The $CI$-index: a new index to characterize the scientific output of researchers

TL;DR

The paper introduces the $CI$-index, a new metric based on the Choquet integral, designed to better differentiate researchers' scientific impact by emphasizing highly cited papers, improving upon existing indices.

Contribution

The $CI$-index is a novel, simple index that improves differentiation among researchers with similar traditional indices by weighting highly cited papers more heavily.

Findings

$CI$-index effectively distinguishes researchers with similar $h$-indices.

The index emphasizes highly cited papers, providing a nuanced measure of scientific impact.

It improves upon the $A$-index and $R$-index in characterizing research output.

Abstract

We propose a simple new index, named the $CI$-index, based on the Choquet integral to characterize the scientific output of researchers. This index is an improvement of the $A$-index and $R$-index and has a notable feature that highly cited papers have highly weights and lowly cited papers have lowly weights. In applications many researchers may have the same $h$-index, $g$-index or $R$-index. The $CI$-index can be provided an effective method of distinguish among such researchers.

Equations

$$w_1+\cdots +w_n=1,0\le w_i\le 1,i=1,2,\cdots ,n.$$

$$f(\alpha x+(1-\alpha )y)\le \alpha f(x)+(1-\alpha )f(y)$$

$$W{S}_Q[X]=w_1{x}_1+\cdots w_n{x}_n,$$

$$w_i=Q\left(1-\frac{i-1}{n}\right)-Q\left(1-\frac{i}{n}\right),i=1,2,\cdots ,n.$$

$$W{S}_Q[X]={\int }_0^{+\infty }Q(S_X(x))dx,$$

$$\displaystyle S_{X}(x)=\left\{\begin{array}[]{lll}1,&{\rm if}\ x<x_{n},\\ 1-\frac{1}{n},&{\rm if}\ x_{n}\leq x<x_{n-1},\\ 1-\frac{2}{n},&{\rm if}\ x_{n-1}\leq x<x_{n-2},\\ \vdots&\vdots\\ 1-\frac{n-1}{n},&{\rm if}\ x_{2}\leq x<x_{1},\\ 0,&{\rm if}\ x\geq x_{1},\\ \end{array}\right.$$

$$\displaystyle S_{X}(x)=\left\{\begin{array}[]{ll}1,&{\rm if}\ x<x_{1},\\ 0,&{\rm if}\ x\geq x_{1},\\ \end{array}\right.$$

$${\rho }_Q[X]\ge {\int }_0^{+\infty }{S}_X(x)dx=E[X],$$

$${\rho }_Q[X]\le {\int }_0^{+\infty }{S}_X(x)dx=E[X].$$

$$C{I}_h[Q]=\sqrt{♯C_h[y_1{w}_1+\cdots +y_{♯C_h}{w}_{♯C_h}]},$$

$$w_j=Q\left(\frac{j}{♯C_h}\right)-Q\left(\frac{j-1}{♯C_h}\right),j=1,2,\cdots ,♯C_h.$$

$$C{I}_h[Q]=\sqrt{h[y_1{w}_1+\cdots +y_h{w}_h]},$$

$$w_j=Q\left(\frac{j}{h}\right)-Q\left(\frac{j-1}{h}\right),j=1,2,\cdots ,h.$$

$$R_m:=C{I}_h[Q]=\sqrt{\sum _{i=1}^{♯C_h}{y}_j},$$

$$♯C_h{w}_j=♯C_h\left(\sqrt{\frac{j}{♯C_h}}-\sqrt{\frac{j-1}{♯C_h}}\right)=\sqrt{♯C_h}\left(\sqrt{j}-\sqrt{j-1}\right),j=1,2,\cdots ,♯C_h.$$

$$C{I}_h[Q]=\sqrt{\sqrt{♯C_h}\sum _{j=1}^{♯C_h}\left(\sqrt{j}-\sqrt{j-1}\right)y_j}.$$

