sum_{r=1}^{12}wt_{r}M_{r}(h\Delta h_{r})}{\sum_{r=1}^{12}wt_{r}}$
(66)

where, $M_{r}(h\Delta h_{r})$ is number of publications that have at least
$h\Delta h_{r}$ citations each.

From the above discussion, we can conclude that neither the e-index nor the
j-index fairly account the excess citation count in the scientific assessment
of scholars. The e-index considers only the excess citation count of h-core
articles and completely ignores the core citation count. Hence, it would be
difficult to imagine the scientific assessment of scholars without the core
citation count. The j-index works on the concept of weighted incremental
approach and how the weights are incremented is one much debatable. In this
context one recent index has been proposed by Bihari and Tripathi, (2017)
named _EM-index_. The EM-index of an author is the square root of sum of the
component of EM-index. The component of the EM-index is the h-index calculated
at multi-level. The first component of EM-index is the original h-index,
second component is the h-index from the excess citation count of h-core
article. Third component is the h-index from second level excess citation
count and so on. Mathematically, the EM-index is defined as:

$EM=\sqrt{\sum_{c=1}^{m}E_{c}}$ (67)

where m is the total number of component of EM-index. This index helps us to
differentiate two different researcher with same h-index. The summary of index
based on the combination of two indices shown in the table 5.

Table 5: Summary of Index based on excess citation count Index |  Definition |  Publication
---|---|---
e-index |  $e^{2}=\sum_{p=1}^{h}(cit_{p}-h)=\sum_{p=1}^{h}cit_{p}-h^{2}$ |  Zhang, 2009a
j-index |  $j=h+\frac{\sum_{r=1}^{12}wt_{r}M_{r}(h\Delta h_{r})}{\sum_{r=1}^{12}wt_{r}}$ |  Todeschini, (2011)
EM-index |  $EM=\sqrt{\sum_{c=1}^{m}E_{c}}$ |  Bihari and Tripathi, (2017)

###  2.7 Indices based on total publication

The scientific product of a scholar is research articles and every article has
its own impact on the research career of a scholar. However, the scientific
assessment of a scholar is done with the help of only a few numbers of high
impact articles and leave many articles that have citation counts near to the
h-index value. It seems that the articles with citation count equal to or a
little less than the h-index value are not used in the scientific assessment
of a scholar. Such articles, referred to as h-tail articles, are obviously
important, however, even the h-tail articles with lesser citations can be
significant in assessing the impact of a scholar, and hence, should not be
ignored. Motivated by this, several studies have been done to incorporate the
impact of all those articles that are cited at least once.

Anderson et al., (2008) proposed a new index called _tapered h-index_ based on
the total number of cited articles. Tapered h-index is defined based on the
Ferrers graph of publications and their citation counts. This graph assigns
positive scores to all publications with non-zero citation count. To score an
h-index value of 1, only one article with one citation count is required, to
score a value of 2, a scholar requires at least 2 publications with 2
citations each. The increment of h-index value from 1 to 2 requires 3 more
citations: 1 for the existing article and 2 for the new article. This
increment is shown in the Ferrer graph with the contribution score of 1/3 in
Fig. 2. The figure also shows the score of all other articles.

Figure 2: Score of article in tapered h-index

In the Ferrer graph, a column represents a partition of citation counts of the
articles and a row represents the total citation count. The largest square in
the Ferrer graph is called Durfee square; its length represents the h-index
value. The score of any citation in the Ferrer graph is defined as:$1/(2l-1)$,
where l is the length of any side in the Ferrer graph. The score of an article
is simply the sum of the score at every level in the Ferrer graph. The score
of the $p^{th}$ article in the Ferrer graph is defined as:

$\displaystyle h_{T}(p)=$ $\displaystyle\frac{cit_{p}}{2p-1},$ if $cit_{p}\leq
p$, (68) $\displaystyle h_{T}(p)=$
$\displaystyle\frac{p}{2p-1}+\sum_{p+1}^{cit_{p}}\frac{1}{2p-1},$ if
$cit_{p}>p$. (69)

where, $cit_{p}$ is the citation count of the $p^{th}$ article. Finally, the
tapered h-index ($h_{T}$) of a scholar is the sum of the $h_{T}$ scores of all
cited articles. Formally, it is defined as:

$h_{T}=\sum_{p=1}^{m}h_{T}(p)$ (70)

This index tries to incorporate all citation count in the scientific
assessment of scholar, but the computation process of this index is little bit
complex.

