Statistical regularities in the rank-citation profile of scientists
Alexander M. Petersen,1, 2 H. Eugene Stanley,2 and Sauro Succi3, 4
1
IMT Lucca Institute for Advanced Studies, Lucca, Italy
Center for Polymer Studies and Department of Physics,
Boston University, Boston, Massachusetts 02215, USA
3
Istituto Applicazioni Calcolo C.N.R., Rome, IT
4
Freiburg Institute for Advanced Studies, Albertstrasse, 19, D-79104, Freiburg, Germany
(Dated: September 29, 2011)
arXiv:1103.2719v2 [physics.soc-ph] 28 Sep 2011
2
Recent “science of science” research shows that scientific impact measures for journals and individual articles have quantifiable regularities across both time and discipline. However, little is known about the scientific
impact distribution at the scale of an individual scientist. We analyze the aggregate scientific production and
impact of individual careers using the rank-citation profile ci (r) of 200 distinguished professors and 100 assistant professors. For the entire range of paper rank r, we fit each ci (r) to a common distribution function that is
parameterized by two scaling exponents. Since two scientists with equivalent Hirsch h-index can have significantly different ci (r) profiles, our results demonstrate the utility of the βi scaling parameter in conjunction with
hi for quantifying individual publication impact. We show that the total number of citations Ci tallied from a
scientist’s Ni papers scales as Ci ∼ hi1+βi . Such statistical regularities in the input-output patterns of scientists
can be used as benchmarks for theoretical models of career progress.
A scientist’s career path is subject to a myriad of decisions
and unforeseen events, e.i. Nobel Prize worthy discoveries
[1], that can significantly alter an individual’s career trajectory. As a result, the career path can be difficult to analyze since there are potentially many factors (e.g. individual,
mentor-apprentice, institutional, coauthorship, field) [2–8] to
account for in the statistical analysis of scientific panel data.
The rank-citation profile, ci (r), represents the number of citations of individual i to his/her paper r, ranked in decreasing
order ci (1) ≥ ci (2) ≥ . . . ci (N ), and provides a quantitative
synopsis of a given scientist’s publication career. Here, we analyze the rank-ordered citation distribution ci (r) for 300 scientists in order to better understand patterns of success and to
characterize scientific production at the individual scale using
a common framework. The review of scientific achievement
for post-doctoral selection, tenure review, award and academy
selection, at all stages of the career is becoming largely based
on quantitative publication impact measures. Hence, understanding quantitative patterns in production are important for
developing a transparent and unbiased review system. Interestingly, we observe statistical regularities in ci (r) that are
remarkably robust despite the idiosyncratic details of scientific achievement and career evolution. Furthermore, empirical regularities in scientific achievement suggest that there are
fundamental social forces governing career progress [9–12].
We group the 300 scientists that we analyze into three sets
of 100, referred to as datasets A, B and C, so that we can
analyze and compare the complete publication careers of each
individual, as well as across the three groups:
[A] 100 highly-profile scientists with average h-index
hhi = 61 ± 21. These scientists were selected using the
citation shares metric [8] to quantify cumulative career
impact in the journal Physical Review Letters (PRL).
[B] 100 additional “control” scientists with average h-index
hhi = 44 ± 15.
[C] 100 current Assistant professors with average h-index
hhi = 14 ± 7. We selected two scientists from each of
the top-50 US physics departments (departments ranked
according to the magazine U.S. News).
In the methods section we describe the selection procedure for
datasets A, B and C in more detail.
There are many conceivable ways to quantify the impact
of a scientist’s Ni publications. The h-index [13] is a widely
acknowledged single-number measure that serves as a proxy
for production and impact simultaneously. The h-index hi
of scientist i is defined by a single point on the rank-citation
profile ci (r) satisfying the condition
ci (hi ) = hi .
(1)
To address the shortcomings of the h-index, numerous remedies have been proposed in the bibliometric sciences [14]. For
example, Egghe proposed the g-index, where the most cited g
papers cumulate g 2 citations overall [15], and Zhang proposed
the e-index which complements the h and g indices quantitatively [16].