$$C{I}_h[Q]=\sqrt{\sqrt{h}\sum _{j=1}^h\left(\sqrt{j}-\sqrt{j-1}\right)y_j}.$$

$$C{I}_g[Q]=\sqrt{g(y_1{u}_1+\cdots +y_g{u}_g)},$$

$$u_j=Q\left(\frac{j}{g}\right)-Q\left(\frac{j-1}{g}\right),j=1,2,\cdots ,g.$$

$$C{I}_g[Q]=\sqrt{\sqrt{g}\sum _{j=1}^g\left(\sqrt{j}-\sqrt{j-1}\right)y_j}.$$

$$C{I}_N[Q]=\sqrt{N(x_1{v}_1+\cdots +x_N{v}_N)},$$

$$v_j=Q\left(\frac{j}{N}\right)-Q\left(\frac{j-1}{N}\right),j=1,2,\cdots ,N.$$

$$C{I}_N[Q]=\sqrt{\sqrt{N}\sum _{j=1}^N\left(\sqrt{j}-\sqrt{j-1}\right)x_j}.$$

$$C{I}_h=\sqrt{\sqrt{3}\times \left[50+\left(\sqrt{2}-1\right)\times 50+\left(\sqrt{3}-\sqrt{2}\right)\times 3\right]}=11.14;$$

$$C{I}_h=\sqrt{\sqrt{4}\times [50+(\sqrt{2}-1)\times 50+(\sqrt{3}-\sqrt{2})\times 3+(\sqrt{4}-\sqrt{3})\times 3]}=12.04;$$

$$C{I}_h=\sqrt{\sqrt{3}\times [70+(\sqrt{2}-1)\times 30+(\sqrt{3}-\sqrt{2})\times 3]}=12.02;$$

$$C{I}_h=\sqrt{\sqrt{4}\times [70+(\sqrt{2}-1)\times 30+(\sqrt{3}-\sqrt{2})\times 3+(\sqrt{4}-\sqrt{3})\times 3]}=12.98;$$

$$C{I}_h=\sqrt{\sqrt{3}\times \left[90+\left(\sqrt{2}-1\right)\times 10+\left(\sqrt{3}-\sqrt{2}\right)\times 3\right]}=12.83;$$

$$C{I}_h=\sqrt{\sqrt{4}\times \left[90+\left(\sqrt{2}-1\right)\times 10+\left(\sqrt{3}-\sqrt{2}\right)\times 3+\left(\sqrt{4}-\sqrt{3}\right)\times 3\right]}=13.85;$$

$$C{I}_h=\sqrt{\sqrt{4}\times \left[40+\left(\sqrt{2}-1\right)\times 30+\left(\sqrt{3}-\sqrt{2}\right)\times 20+\left(\sqrt{4}-\sqrt{3}\right)\times 13\right]}=11.16;$$

Full text

The $CI$-index: a new index to characterize the scientific output of researchers

Xuehua Yin,   Xiuyan Sha,   Chuancun Yin

*School of Statistics, Qufu Normal University

Shandong 273165, China

*e-mail: ccyin@qfnu.edu.cn

Abstract We propose a simple new index, named the $CI$-index, based on the Choquet integral to characterize the scientific output of researchers. This index is an improvement of the $A$-index and $R$-index and has a notable feature that highly cited papers have highly weights and lowly cited papers have lowly weights. We proposed a ranking method of distinguish researchers to compare their scientific outputs. In applications many researchers may have the same $h$-index, $g$-index or $R$-index, however, the $CI$-index can be provided an effective method of distinguish among such researchers.