In this way, García-Pérez, (2009) proposed the _multidimensional h-index_ to
account for the importance of all cited articles. To compute the
multidimensional h-index, first the citation count of all those articles which
has at least one citation count is stored in an m-dimensional vector in
descending order. Then, the multidimensional h-index is defined as:
H=($h_{1}$, $h_{2}$, $h_{3}$, ……, $h_{p}$), where $h_{1}$ is the first h-index
from all $m$ publications, $h_{2}$ is the second subsequent h-index from first
tail (m-$h_{1}$) publications, and so on, until the entire publication
citations are exhausted. The component of multidimensional h-index should be
such that $h_{1}~{}\geq~{}h_{2}~{}\geq h_{3}$…. . The $k^{th}$ component of
the multidimensional h-index is defined as:

$h_{k}=p-\sum_{m=0}^{k-1}h_{m}$ (71)

where, $p$ is the largest integer in the range $\sum_{m=0}^{k-1}h_{m}+1\leq
p\leq k$ that satisfies $cit_{p}\geq p-\sum_{m=0}^{k-1}h_{m}$ (where
$h_{0}=0$).

The total number of components in multidimensional h-index is always less than
or equal to the total cited publication count. If all articles have a single
citation, then the total number of components in the multidimensional h-index
is equal to the total number of publications, otherwise, it is always less
than the total cited publication count. The number of components and their
values vary from scholar to scholar. This shows the significant difference
between scholars.

For example, let a scholar publish 20 articles. Out of these 20, let 17
articles have at least one citation count: Cit=(100, 20, 20, 17, 10, 10, 9, 9,
8, 8, 5, 4, 4, 3, 2, 2, 1). Then the components of the multi-dimensional
h-index are {8, 4, 2, 2, 1}. If we take the reasonable minimum component value
to be 4, then the components of the multi-dimensional h-index are {8, 4}. The
sum of the component of the multidimensional h-index is 17 equal to the total
number of cited publications. But this index does not produce a global index
value to show the overall impact of scholars. To overcome this limitation,
Todeschini and Baccini, (2016) defined a new index called iteratively weighted
h-index. The iteratively weighted h-index of a scholar is defined as:

$iw(h)=\sum_{c=1}^{p}\frac{h_{c}}{c}$ (72)

where, $p$ is the total number of components and $h_{c}$ is the $c^{th}$
component of the multidimensional h-index.

Franceschini and Maisano, 2010b  proposed a new variant with additional sign
to account all the cited publications in scientific assessment. The additional
sign is the total time-range of all those articles that have at least one
citation count. It is defined as:

$f=range_{k\in p_{all}}(yr_{1},yr_{2},yr_{3},........,yr_{k})+1$ (73)

where, $P_{all}$ is the set of all cited publications and $yr_{k}$ is the
publication year of $k^{th}$ article.

The main feature of this indicator is that it never decreases and is never
more than the total publication career. For example, to compute the f-range,
consider an artificial example of the publication history of scholar $A$ as
given in Table 6. Here, $A$ publishes 10 articles such that out of these 10, 8
articles have at least 1 citation count.

Table 6: Example of f-value Year | 2000 | 2000 | 1989 | 2001 | 2005 | 2009 | 2010 | 2010 | 1990 | 2014
---|---|---|---|---|---|---|---|---|---|---
Citation Count | 40 | 30 | 29 | 12 | 5 | 5 | 5 | 1 | 0 | 0
Rank | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10

The h-index of scholar is 5 and the f-value of scholar A = {2000, 1989, 2001,
2005, 2009, 2010} + 1 = 7. Here, +1 is done to account for the preparation of
the first article. The scientific impact of a scholar with this additional
sign is categorized in the following four categories:

  1. 1.

Low h-index and low f-value: It indicates new scholars.

  2. 2.

Low h-index and high f-value: It indicates the scholar working from a long
time but their research impact is relatively less.

  3. 3.

High h-index and low f-value: It indicates a young prominent scholar.

  4. 4.

High h-index and high f-value: It indicates the scholar worked hard in the
research community from long time.

This index helps in comparing the scientific impact of scholars having
different research careers.

Smith, (2015) proposed the _Platinum H_ -index that covers the total citation
count, total research career and the total publication count. Formally, it is
defined as:

$PlatinumH=\frac{H}{CL}\times\frac{Cit_{all}}{Pub_{count}}$ (74)

where, $H$ is the h-index, $CL$ is the career length, $Cit_{all}$ is the total
citation count and $pub_{count}$ is the publication count. The computation
process of the platinum H-index is very simple and comparable to that of other
than the h-index. The Platinum H-index considers the total number of articles;
these articles can be either cited or otherwise. However, the logic behind
inclusion of non-cited publications in scientific assessment is not clear.
Motivated by this, Yaminfirooz and Gholinia, (2015) proposed a new index
called multiple h-index (mh-index).