To justify the importance of analyzing the entire profile
ci (r), consider a scientist i = 1 with rank-citation profile
c1 (r) ≡ [100, 50, 33, 25, 20, 16, 14, 12, 11, 10, 9...] and
a scientist i = 2 with c2 (r) ≡ [10, 10, 10, 10, 10, 10, 10,
10, 10, 10, 9 ...]. Both scientists have the same h-index value
h = 10, although c1 (r) tallies 2.9 times as many citations as
c2 (r) from his/her most-cited 10 papers. Hence, an additional
parameter βi is necessary in order to distinguish these two example careers. Specifically, the βi parameter quantifies the
scaling slope in ci (r) for the high-rank papers corresponding
to small r values. In this simple illustration, β1 ≈ 1 while
β2 ≈ 0.
In Fig. 1 we plot ci (r) for 5 extremely high-impact scientists. The individuals EW, ACG, MLC, and PWA are physicists with the largest hi values in our data set; BV is a prolific
molecular biologists who we include in this graphical illustration in order to demonstrate the generality of the statistical
regularity we find, which likely exists across discipline. To
demonstrate how the singe point ci (hi ) is an arbitrary point
2
4
citations, ci(r)
10
10
βi
3
H80
2
10
1
10
H20
EW (hi = 121)
ACG
(108)
MLC
(107)
PWA
(103)
BV
(176)
H5
H2
H1
0
10 0
10
1
10
paper rank, r
10
2
10
3
FIG. 1: The citation distribution of individual scientists is heavy-tailed. We show 5 empirical rank-citation ci (r) profiles corresponding
to extremely high-impact scientists whose initials and h-index (as of Jan. 2010) are listed in the legend. The hierarchical scaling pattern
in ci (r) for small r values indicate that the pillar contributions of top scientists are “off-the-charts” since they have no characteristic scale.
Furthermore, this statistical regularity demonstrates the utility of the βi scaling exponent in characterizing the highly cited papers of a given
scientist i. Interestingly, each scientist has coauthored a significant number of papers that are significantly lower impact than their ci (1) pillar
paper. The ci (r) distributions show significant variability in both the high-rank (β) and low-rank (γ) regimes. Moreover, for ci (r) with similar
h values, the h-index (a single point on each curve) is insufficient to adequately distinguish career profiles. The solid curves are the best-fit
DGBD functions (see Eq. 3) for each corresponding ci (r) over the entire rank range in each case. The intersection of ci (r) with the line Hp (r)
corresponds to the generalized h-index hp , which together uniquely quantify the ci (r) profile. Five Hp (r) lines are provided for reference,
with p = {1, 2, 5, 20, 80}.
along the ci (r) curve, we also plot the lines Hp (r) ≡ p r for 5
values of p = {1, 2, 5, 20, 80}. The value p ≡ 1 recovers the
h-index h1 = h proposed by Hirsch. The intersection of any
given line Hp (r) with ci (r) corresponds to the “generalized
h-index” hp ,
scientific careers are like “startup” companies that need appropriate venture funding to support the career trajectory through
lows as well as highs [12].
I.
c(hp ) = php ,
RESULTS
(2)
proposed in [17] and further analyzed in [18], with the relation
hp ≤ hq for p > q. Since the value p ≡ 1 is chosen somewhat
arbitrarily, we take an alternative approach which is to quantify the entire ci (r) profile at once (which is also equivalent
to knowing the entire hp spectrum). Surprisingly, because we
find regularity in the functional form ci (r) for all 300 scientists analyzed, we can relate the relative impact of a scientist’s
publication career using the small set of parameters that specify the ci (r) profile for the entire set of papers ranging from
rank r = 1...Ni . Using a much smaller parameter space than
the hp spectrum, we can begin to analyze the statistical regularities in the career accomplishments of scientists.
The aim of this analysis is not to add another level of
scrutiny to the review of scientific careers, but rather, to highlight the regularities across careers and to seed further exploration into the mechanisms that underlie career success. The
aim of this brand of quantitative social science is to utilize the
vast amount of information available to develop an academic
framework that is sustainable, efficient and fruitful. Young
A.