Key words: Hirsch $h$-index; $h$-index variants; $g$-index; Choquet integral; $CI$-index; $R$-index

1 Introduction

Since the physicist Hirsch (2005) introduced the so-called $h$-index which as a new indicator for measure the impact of a researcher’s scientific research output, it has got a lot of attention from both in the scientific community and the scientometrics (informetrics) literature for its good properties to measure the scientific production of researchers, more than 8430 of articles have been written on the $h$-index (Data from Google Scholar as of March 15, 2019). The result was reviewed in Nature (Ball (2005)) and Science (Anon (2005)) as well. A large part of the literature building on Hirsch’s work is concerned with introducing variants, extensions, and generalizations of the $h$-index. In the study of Bornmann et al. (2011), no less than 37 variants of the $h$-index were listed. In a more recent study (Bornmann (2014)) says that there are around 50 variants of the $h$-index. The $h$-index is a simple single number incorporating both quantitative and qualitative aspects. The $h$-index is also robust in the sense that it is insensitive to a set of uncited (or lowly cited) papers but also it is insensitive to one or several outstandingly highly cited papers. This last aspect can be considered as a drawback of the $h$-index, for more details we refer to Egghe (2006a). For more advantages and disadvantages of the $h$-index see also Hirsch (2005, 2007) and Jin et al. (2007). In order to overcome some of these limitations scientists have proposed several new indicators based on the $h$-index with the intention of either replacing or complementing the original $h$-index. To overcome the well-known problem of the insensitivity of the h-index to the number of citations received by highly cited paper, Egghe (2006a, 2006b) first developed the $g$-index, Jin (2006) followed with the $A$-index, Jin et al. (2007) suggested correcting the $h$-index for the aging of papers using the $AR$-index, Egghe and Rousseau (2008) proposed a citation-weighted $h$-index, Anderson et al. (2008) describe a new version of the $h$-index, the “tapered h-index”, which positively scores all of an author’s citations, accounting for the tapered distribution of citations associated with highly cited papers rather than using a cut-off at $h$. Hirsch (2010) proposed an index to quantify an individual’s scientific research output that takes into account the effect of multiple coauthorship. Alonso et al. (2010) proposed a new index, called $hg$-index, to characterize the scientific output of researchers which is based on both $h$-index and $g$-index to overcome the limitations of both indices. Recall that $A$-index is simply defined as the average number of citations received by the publications included in the Hirsch core. Mathematically, this is $A=h^{-1}\sum_{i=1}^{h}y_{i},$ where the numbers of citations $y_{i}$’s are ranked in decreasing order. Note that all the $y_{i}$ have the same weight $\frac{1}{h}$. A closer related to $h$-index is $R$-index introduced in Jin et al. (2007) which is defined as $R=\sqrt{hA}$. Recently, Perry and Reny (2016) introduced Euclidean index $l_{E}$, which is the Euclidean length of $(x_{1},\cdots,x_{n})$. Note that the Euclidean index, to some extent, avoids several shortcomings of the $h$-index and its successors. The only drawback may be that the weight of highly cited papers is too high. For example, consider three researchers $A$, $B$ and $C$, $A$ has 100 papers, each has 10 citations ($h$= 10), $B$ has 1 paper with 100 citations ($h$= 1) and $C$ has 10000 papers, each has 1 citation ($h$= 1). Then $A$, $B$ and $C$ have the same Euclidean index 100.

The aim of this paper is to present a new index—called the Choquet integral index ($CI$-index for short)- to characterize the scientific output of researchers. This index is an improvement of the $A$-index and has a notable feature that highly cited papers have highly weights and lowly cited papers have lowly weights. To our best knowledge, such a index has not been studied before. In the following, we first recall the definitions of distortion function and distortion expectation or Choquet integral, then we introduce three $CI$-indices, namely, $CI_{h}$-index in the $h$-core, $CI_{g}$-index in the $g$-core and $CI_{N}$-index in the $N$-core, where $N$ stands for all citations. After that we put forward a method to compare two researchers.

2 Preliminaries

Definition 2.1**.**

A vector $W=(w_{1},\cdots,w_{n})$ is called a weight if $W$ having the properties

[TABLE]

Moreover, if $w_{i}\leq w_{j}\;\forall\;i<j$, then we call $W$ is increasing, on the other hand if $w_{i}\geq w_{j}\;\forall\;i<j$, we call it is decreasing.