_“The mh-index of a scholar is the square root of the product of h-index at
every level and the square of citation count of an article is divided by the
age of the publication.”_

Mathematically, it is defined as:

$mh=\sqrt{\sum_{p=1}^{n}}\sum_{k=1}^{h_{p}}\frac{h_{p}cit_{k}^{2}}{age_{k}}$
(75)

where,
$h_{p}$ is the scholar’s $p^{th}$ h-index,
$cit_{k}$ is the citation count of $k^{th}$ article, and
$age_{k}$ is the age of $k^{th}$ article.

To calculate the mh-index, first calculate the h-index from all those articles
that have at exactly one citation count. Then calculate the h-index from all
those articles that have 2 citation count, and so on. Finally, calculate the
mh-index.

All of the indices which considers the total number publications do not
account the importance of citation count of all articles. To overcome this
limitations Bihari and Tripathi, (2017) proposed a new index called
$EM^{{}^{\prime}}$-index. The $EM^{{}^{\prime}}$-index of an author is the
square root of sum of the component of $EM^{{}^{\prime}}$-index. The component
of the $EM^{{}^{\prime}}$-index is the h-index, calculated at multi-level. The
first component of $EM^{{}^{\prime}}$-index is the original h-index, after
that subtract the h-index value from core-publications citation count, then
the second component of $EM^{{}^{\prime}}$-index is h-index value from updated
citation data set. This process will continue until the all citation count
exhausted or the all publications citation remain one or only one article
citation count is more than zero. Mathematically, the $EM^{{}^{\prime}}$-index
is defined as:

$EM^{{}^{\prime}}=\sqrt{\sum_{c=1}^{m}E_{c}^{{}^{\prime}}}$ (76)

where m is the total number of component in $EM^{{}^{\prime}}$-index.

In this section, we have discussed a number of indices that consider the total
number of publications that have at least one citation count in the scientific
assessment of scholars. Every indicator gives a perspective to consider the
impact of the total number of articles, but none of the indices consider the
total number of scholars and the total citation counts with the total number
of publications. This may provide a good alternative in the scientific
assessment of scholars. The summary of index based on the total publication
count shown in the table 7.

Table 7: Summary of Index based on total publication count Index |  Definition |  Publication
---|---|---
Multidimensional h-index |  _Multidimensional h-index is the set of h-index. It captures all publication which has at least one citation count. Multidimensional h-index is defined as: H=( $h_{1}$, $h_{2}$, $h_{3}$, ……, $h_{p}$). Where $h_{1}$ is first h-index, $h_{2}$ is subsequent h-index of k-$h_{1}$ papers and so on._ |  García-Pérez, (2009)
Iteratively weighted h-index |  $iw(h)=\sum_{c=1}^{p}\frac{h_{c}}{c}$, where, $p$ is the total number of components and $h_{c}$ is the $c^{th}$ component of the multidimensional h-index. |  Todeschini and Baccini, (2016)
Platinum H-index |  $PlatinumH=\frac{H}{CL}\times\frac{Cit_{all}}{Pub_{count}}$, where, $H$ is the h-index, $CL$ is the career length, $Cit_{all}$ is the total citation count and $pub_{count}$ is the publication count. |  Smith, (2015)
mh-index |  _“The mh-index of a scholar is the square root of the product of h-index at every level and the square of citation count of an article is divided by the age of the publication.”_ |  Yaminfirooz and Gholinia, (2015)
$EM^{{}^{\prime}}-index$ |  $EM^{{}^{\prime}}=\sqrt{\sum_{c=1}^{m}E_{c}^{{}^{\prime}}}$ |  Bihari and Tripathi, (2017)

###  2.8 Indices based on other variants

Instead of the total number of publications, the age of publications, the
number of co-authors, the excess citation count, etc., several indices
consider other parameters such as a successful paper, the total number of
citers (reader), the citation curve and many more to assess the scientific
impact of scholars.

####  2.8.1 Based on core & tail citation

Generally, the citation count of a scholar’s articles is categorized into
three categories: (i) the citation count used in h-index computation, (ii)
excess citation count and (iii) the citation count not used in the h-index
computation (tail-citations). Bornmann et al., (2010) referred to all those
categories as the center (the h-index citation count), upper (the excess
citation count) and the lower (the tail articles citation count). Based on
this distribution of citation count in citation curve, the author proposed
three different measures to evaluate the scientific impact of scholars:
_$h^{2}$ -lower, $h^{2}$-center and $h^{2}$-upper_.