A Quantitative Model for ci (r)
For each scientist i, we find that ci (r) can be approximated
by a scaling regime for small r values, followed by a truncated
scaling regime for large r values. Recently a novel distribution, the discrete generalized beta distribution (DGBD)
ci (r) ≡ Ai r−βi (Ni + 1 − r)γi
(3)
has been proposed as a model for rank profiles in the social
and natural sciences that exhibit such truncated scaling behavior [19, 20]. The parameters Ai , βi , γi and Ni are each defined
for a given ci (r) corresponding to an individual scientists i,
however we suppress the index i in some equations to keep the
notation concise. We estimate the two scaling parameters βi
and γi using Mathematica software to perform a multiple linear regression of ln ci (r) = ln Ai − βi ln r + γi ln(Ni + 1 − r)
in the base functions ln r and ln(Ni + 1 − r). In our fitting
procedure we replace N with r1 , the largest value of r for
which c(r) ≥ 1 (we find that r1 /Ni ≈ 0.84 ± 0.01 for careers
3
c(r) ∼ r−β [β ≈ 0.92 ± 0.01]
(4)
for 100 ≤ r ≤ 102 . Interestingly, this β value is similar
to the scaling exponents calculated for other rank-size (Zipf)
distributions in the social and economic sciences, e.g. word
frequency [22] and city size [19, 20, 23].
We calculate each βi value using a multilinear least-squares
regression of ln ci (r) for 1 ≤ r ≤ r1 using the DGBD model
defined in Eq. [3]. To properly weight the data points for better regression fit over the entire range, we use only 20 values
of ci (r) data points that are equally spaced on the logarithmic scale in the range r ∈ [1, r1 ]. We elaborate the details
of this fitting technique in the methods section. We plot five
empirical ci (r) along with their corresponding best-fit DGBD
functions in Fig. 1 to demonstrate the goodness of fit for the
entire range of r.
In order to demonstrate the common functional form of the
a 10
5
Average c(r)
citations, c(r)
10
4
3
! = 0.92
10
10
2
H80
1
H2
10
0
10
0
10
10
1
2
paper rank, r
3
10
10
"
b
c(r') = c(r)/A (N+1-r)
in datasets A and B). Figs. 1 and 2 demonstrate the utility of
the DGBD to represent ci (r), for both large and small r. The
regression correlation coefficient Ri > 0.97 for all ln ci (r)
profiles analyzed.
The DGBD proposed in [19] is an improvement over the
Zipf law (also called the generalized power-law or Lotka-law
[21]) model and the stretched exponential model [13] since
it reproduces the varying curvature in ci (r) for both small
and large r. Typically, an exponential cutoff is imposed in
the power-law model, and justified as a finite-size effect. The
DGBD does not require this assumption, but rather, introduces
a second scaling exponent γi which controls the curvature in
ci (r) for large r values. The DGBD has been successfully
used to model numerous rank-ordering profiles analyzed in
[19, 20] which arise in the natural and socio-economic sciences. The relative values of the βi and γi exponents are
thought to capture two distinct mechanisms that contribute
to the evolution of ci (r) [19, 20]. Due to the data limitations in this study, we are not able to study the dynamics in
ci (r) through time. Each ci (r) is a “snapshot” in time, and
so we can only conjecture on the evolution of ci (r) throughout the career. Nevertheless, we believe that there is likely a
positive feedback effect between the “heavy-weight” papers
and “newborn” papers, whereby the reputation of the “heavyweight” papers can increase the exposure and impact the perceived significance of “newborn” papers during their infant
phase. Moreover, the 2-regime power-law behavior of ci (r)
suggests that the reinforcement dynamics can be quantified
by the scale-free parameters β and γ.
The βi value determines the relative change in the ci (r) values for the high-rank papers, and thus it can be used to further
distinguish the careers of two scientists with the same h-index.
In particular, smaller β values characterize flat profiles with
relatively low contrast between the high and low-rank regions
of any given profile, while larger β values indicate a sharper
separation between the two regions.
In Fig. 2(a) we plot ci (r) for each scientist from dataset
[A] as well as the average of the 100 individual curves c(r) ≡
P100
1
i=1 ci (r) (see Figs. S1 and S2 for analogous plots for
100
datasets [B] and [C]). We find robust power-law scaling
10
0
1
-2
10
0
10
10
1
2
3
10
scaled paper rank, r' = r
10
!