Definition 2.2**.**

A distortion function is a non-decreasing function $g:[0,1]\rightarrow[0,1]$ such that $g(0)=0,g(1)=1$.

The notion of distortion function was proposed by Yaari (1987) in dual theory of choice under risk, since then many different distortions $g$ have been proposed in the literature. The distortion function is also called regular increasing monotone quantifier in computer science and artificial intelligence literature, see Yager (1996). Here we list some commonly used distortion functions:

$\bullet$ Incomplete beta function $g(x)=\frac{1}{\beta(a,b)}\int^{x}_{0}t^{a-1}(1-t)^{b-1}dt$, where $a>0$ and $b>0$ are parameters and $\beta(a,b)=\int^{1}_{0}t^{a-1}(1-t)^{b-1}dt$. Setting $b=1$ gives the power distortion $g(x)=x^{a}$; setting $a=1$ gives the dual-power distortion $g(x)=1-(1-x)^{b}.$

$\bullet$ The Wang distortion $g(x)=\Phi(\Phi^{-1}(x)+\Phi^{-1}(p)),0<p<1,$ where $\Phi$ is the distribution function of the standard normal.

$\bullet$ The lookback distortion $g(x)=x^{p}(1-p\ln x),p\in(0,1].$

Let us recall the standard definitions of convexity and concavity of functions.

Definition 2.3**.**

Let $I$ be an interval in real line $\mathbb{R}$. Then the function $f:I\rightarrow\mathbb{R}$ is said to be convex if for all $x,y\in I$ and all $\alpha\in[0,1]$, the inequality

[TABLE]

holds. If this inequality is strict for all $x\neq y$ and $\alpha\in(0,1)$, then $f$ is said to be strictly convex. A closely related concept is that of concavity: $f$ is said to be (strictly) concave if, and only if, $-f$ is (strictly) convex.

Assume that $x_{1}\geq x_{2}\geq\cdots\geq x_{n}$ are $n$ positive numbers, given a distortion function $Q$, considering the following weighted sum

[TABLE]

where $w_{i}$ is the weights generated by $Q$ as follows

[TABLE]

Because of the nondecreasing nature of $Q$ it follows that $w_{i}>0,i=1,2,\cdots,n$. Furthermore, from $Q(1)=1$ and $Q(0)=0$, it follows that $\sum_{i=1}^{n}w_{i}=1$. If $Q$ is convex, then $\{{w}_{i}\}$ is monotonic decreasing; if $Q$ is concave, then $\{{w}_{i}\}$ is monotonic increasing (see, e.g., Sha et al. (2018) for details). We remark that (2.1) can be written as a Choquet integral (see, e.g. Denneberg (1994)) of a random variable $X$ with probability distribution $P(X=x_{i})=\frac{1}{n},i=1,2,\cdots,n$:

[TABLE]

where $S$ is the decumulative distribution function of $X$ with probability distribution $P(X=x_{i})=\frac{1}{n},i=1,2,\cdots,n$. For example,

[TABLE]

for the case of $x_{1}>x_{2}>\cdots>x_{n}$, and

[TABLE]

for the case of $x_{1}=x_{2}=\cdots=x_{n}$.

Obviously, the identity function is the smallest concave distortion function and also the largest convex distortion function; Any concave distortion function $Q$ gives more weight to the tail than the identity function $Q(x)=x$, whereas any convex distortion function $Q$ gives less weight to the tail than the identity function $Q(x)=x$. If $Q(x)=x$, then $\rho_{Q}[X]=E[X]$, the expectation of $X$. If $Q$ is concave, then

[TABLE]

and if $Q$ is convex, then

[TABLE]

Clearly, if $Q_{1}(x)\leq Q_{2}(x)$ for $x\in[0,1]$, then $\rho_{Q_{1}}[X]\leq\rho_{Q_{2}}[X]$ for any random variable $X$.