These are defined as:

$\displaystyle
h^{2}upper=\frac{\sum_{k=1}^{h}(cit_{k}-h)}{\sum_{k=1}^{m}cit_{k}}\times
100=\frac{e^{2}}{\sum_{k=1}^{m}cit_{k}}\times 100$ (77)
$h^{2}center=\frac{h\times h}{\sum_{k=1}^{m}cit_{k}}\times 100$ (78)
$h^{2}lower=\frac{\sum_{k=h+1}^{m}(cit_{k}-h)}{\sum_{k=1}^{m}cit_{k}}\times
100$ (79)

where,
$h$ is the h-index,
$cit_{k}$ is the citation count of the $k^{th}$ article,
$e^{2}$ is the excess citation, and
$m$ is the total number of articles.

The main objective of this index is to assess the scientific impact of
scholars at three different levels, which resolve the excess and tail-citation
issue of h-index. A scholar having high value in the upper section shows the
high impact of the scholar and is called as perfectionist scientist. While a
scholar having a low value in the upper part and high value in the lower part
shows the low impact of the scholar (huge number of publication with less
citation). A scholar having high center value shows the average impact of the
scholar in the community. Instead of three different measures, the only one
measure should be used in the scientific assessment of scholars Zhang, (2013).
To do in this way, Zhang, (2013) proposed an improved version of
$h^{2}$-lower,$h^{2}$-center and $h^{2}$-upper indices called
$h^{{}^{\prime}}$-index. The $h^{{}^{\prime}}$-index of a scholar is defined
as:

$h^{{}^{\prime}}=Rh=\frac{eh}{t}$ (80)

where $R$ represents the head-tail ratio of e and t-index.

Fred and Rousseau, (2010) proposed the _k-index_ that covers the core-tail
citation ratio of a scholar. It is also called the tail-core ratio. Formally,

$k=\frac{(Cit_{all}/Pub_{count})}{(Cit_{T}/Cit_{H})}$ (81)

where,
$Cit_{all}$ is the total citation count,
$Pub_{count}$ is the total publication count,
$Cit_{T}$ is the total citation count of h-tail articles, and
$Cit_{H}$ is the total citation count of h-core articles.

The k-index of a scholar at time $t$ is defined as:

$k(t)=\frac{(Cit_{all}(t)/Pub_{count}(t))}{(Cit_{T}(t)/Cit_{H}(t)}$ (82)

After a slight rearrangement, the above can be defined as:

$k(t)=\frac{(Cit_{all}(t)Cit_{h}(t))}{Pub_{all}(t)(Cit_{T}(t)-Cit_{H}(t))}$
(83)

This index suffers with divide by zero problem when h-tail is zero and also it
gives an average impact of scholarsRousseau and Leydesdorff, (2011). To
overcome the divide by zero problem of k-index, Chen et al., (2013) modified
the k-index as the $k^{{}^{\prime}}$-index:

$k^{{}^{\prime}}=\frac{Cit_{all}-pub_{count}}{Cit_{T}-Cit_{H}}$ (84)

The _Two sided h-index_ by García-Pérez, (2012) considers the core-tail ratio
and states that the citation curves of two different scholars having equal or
similar h-index value are different because of the variation of citation count
in their h-core and h-tail publications.The two sided h-index of a scholar for
length $m$ is defined as:

$h\pm m=(h_{-m},........,h_{-1},h_{0},h_{1},..........,h_{m})$ (85)

where,
$h_{0}$ represents the original h-index,
the negative script represents the citations of the h-core articles with
consecutive square up the top of the citation curve, and
the positive script represents the citations of the h-tail articles with
consecutive square out of the tail of citation curve.

The negative scripted element is calculated by finding the highest rank $R$ in
h-core for $R\leq h_{m-1}$ satisfying $cit_{R}\geq R+\sum_{k=0}^{m-1}h_{-k}$.
Then $h_{-m}=R$ and the positive subscripted element is calculated by finding
the largest rank $R$ for $\sum_{k=0}^{m-1}h_{k}+1\leq R\leq N$ satisfying
$cit_{R}\geq R-\sum_{k=0}^{m-1}h_{k}$, then $h_{m}=R-\sum_{k=0}^{m-1}h_{k}$.

In this subsection, we have discussed all those indices that estimate the
scientific impact of scholars based on the ratio of h-core and h-tail articles
citation count. But most of the indices are very difficult to compute when
compared with the h-index.

####  2.8.2 Based on improvement of h-index to higher value

The h-index considers only a few highly influential articles and can be
limited due to the complete ignorance of the citations of the h-tail articles.
Ignoring h-tail articles is not good because some of the articles have equal
or little less citation counts than the h-index value. Such h-tail articles
are obviously important in the improvement of next higher h-index. Motivated
by this, Ruane and Tol, (2008) proposed a new variant of h-index named
_rational h-index ( $h_{rat}$)_. It is continuous in nature and not only
considers the number of significant articles (h-index), but also the
fractional progress of the scientific impact to reach the next higher value of
h-index. Formally, it is defined as:

$h_{rat}=h+1-\frac{k}{2h+1}$ (86)

where, $h$ is the h-index and $k$ is the number of citations required to reach
$h+1$ h-index value.