FIG. 2: Data collapse of each ci (r) along a universal curve. A
comparison of 100 rank-citation profiles ci (r) demonstrates the statistical regularity in career publication output. Each scientist produces a cascade of papers of varying impact between the ci (1) pillar
paper down to the least-known paper ci (Ni ). (a) Zipf rank-citation
profiles ci (r) for 100 scientists listed in dataset [A]. For reference,
we plot the average c(r) of these 100 curves and find c(r) ∼ r−β
with β = 0.92 ± 0.01. The solid green line is a least-squares fit
to c(r) over the range 1 ≤ r ≤ 100. We also plot the H2 (r) and
H80 (r) lines for reference. (b) We re-scale the curves in panel (a),
plotting ci (r0 ) ≡ ci (r)/A(r1 + 1 − r)γ , where we use the best-fit γi
and Ai parameter values for each individual ci (r) profile. Using the
rescaled rank value r0 ≡ rβi , we show excellent data collapse onto
the expected curve c(r0 ) = 1/r0 . (see Figs. S1 and S2 for analogous
plots for dataset [B] and [C] scientists). Green data points correspond
to the average c(r0 ) value with 1σ error bars calculated using all 100
ci (r0 ) curves separated into logarithmically spaced bins.
DGBD model, we collapse each ci (r) along a universal scaling function c(r0 ) = 1/r0 , by using the rescaled rank values
r0 ≡ rβi defined for each curve. In Figs. 2(b), S1(b) and
S2(b), we plot the quantity ci (r0 ) ≡ ci (r)/A(r1 + 1 − r)γ , using the best-fit γi and Ai parameter values for each individual
ci (r) profile. While the curves in Fig. 2(a) are jumbled and
distributed over a large range of c(r) values, the rescaled ci (r)
curves in Fig. 2(b) all lie approximately along the predicted
curve c(r0 ) = 1/r0 .
4
A main advantage of the h-index is the simplicity in which
it is calculated, e.g. ISI Web of Knowledge [24] readily
provides this quantity online for distinct authors. Another
strength of the h-index is its stable growth with respect to
changes in ci (r) due to time and information-dependent factors [25]. Indeed, the h-index is a “fixed-point” of the citation
profile. This time stability is evident in the observed growth
rates of h for scientists. Average growth rates, calculated here
as h/L, where L is the duration in years between a given author’s first and most recent paper, typically lie in the range
of one to three units per year (this annual growth rate corresponds to the quantity m introduced by Hirsch [13]). Annual
growth rates h/L ≈ 3 correspond to exceptional scientists
(for the histogram of P (h/L) see the Fig. S3 and for h/L
values see the SI text (Tables S1-S4)). As a result, h/L is a
good predictor for future achievement along with h [26].
It is truly remarkable how a single number, hi , correlates
with other measures of impact. Understandably, being just a
single number, the h-index cannot fully account for other factors, such as variations in citation standards and coauthorship
patterns across discipline [27–29], nor can hi incorporate the
full information contained in the entire ci (r) profile. As a result, it is widely appreciated that the h-index can underrate the
value of the best-cited papers, since once a paper transitions
into the region r ≤ hi , its citation record is discounted, until
other less-cited papers with r > hi eventually overcome the
rank “barrier” r = hi . Moreover, as noted in [13], the papers
for which r > hi do not contribute any additional credit.
Instead of choosing an arbitrary hp as an productivityimpact indicator, we use the analytic properties of the DGBD
to calculate a crossover value ri∗ . In the methods section,
we derive an exact expression for ri∗ which highlights the
distinguished papers of a given author. To calculate ri∗ , we
use the logarithmic derivative χ(r) ≡ d ln c(r)/dr to quantify the relative change in ci (r) with increasing r. We defined papers as “distinguished” if they satisfy the inequality
ci (r)/ci (r + 1) > exp(χ), where χ is the average value of
χ(r) over the entire range of r values. This inequality selects the peak papers which are significantly more cited than
their neighbors. The peak region r ∈ [1, ri∗ ] corresponds to a
“knee” in ci (r) when plotted on log-linear axes. The dependence of χ and ri∗ on the three DGBD parameters βi , γi and
Ni are provided in the methods section.