From (2.2) we see that the Choquet integral satisfy the following properties:

a) Positive homogeneity: $\rho_{Q}[aX]=a\rho_{Q}[X]$ for any non-negative constant $a$;

b) Translation invariance: $\rho_{Q}[X+b]=\rho_{Q}[X]+b$ for any constant $b$;

c) Monotonicity: $\rho_{Q}[X]\leq\rho_{Q}[Y]$ for any two random variables $X$ and $Y$, where $X\leq Y$ with probability 1.

3 The indices for different datasets

3.1 $CI$-index in the $h$-core

A paper belongs to the $h$-core of a scientist if it has $\geq h$ citations (Hirsch 2010). Hence the $h$-core may contain more than $h$ elements and $h$-core contains exactly $h$ elements if only one paper has $h$ citations. We will use $C_{h}$ standard for the set of $h$-core and the number of elements in $C_{h}$ is denoted by $\sharp C_{h}$. Note that $C_{h}$ is a multiset, which, unlike a set, allows for multiple instances for each of its elements. The number or cardinality of a multiset is constructed by summing up the multiplicities of all its elements. For example, $C(h)=[1,1,6]$, the element 1 has multiplicity 2, 6 has multiplicity 1 and $\sharp C_{h}=3$.

Let $y_{1},\cdots,y_{\sharp C_{h}}$ be the elements of $h$-core $C_{h}$ which are ranked in decreasing order, where $h$ is the $h$-index. Note that $y_{h}=\cdots=y_{\sharp C_{h}}$. The $CI$-index in the $h$-core is defined as

[TABLE]

where

[TABLE]

In particular, when $\sharp C_{h}=h$,

[TABLE]

where

[TABLE]

Here $Q$ is a distortion function. The reason that taking the root is to prevent the number being too large. The distinguishability of $CI_{h}[Q]$ and $(CI_{h}[Q])^{2}$ are same, since the function $f(x)=\sqrt{x}$ is strictly increasing.

If $Q$ is a concave distortion function, then by (2.3) we get $CI_{h}[Q]\geq R$, where $R$ is the $R$-index which defined by $R=\sqrt{\sum_{i=1}^{h}y_{j}}$ (see e.g. Jin et al. (2007)). Taking $Q(x)=x$ in (3.1) yields

[TABLE]

which can be seen as the modified version of $R$-index. The $R_{m}$-index are well-defined regardless of $\sharp C_{h}=h$ or not.

If $Q(x)=\sqrt{x}$, then

[TABLE]

Therefore,

[TABLE]

In particular, if $\sharp C_{h}=h$, then

[TABLE]

3.2 $CI$-index in the $g$-core

In order to give more weight to highly cited articles, Egghe (2006b) proposed the $g$-index. The $g$-index was presented by Egghe (2006a, b) as a simple variant of the $h$-index. A set of papers has a $g$-index $g$ if $g$ is the highest rank such that the top $g$ papers have, together, at least $g^{2}$ citations. This also means that the top $g+1$ papers have less than $(g+1)^{2}$ cites. Egghe and Rousseau (2008) pointed out that a small variant of the $g$-index is possible by not limiting it to $g\leq T$, where $T$ stands for total number of papers. This means that, in these cases, fictitious articles with 0 citations have to be added. We will use $C_{g}$ stands for the set of $g$-core and the number of elements in $C_{g}$ is denoted by $\sharp C_{g}$. Obviously, $\sharp C_{g}=g$.

Let $y_{1},\cdots,y_{g}$ be the elements of $g$-core $C_{g}$ which are ranked in decreasing order, where $g$ is the $g$-index. One can define an analogous quantity of $CI$-index in the $g$-core

[TABLE]

where

[TABLE]

Here $Q$ is a distortion function.