The maximum number of citations required to reach the next higher h-index is
2h+1. This index is almost similar to the h-index and the only difference is
the fractional part of the index value.

Another similar approach has been discussed by Guns and Rousseau, (2009)
called _real h-index_. Mathematically, the real h-index ($h_{r}$) is defined
as:

$h_{r}=\frac{(h+1)cit_{h}-h.cit_{h+1}}{1-cit_{h+1}+cit_{h}}$ (87)

where, h is the h-index and $cit_{h}$ is the citation count of $h^{th}$
article.

Wu, (2010) proposed the w(q)-index, where $q$ is the minimum number of
additional citations required to enhance the w-index to w+1. Theoretically, it
is almost similar to rational h-index, but the improvement is sought of
w-index and not to the h-index. Further, it is not restricted to only w+1,
w+2, w+3 and many more also possible. Apart from these indices, Nair and
Turlach, (2012) used the probability theory on the total minimum citation
count required to reach the next level. They proposed the _stochastic
h-index_. The stochastic h-index ($h_{s}$) of a scholar is defined as:

$h_{s}=h+P$ (88)

where, $P$ is the probability of articles to become a part of h-core based on
current publication to increase the index value to the next within a specific
time period (e.g., within a year). Mathematically, it is defined as:

$h_{s}=h_{0}+p=h_{0}+1-r=h_{0}+1-\sum_{p=0}^{m}r_{p}$ (89)

where, $r_{p}$ is the probability of the $p^{th}$ article from h-tail with
citation count at least $h_{0}$ +1 within one time unit as computed based on
Poisson distribution with distribution rate $\lambda_{p}$. It is defined as:

$\lambda_{p}=\frac{cit_{p}}{Y_{c}-Y_{pub}(p)+1}$ (90)

where, $Y_{c}$ and $Y_{pub}$ represent the current and the publication year of
$p^{th}$ article respectively. Then, $r_{p}$ is defined as:

$r_{p}=1-P_{k}=P(cit_{k}\leq ct_{k})$ (91)

where, $P(cit_{k}\leq ct_{k})$ represents the probability of additional
citation count to reach the next h-index.

In this subsection, we have discussed all those indices that try to gauge how
the index value will increase to the next higher value in near future. Each of
the indices gives its own mechanism to improve the index value to the next
higher value.

####  2.8.3 Based on variants of citation process

The impact of an article is measured in terms of the citation count, which
refers to all those articles that cite the given article. However, it seems
that a single article is cited by multiple articles of a scholar. Hence,
instead of the total citation count, the total number of unique citers
(readers) Ajiferuke and Wolfram, (2010) can be an effective alternative to
assess the scientific impact of an article. Motivated by this, Isola and
Wolfram, (2009) used the total number of unique citers to assess the
scientific impact of an article and proposed a new index called _ch-index_.
Formally, it is defined as:

_“The ch-index of a scholar is k if k of his/her articles have at least k
citers each.”_

To compute the ch-index, first we calculate the citers of every article. Then,
we sort the publications in the descending order of their citer-count. The ch-
index is the largest rank corresponding to their citer count. But one of the
main difficulty with the ch-index is to find the total number of citers of an
article because none of the publications databases have citer information. So,
it is a very difficult task to do.

Katsaros et al., (2009) give an alternative way to measure the total impact of
an article by the number of distinct cited scholars called the _coterminal
citation count_ and proposed a new index called _f-index_. The f-index of a
scholar is defined as :

_“The f-index of a scholar is k if k of his/her articles have at least k
coterminal citation each.”_

The coterminal citation count of an article is the dot product of
m-dimensional quantities and $S$. The quantities of an article is the ratio of
the cardinalities of the set of authors of the citing article and the total
number of distinct authors of all the cited article. $S$ is the vector
containing ${m,m-1,m-2,.....,1}$. Where $m$ is the total number of citing
articles. The series of the set of the author is defined as:

$S_{K}^{A}=\\{A_{p}:author~{}A_{p}~{}present~{}precisely~{}in~{}K^{th}~{}citing~{}article\\}$
(92)

The quantities of citing article are defined as:

$F_{k}^{A}=\frac{|S_{K}^{A}|}{Total~{}distinct~{}citing~{}author}$ (93)

For calculation of the f-index, we first calculate the co-terminal citations
of all article, then arrange all articles in descending order of their
coterminal citation count. Then the f-index of a scholar is the largest rank
in which the coterminal citation count is greater than or equal to the rank.
This index tries to remove the author reparation from the citation process.
But this index is a little bit difficult to compute.