The advantage of ri∗ is that this characteristic rank value is a
comprehensive representation of the stellar papers in the highrank scaling regime since it depends on the DGBD parameter
values βi , γi and Ni , and thus probes the entire citation profile. Fig. 3 shows a scatter plot of the “c-star” c∗i ≡ ci (ri∗ ) and
hi values calculated for each scientist and demonstrates that
there is a non-trivial relation between these two single-value
indices. It also shows that for scientists within a small range
of c∗ there is a large variation in the corresponding h values, in
some cases straddling across all three sets of scientists. Also,
there are several c∗i values which significantly deviate from
the trend in Fig. 3, which is plotted on log-log axes. These
results reflect the fact that the h-index cannot completely in-
a
10
c* = c (r*)
Using ci (r) to quantify career production and impact
10
3
2
10
Set A 100
Set B 100
Set C 100
1
10
b
!
B.
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
100
h1
Set A100
Set B100
Set C100
20
40
60
h1
80
100
120
FIG. 3: Limitations to the use of the h-index alone. The h-index
can be insufficient in comprehensively representing ci (r). (a) The
h-index does not contain any information about ci (r) for r < hi ,
and can shield a scientist’s most successful accomplishments which
are the basis for much of a scientist’s reputation. This is evident
in the cases where c(ri∗ ) hi , in which case the h-index cannot
account for the stellar impact of the papers. (b) For a given hi value,
prolific careers are characterized by a large βi value, as it is harder
to maintain large βi values for large hi . As a result, the βi vs hi
parameter space can be used to identify anomalous careers and to
better compare two scientists with similar hi indices. We find that a
third career metric Ci , the total number of citations to the papers of
author i, can be calculated with high accuracy by the scaling relation
i
Ci ∼ h1+β
, which we illustrate in Fig. 4(b).
i
corporate the entire ci (r) profile. We plot the histogram of c∗i
and ri∗ values in Figs. S4 and S5, respectively.
To further contrast the values of c∗i and the h-index, we
propose the “peak indicator” ratio Λi ≡ c∗i /hi , which corrects
specifically for the h-index penalty on the stellar papers in
the peak region of ci (r). Thus, all papers in the peak region
of ci (r) satisfy the condition ci (r) ≥ hi Λi . In an extreme
example, R. P. Feynman has a peak value Λ ≈ 36, indicating
that his best papers are monumental pillars with respect to his
other papers which contribute to his h-index. Fig. S6 shows
5
Ci =
N
X
a
ci (r) ,
4
(5)
Ar−β ≈ h1+β
r=1
b
(6)
where HN 0 ,β is the generalized harmonic number and is of order O(1) for β ≈ 1. We neglect the γi scaling regime since the
low-rank papers do not significantly contribute to an author’s
Ci tally. We approximate the coefficient A in Eq. [6] using the
definition c(h) ≡ h, which implies that A/hβ ≈ h. We use
the value N 0 ≡ 3h, so that Cβ,h can be approximated by only
the two parameters hi and βi for any given author. We justify
this choice of N 0 by examining the rescaled ci (r/h), which
we consider to be negligible beyond rank r = 3hi for most
scientists. In Fig. 4(a), we plot for each scientist the predicted
Cβ,h value versus the empirical Ci value, and we find excellent agreement with our theoretical prediction Ci ∼ hi1+βi
given by Eq. [6]. In Fig. 4(b), we
for each scientist the
Prplot
1
total number of citations Cm = r=1
cm (r) using the bestfit DGBD model cm (r) ≡ ci (r; βi , γi , Ai , r1 ) to approximate
ci (r). The excellent agreement demonstrates that the fluctuations in the residual difference cm (r) − ci (r) cancel out on the
aggregate level. Furthermore, a comparison of the quality of
agreement between the theoretical Ci values and the empirical Ci values in Fig. 4(a) and (b) shows the importance of the
additional γi scaling regime in the DGBD model.
DISCUSSION
We use the DGBD model to provide an analytic description of ci (r) over the entire range of r, and provide a deeper
quantitative understanding of scientific impact arising from an
author’s career publication works. The DGBD model exhibits
scaling behavior for both large and small r, where the scaling
for small r is quantified by the exponent βi , which for many
scientists analyzed, can be approximated using only two values of the generalized h-index hp (see SI text). In particular,
we show that for a given h-value, a larger βi value corresponds
i
to a more prolific publication career, since Ci ∼ h1+β
.
i
Many studies analyze only the high rank values of generic
Zipf ranking profiles c(r), e.g. computing the scaling regime
10
3
4
C! , h
5
5
10
10
10
r−β = h1+β HN 0 ,β ∼ h1+β
r=1
II.