In particular, if $y_{1}=\cdots=y_{g}$, then $CI_{g}[Q]=\sqrt{gy_{1}}$. If $Q$ is a concave distortion function, then by (2.3) we get $CI_{g}[Q]\geq R_{g},$ where $R_{g}=\sqrt{\sum_{i=1}^{g}y_{j}}$, which is closely related to $A_{g}$-index (see, Schreiber (2010)). Taking $Q(x)=x$ in (3.3) yields $CI_{g}[Q]=R_{g}$.

If $Q(x)=\sqrt{x}$, then

[TABLE]

3.3 $CI$-index in the core of all citations

Highly cited papers are, of course, important for the determination of the values of $CI_{h}$-index and $CI_{g}$-index. However, it is not to take into account the “tail” papers (with low number of citations). Thus, maybe many citations that accompany the most highly cited papers effectively contribute zero. A bibliometric measure of publication output should be assign a positive score to each new citation as it occurs. It is necessary to consider the $CI$-index in the core of all citations.

Let $n$ denote the number of published articles by a scientist, and let $x_{i},i=1,2,\cdots,n$, denote the number of citations of the $i$-th most cited article, so that $x_{1}\geq x_{2}\geq\cdots\geq x_{n}>0$. Assume that $N=x_{1}+x_{2}+\cdots+x_{n}$ represents the total number of citations received. The $CI$-index in the core of all citations is defined as

[TABLE]

where

[TABLE]

Here $Q$ is a distortion function. If $Q$ is a concave distortion function, then by (2.3) $CI_{N}[Q]\geq R_{N},$ where $R_{N}=\sqrt{\sum_{i=1}^{N}x_{i}}$.

If $Q(x)=\sqrt{x}$, then

[TABLE]

4 Ranking method of distinguish researchers

Note that $CI_{h}[Q],CI_{g}[Q]$ and $CI_{N}[Q]$ are the increasing functions of $h,g,N$ and $y_{i}^{\prime}s$. Thus a method can be proposed to compare the scientific outputs of two researchers $R_{1}$ and $R_{2}$ as follows: Assume the researcher $R_{i}$ have $h$-indices $h_{i}$, $g$-indices $g_{i}$ and the total number of citations $N_{i}$, respectively.

(1)  If $CI_{h_{1}}[Q]>CI_{h_{2}}[Q]$, then $R_{1}\succ R_{2}$;

(2)  $CI_{h_{1}}[Q]<CI_{h_{2}}[Q]$, then $R_{1}\prec R_{2}$;

(3)  $CI_{h_{1}}[Q]=CI_{h_{2}}[Q]$ and $CI_{g_{1}}[Q]>CI_{g_{2}}[Q]$, then $R_{1}\succ R_{2}$;

(4)  $CI_{h_{1}}[Q]=CI_{h_{2}}[Q]$ and $CI_{g_{1}}[Q]<CI_{g_{2}}[Q]$, then $R_{1}\prec R_{2}$;

(5)  $CI_{h_{1}}[Q]=CI_{h_{2}}[Q]$, $CI_{g_{1}}[Q]=CI_{g_{2}}[Q]$ and $CI_{N_{1}}[Q]>CI_{N_{2}}[Q]$, then $R_{1}\succ R_{2}$;

(6)  $CI_{h_{1}}[Q]=CI_{h_{2}}[Q]$, $CI_{g_{1}}[Q]=CI_{g_{2}}[Q]$ and $CI_{N_{1}}[Q]<CI_{N_{2}}[Q]$, then $R_{1}\prec R_{2}$;

(7)  $CI_{h_{1}}[Q]=CI_{h_{2}}[Q]$, $CI_{g_{1}}[Q]=CI_{g_{2}}[Q]$ and $CI_{N_{1}}[Q]=CI_{N_{2}}[Q]$, then $R_{1}\sim R_{2}$;

where $\prec$ means “is ranked worse than”, $\succ$ means “is ranked good than” and $\sim$ means “is equivalent to”.

In order to understand this easily, below we will give two toy examples.

Example 4.1. Suppose that $Q(x)=\sqrt{x}$.