An article is written with the help of other articles it refers to. In this
way, the impact of an article can be based on the impact of the refereed
articles. If an article refers to the highly influential article, then it has
good chances to gain more higher impact in future. Along these lines,
Schubert, (2008) proposed the _Single publication h-index_. The single
publication h-index is based on highly cited referred articles. For
calculation of single publication h-index, first we arrange the referred
articles in the descending order of their citation counts. The single
publication h-index is the h-index value from the reference list. Using the
single publication h-index, Egghe, 2011a  proposed the _Indirect h-index_. For
the calculation of the indirect h-index, we first calculate the single
publication h-index of all publications. Then, we arrange all the publications
in the descending order of their single publication h-index. The indirect
h-index of a scholar is defined as:

_“The indirect h-index of a scholar is m, if m of his/her articles have at
least m single publication h-index each.”_

Egghe, 2011b  discussed the indirect h-index and improved this index based on
Lotka’s principle:

$IH=h^{\frac{\alpha}{1+\alpha^{{}^{\prime}}(\alpha-1)}}$ (94)

where $\alpha$ and $\alpha^{{}^{\prime}}$ are Lotka parameters defined as
$\alpha$ $>$ 1 and $\alpha^{{}^{\prime}}$ is:

$\alpha^{{}^{\prime}}=\frac{ln(h)}{ln(H_{h})}$ (95)

Another index based on single publication h-index is the _average h-index_ :

_“The average h-index of a scholar is the average of the single publication
h-index of h-core articles.”_
Mathematically, it is defined as:

$\overline{H}=\frac{1}{h}\sum_{k=1}^{h}H_{k}$ (96)

But the single publication h-index is very difficult to calculate because none
of the publications databases has any information related to reference
citation count.

The citation count of an article is influenced by the self-citation count,
citations from co-authors articles and from institutional colleagues. All
those types of citations are treated as self-citation. Instead of the total
number of citation count, the total number of international recognition is a
much better way to know the scientific impact of an article. Based on this
international recognition Kosmulski, 2010b  proposed the _$h_{int}$_ -index.
The $h_{int}$-index of a scholar is defined as:

_“The $h_{int}$-index of a scholar is k, if k of his/her articles have at
least k number of international recognition each.”_

The international recognition of an article is the sum of all those articles
that refer to other country scholars’ publications. To compute the
$h_{int}$-index, first calculate the total number of international recognition
of each article. Then sort the publication data into the descending order of
their total number of international recognition. The $h_{int}$-index of a
scholar is the largest rank in which the corresponding international
recognition value is greater or equal than the rank.

In a research article, it seems that the some of the references are cited at
once while some are cited multiple times. In both cases, the citation count of
an article remains the same. So, instead of total citation count, the total
number of the distinct author or the total number of international
recognition, the total number of citation mention could be used an effective
alternative to assess the impact of an article. To incorporate the citation
mentioned, Wan and Liu, (2014) proposed a new variant of h-index called _wl-
index_.

_“The wl-index of a scholar is the largest integer k, if k of his/her articles
have at least k citation mentioned each.”_

In this subsection, we have discussed all those indices that are based on
alternative citation processes of an article. Every index gives its own
benchmark to know the scientific impact of articles and scholars.

####  2.8.4 Miscellaneous indices

Many other indices that have not been covered under the above mentioned
categories have been covered in this section. They are also important in
scientific assessment of scholars. All such indices mentioned in this section
are trivial to calculate than the other indices mentioned previously.

Van Eck and Waltman, (2008) mentioned that the h-index definition is somewhat
arbitrary Lehmann et al., (2006, 2008) being based on the number of highly
cited articles. A single arbitrary number is not fair in assessing the
scientific impact of scholars and while comparing two different scholars. To
overcome this limitation Van Eck and Waltman, (2008) proposed a new index
called _$h_{\alpha}$ -index_, which depends on a parameter $\alpha$.

_“The $h_{\alpha}$-index of a scholar is k, if k of his/her articles have at
least $\alpha k$ citations each.”_

where $\alpha>1$.

For example, let scholar $X$ publish 20 articles with 20 citations each. Let
scholar $Y$ publish 10 articles with 100 citations each. Then, the h-index of
scholar $X$ is 20, while for scholar $Y$, it is 10. But the scientific impact
of author $Y$ is more than author $X$. If the value of $\alpha$ is 10, then
the $h_{\alpha}$ is 2 and 10 respectively for authors X and Y. If we set
$\alpha$ = 5, then the $h_{\alpha}$ is 4 and 10 respectively for authors $X$
and $Y$. Finally, we can say that the $h_{\alpha}$-index gives more fine
grained value than the other h-types indices. But, the main limitations of
this index is the determination of the value of $\alpha$.