2
10 2
10
4
10
Set A 100
Set B 100
Set C 100
C
Cβ,h ≡
m=1
3
10
which incorporates
√ the entire ci (r) profile. However, it has
been shown that Ci correlates well with hi [30], a result
which we will demonstrate in Eq. [6] to follow directly from
a ci (r) with βi ≈ 1.
We test the aggregate properties of ci (r) by calculating the
aggregate number of citations Cβ,h for a given profile,
N0
X
Set A 100
Set B 100
Set C 100
10
r=1
N
X
5
10
C
the histogram of Λi values, with typical values for dataset [A]
scientists hΛi ≈ 3.4±3.9, and for dataset [B] scientists hΛi ≈
2.2±1.1. This indicator can only be used to compare scientists
with similar h values, since a small hi can result in a large Λi .
An alternative “single number” indicator is Ci , an author’s
total number of citations
m=1
3
10
2
10 2
10
10
3
4
Cm
10
5
10
FIG. 4: Aggregate publication impact C. The total number of cii
tations Ci ∼ h1+β
is also comprehensive productivity-impact meai
sure. For most best-fit DGBD model curves, the Ci value is preserved with high precision. This shows that the difference between
a given ci (r) and the corresponding best-fit DGBD model function
are negligible on the macroscopic scale. (a) The exact aggregate
number of citations Ci , calculated from ci (r) using Eq. [5], can
be analytically approximated by Cβ,h using Eq. [6] which depends
only on the scientist’s βi and hi values. (b) We justify the use of
the DGBD model defined in Eq. [3] for the approximation of ci (r)
by comparing the aggregate
citations Ci with the expected aggregate
Pr1
citations Cm =
r=1 cm (r) calculated from the best-fit DGBD
model cm (r). Including the extra scaling-parameter, as in the DGBD
model, improves the agreement between the theoretical and empirical Ci values in (a) and (b). We plot the line y = x (dashed-green
line) for visual reference.
for r < rc below some some rank cutoff rc . However, these
studies cannot quantitatively relate the large observations to
the small observations within the system of interest. To account for this shortcoming, our method for calculating the
crossover values ri∗ ≡ r− , rx , and r+ , which we elaborate in
the methods section, can be used in general to quantitatively
distinguish relatively large observations and relatively small
observations within the entire set of observations. Moreover,
the DGBD model has been shown to have wide application in
6
quantifying the Zipf rank profiles in various phenomena [20].
To measure the upward mobility of a scientist’s career, in
the SI text we address the question: given that a scientist has
index h, what is her/his most likely h-index value ∆t years in
the future? In consideration of the bulk of ci (r), and following
from the regularity of ci (r) for r ≈ h, we propose a modelfree gap-index G(∆h) as both an estimate and a target for
future achievement which can be used in the review of career
advancement. The gap index G(∆h), defined as a proxy for
the total number of citations a scientist needs to reach a target
value h + ∆h, can detect the potential for fast h-index growth
by quantifying ci (r) around h. This estimator differs from
other estimators for the time-dependent h-index [32–34] in
that G(∆h) is model independent.
Even though the productivity of scientists can vary substantially [8, 35–38], and despite the complexity of success in
academia, we find remarkable statistical regularity in the functional form of ci (r) for the scientists analyzed here from the
physics community. Recent work in [7, 8, 39] calculates the
citation distributions of papers from various disciplines and
shows that proper normalization of impact measures can allow for comparison across time and discipline. Hence, it is
likely that the publication careers of productive scientists in
many disciplines obey the statistical regularities observed here
for the set of 300 physicists. Towards developing a model for
career evolution, it is still unclear how the relative strengths
of two contributing factors (i) the extrinsic cumulative advantage effect [2, 3, 8] versus (ii) the intrinsic role of the ”sacred
spark” in combination with intellectual genius [36] manifest
in the parameters of the DGBD model.