(1) If $y_{1}=y_{2}=50,y_{3}=3,y_{4}=1$, then $h=3$, $\sharp C_{h}=3$ and

[TABLE]

(2) If $y_{1}=y_{2}=50,y_{3}=3,y_{4}=3,y_{5}=1$, then $h=3$, $\sharp C_{h}=4$ and

[TABLE]

(3) If $y_{1}=70,y_{2}=30,y_{3}=3,y_{4}=1$, then $h=3$, $\sharp C_{h}=3$ and

[TABLE]

(4) If $y_{1}=70,y_{2}=30,y_{3}=3,y_{4}=3,y_{5}=1$, then $h=3$, $\sharp C_{h}=4$ and

[TABLE]

(5) If $y_{1}=90,y_{2}=10,y_{3}=3,y_{4}=1$, then $h=3$, $\sharp C_{h}=3$ and

[TABLE]

(6) If $y_{1}=90,y_{2}=10,y_{3}=3,y_{4}=3,y_{5}=1$, then $h=3$, $\sharp C_{h}=4$ and

[TABLE]

(7) If $y_{1}=40,y_{2}=30,y_{3}=20,y_{4}=13,y_{5}=4$, then $h=4$, $\sharp C_{h}=4$ and

[TABLE]

(8) If $y_{1}=\cdots=y_{10}=10,y_{11}=3$, then $h=10$, $\sharp C_{h}=10$ and $CI_{h}=\sqrt{10\times 10}=10;$

(9) If $y_{1}=100,y_{2}=3,y_{2}=1$, then $h=2$, $\sharp C_{h}=2$ and

[TABLE]

(10) If $y_{1}=103,y_{2}=1$, then $h=1$, $\sharp C_{h}=1$ and $CI_{h}=\sqrt{103}=10.15.$

Note that the cases (1),(3) and (5) have the same $h=3$, $\sharp C_{h}=3$ and $R=\sqrt{103}$, but with different $CI_{h}$-index; the cases (2),(4) and (6) have the same $h=3$, $\sharp C_{h}=4$ and $R_{m}=\sqrt{106}$, but with different $CI_{h}$-indices. Finally, we get that

[TABLE]

Example 4.2. Suppose that $Q(x)=\sqrt{x}$.

(1) If $y_{1}=\cdots=y_{10}=10,y_{11}=0$, then $h=10$, $\sharp C_{h}=10$, $g=10$ and

[TABLE]

(2) If $y_{1}=\cdots=y_{4}=25,y_{5}=0$, then $h=4$, $\sharp C_{h}=4$, $g=10$ and

[TABLE]

[TABLE]

(3) If $y_{1}=y_{2}=50,y_{3}=0$, then $h=2$, $\sharp C_{h}=2$, $g=10$ and

[TABLE]

[TABLE]

(4) If $y_{1}=100,y_{2}=0$, then $h=1$, $\sharp C_{h}=1$, $g=10$, $CI_{h}=10$ and $CI_{g}=17.78.$

From the calculation results above, we see that the cases (1)-(4) have the same $CI_{h}$, but with different $CI_{g}$. More precisely, we find that $R_{1}\prec R_{2}\prec R_{3}\prec R_{4}$.

5 Concluding remarks

Based on the Choquet integral and the foundation of the $h$-index and $g$-index we have introduced the $CI$-indices within the $h$-core and $g$-core, and then we consider the $CI$-index in the core of all citations. These new indices eliminate some of the disadvantages of the $h$-index, $g$-index, $A$-index and $R$-index and has a notable feature that highly cited papers have highly weights and lowly cited papers have lowly weights. After that, we propose a new method to compare the academic achievements of two researchers. This research has not taken into account the effect of multiple authorship as in Hirsch (2010, 2019) and the effect of self-citation as in Bartneck and Kokkelmans (2011) and others which could be an excellent direction for further research. We hope that this new $CI$-index will be further studied and used in practical assessments.

Acknowledgements.  The research was supported by the National Natural Science Foundation of China (No. 11571198).