The citation based indices are designed based on the number of articles having
significant amount of influence. The articles being considered are either
journal articles, conference articles or book chapters. Further, all of the
articles are treated equally for the scientific assessment. But every type of
publications has its own impact and definitions. So, we can not treat all
equally. To overcome this limitation, Vinkler, (2009) proposed a new index
based on an elite set of articles called _$\pi$ -index_. The elite set of
highly cited journal article is the square root of total article count
($A_{\pi}=\sqrt{A}$). The $\pi$-index of a scholar is the $100^{th}$ of the
number of citation count of an elite set of highly cited journal articles that
are sorted in descending order of their citation count. Mathematically, it is
defined as:

$\pi-index=0.01\times cit(A_{\pi})$ (97)

where, $cit(A_{\pi})$ is the citation count of an elite set of articles.

Bornmann and Daniel, (2010); Schubert and Glänzel, (1986); Van Dalen and
Henkens, (2005) use the speed of the citation count of an article to evaluate
the scientific impact of scholars. The speed of the citation count is measured
at which the first citation count of an article is gained. For the calculation
of the citation speed index of a scholar, first the total number of months
after the article earns its first citation count is calculated for each of the
publications. Then, the publication list is arranged in the descending order
of their month count. The citation speed index of a scholar is then defined
as:

_“The citation speed index of a scholar is k, if k of his/her articles got
first citation at least k months ago.”_
Mathematically, it is defined as:

$SI=\underset{k}{max}(Month_{k}\geq k)$ (98)

where $k$ is the largest rank and $Month_{k}$ is the total number of months in
which the first citation count of $k^{th}$ article was earned.

This index penalizes all those articles that have earned citations earlier.

Instead of citation count of highly cited articles, the citation count of all
those articles that have a significant amount of influence in the proportion
of collaboration distance between the cited and the citing scholars, have been
used for the scientific assessment of scholars. Following this, Domingo-
Ferrera and Torrab, (2011); Bras-Amorós et al., (2011) considered the impact
of collaboration distance between the cited and the citing scholars and
proposed the _c-index_. The collaboration distance between cited and citing
scholar is measured in two categories: (i) Classical collaboration distance
and (ii) Refined collaboration distance.

The classical collaboration distance is the total number of scholars who are
present between the citing and the cited scholars. This collaboration distance
requires at least one collaboration between any member of the citing and the
cited scholars. For example, let scholar A1, A2, and A3 be the citing
scholars, and scholar A7, A5, and A4 be the cited scholars. Let one
collaboration between A2 and A5 be present, hence, the collaboration distance
between scholar A1 & A4 is 4. The collaboration distance between cited and
citing article is 1 because only one collaboration distance is present between
these two articles.

In the refined collaboration distance, the distance between two scholars is
the total collaboration path length, defined as:

$\sum_{r=0}^{m-1}\frac{1}{|pub(A_{r})\cap pub(A_{r+1})|}$ (99)

where, $m$ is the total number of scholars between the cited and the citing
scholars.

Based on these two types of collaboration distances, the c-index Domingo-
Ferrera and Torrab, (2011); Bras-Amorós et al., (2011) is defined as:

$C_{\alpha}=max(min(\alpha k,Qt(p_{k}))):k\in\\{1,2,...,m\\}$ (100)

where, $\alpha$ is the slope in citation graph and $Qt(p_{k})$ is quality
function of $p_{k}$.

In the case of the c-index of an article, the quality function $Qt(p_{k})$ is
the collaboration distance between the cited and the citing article. In case
of the c-index of a scholar, the quality function $Qt(p_{k})$ is the citation
distance between the scholars of the citing and the cited articles. If a
scholar published all papers by self without any co-author, then the
collaboration distance is $\infty$ and the c-index is $\alpha$m.

One another variant of the h-index is based on the number of success articles
Moed, (2010) and was defined by Kosmulski, (2011). If the total citation count
of an article is more than the total number of references, then the article is
called a success article. The score (SC) of the success of an article is
defined as:

$SC_{A}=\begin{cases}1,&\textrm{if $cit_{A}>RC_{A}$}\\\
0,~{}~{}~{}&\textrm{Otherwise}\end{cases}$ (101)

where,
$SC_{A}$ represents the score of article $A$,
$cit_{A}$ is the citation count of article $A$, and
the $RC_{A}$ is the total reference count of article $A$.

The success article penalizes all review articles because the review articles
contain a huge number of references. This method also penalizes the newly
published articles. Suppose an article earns 5 citation counts and contains 4
references, then this article called a success article. But an article that
earns 100 citation count and contains 120 references, does not count as a
success article. To overcome this limitation, a new index was proposed in
reference Kosmulski, (2011) to calculate the score of an article. Formally, it
is defined as:

$SC_{A}=\begin{cases}\frac{cit_{A}}{RC_{A}+1},&\textrm{if $cit_{A}>RC_{A}$}\\\
0,~{}~{}~{}&\textrm{Otherwise}\end{cases}$ (102)

The total success score of a scholar is the sum of the success score of all
articles.

$NSA=\sum_{A=1}^{m}SC_{A}$ (103)

where,
$NSA$ represents the number of success articles,
$m$ is the total number of articles published by the scholar, and
$SC_{A}$ is the success score of an article.

In case of multi-authored articles, the success score of an article is defined
as:

$SC_{A}=\begin{cases}\frac{1}{M_{A}},&\textrm{if $cit_{A}>RC_{A}$}\\\
0,~{}~{}~{}&\textrm{Otherwise}\end{cases}$ (104)

where, $M_{A}$ is the total number of authors in article $A$.

The citation of an article increases over time, so age of an article is an
important factor to evaluate the scientific impact of an article. Based on
this factor the score of an article is defined as:

$SC_{A}=\begin{cases}\frac{1}{CY+1-PY},&\textrm{if $cit_{A}>RC_{A}$}\\\
0,~{}~{}~{}&\textrm{Otherwise}\end{cases}$ (105)

where, $CY$ and $PY$ represent current year and publication year respectively.

Further, Franceschini et al., (2012) discussed the success paper based index
and mentioned that it also suffers with excess citation count (as discussed in
Franceschini and Maisano, 2010a ). To overcome this limitation of NSP (number
of success paper) based indicator, Franceschini et al., (2012) proposed a new
index called _Success-index_. The success-index of a scholar is the sum of the
score of all articles. Formally, it is defined as:

$Success-index(A)=\sum_{k=1}^{m}SC_{k}$ (106)

where, $SC_{k}$ is score of the $k^{th}$ success article (Kosmulski, (2011)).

An article consists of two components: (i) article citation count and (ii)
publication impact factor. However, generally only the number of citations is
used and the impact factor of publications is completely ignored. Further, the
citation count of an article is categorized into the following categories:

  1. 1.

A citation by scholars who have never worked with the cited scholar.

  2. 2.

A citation by scholars who have never worked with the cited scholar, but have
collaborated with an intermediate scholar(s).

  3. 3.

A citation by the scholars who are co-authors in another publication.

  4. 4.

A citation by the co-author except for researchers whom the individual
scientific activity is established.

  5. 5.

Self-citation.

By using the impact of different types of citation count and the publication
impact factor, Mikhailov, (2012) proposed a new index named _Summary Citation
Index_. The value of the reference is determined by the impact factor of
publication venue Erren and Gro$\beta$, (2016). The value coefficient ($\phi$)
is introduced to assign weight to the impact of different references. In all
those five citation sources discussed above, the first one is more valuable
than others. The value coefficient is 1 for the first type of citation and
0.9, 0.75, 0.5 & 0.25 respectively for other types of citations. The product
of impact factor and the value coefficient is distributed equally to all
scholars in multi-authored articles. The citation count of an article for a
scholar is defined as:

$OC=\sum_{p=1}^{m}\frac{\Phi(\phi_{p}IF_{p})}{A_{p}}$ (107)

where,
$OC$ represents the summary citation count,
$\Phi$ represents correction factor,
$\phi_{p}$ & $IF_{p}$ are the value coefficient and impact factor of $p^{th}$
article,
$A_{p}$ represents the number of scholars in $p^{th}$ article, and
$m$ is the total number of articles.

The Edition Citation (EC) with additional coefficient $a$ is defined as:

$EC=a\sum_{p=1}^{m}\frac{\Phi(IF_{p})}{A_{p}}$ (108)

The value of $a$ is typically set as 100.

Finally, the author mentioned that $OC$ and $EC$ are additive values. By using
the citation count of an article and the impact factor of the publication
venue, the summary citation (SC)-index is defined as the sum of $OC$ and $EC$
parameters. Formally,

$SC=OC+EC=\sum_{p=1}^{m}\frac{\Phi(\phi_{p}IF_{p})}{A_{p}}+a\sum_{p=1}^{m}\frac{\Phi(IF_{p})}{A_{p}}$
(109)

Generally, all of the indices produce only a single number to evaluate the
scientific impact of scholars. In the process, they fail (i) to cover the
yearly impact of scholars, and (ii) to compare the scientific impact of
scholars. Motivated by this, Liu and Yang, (2014) discussed the h-index
sequences and proposed the L-sequence. The L-sequence considers the entire