With little calculation, the βi metric developed here, used
in conjunction with the hi , can better answer the question,
“How popular are your papers?” [40]. Since the cumulative impact and productivity of individual scientists are also
found to obey statistical laws [8, 10], it is possible that the
competitive nature of scientific advancement can be quantified
and utilized in order to monitor career progress. Interestingly,
there is strong evidence for a governing mechanism of career
progress based on cumulative advantage [8, 10, 41] coupled
with the the inherent talent of an individual, which results in
statistical regularities in the career achievements of scientists
as well as professional athletes [10, 42, 43]. Hence, whenever data are available [44, 45], finding statistical regularities
emerging from human endeavors is a first step towards better
understand the dynamics of human productivity.
III.
A.
METHODS
Selection of scientists and data collection
We use disambiguated “distinct author” data from ISI Web
of Knowledge. This online database is host to comprehensive
data that is well-suited for developing testable models for scientific impact [8, 31, 39] and career progress [10]. In order to
approximately control for discipline-specific publication and
citation factors, we analyze 300 scientists from the field of
physics.
We aggregate all authors who published in Physical Review
Letters (PRL) over the 50-year period 1958-2008 into a common dataset. From this dataset, we rank the scientists using
the citations shares metric defined in [8]. This citation shares
metric divides equally the total number of citations a paper
receives among the n coauthors, and also normalizes the total
number of citations by a time-dependent factor to account for
citation variations across time and discipline.
Hence, for each scientist in the PRL database, we calculate a cumulative number of citation shares received from only
their PRL publications. This tally serves as a proxy for his/her
scientific impact in all journals. The top 100 scientists according to this citation shares metric comprise dataset [A]. As a
control, we also choose 100 other dataset [B] scientists, approximately randomly, from our ranked PRL list. The selection criteria for the control dataset [B] group are that an author
must have published between 10 and 50 papers in PRL. This
likely ensures that the total publication history, in all journals,
be on the order of 100 articles for each author selected. We
compare the tenured scientists in datasets A and B with 100
relatively young assistant professors in dataset [C]. To select
dataset [C] scientists, we chose two assistant professors from
the top 50 U.S. physics and astronomy departments (ranked
according to the magazine U.S. News).
For privacy reasons, we provide in the SI tables only the
abbreviated initials for each scientist’s name (last name initial,
first and middle name initial, e.g. L, FM). Upon request we
can provide full names.
We downloaded datasets A and B from ISI Web of Science
in Jan. 2010 and dataset C from ISI ISI Web of Science in Oct.
2010. We used the “Distinct Author Sets” function provided
by ISI in order to increase the likelihood that only papers published by each given author are analyzed. On a case by case
basis, we performed further author disambiguation for each
author.
B.
Statistical significance tests for the c(r) DGBD model
We test the statistical significance of the DGBD model fit
using the χ2 test between the 3-parameter best-fit DGBD
cm (r) and the empirical ci (r). We calculate the p-value for
the χ2 distribution with r1 − 3 degrees of freedom and find,
for each data set, the number N>pc of ci (r) with p-value > pc :
N>pc = 4 [A], 19 [B], 22 [C] for pc = 0.05, and 8 [A], 22 [B],
37 [C] for pc = 0.01.
The significant number of ci (r) which do not pass the χ2
test for pc = 0.05, results from the fact that the DGBD is
a scaling function over several orders of magnitude in both
r and ci (r) values, and so the residual differences [ci (r) −
cm (r)] are not expected to be normally distributed since there
is no characteristic scale for scaling functions such as the
DGBD. Nevertheless, the fact that so many ci (r) do pass the
χ2 test at such a high significance level, provides evidence for
the quality-of-fit of the DGBD model. For comparison, none
of the ci (r) pass the χ2 test using the power-law model at the
pc = 0.05 significance level. In the next section, we will also
compare the macroscopic agreement in the total number of
7
citations for each scientist and the total number of citations
predicted by the DGBD model for each scientist, and find excellent agreement. s
Derivation of the characteristic DGBD r values
C.
χ = −m
Here we use the analytic properties of the DGBD defined
in Eq. [3] to calculate the special r values from the parameters β, γ and N which locate the two tail regimes of c(z),
and in particular, the distinguished paper regime. The scaling
features of the DGBD do not readily convey any characteristic scales which distinguish the two scaling regimes. Instead,
we use the properties of ln ci (r) to characterize the crossover
between the high-rank and the low-rank regimes of ci (r).
We begin by considering ci (r) under the centered rank
transformation z = r − z0 , where z0 = (N + 1)/2, then
c(z) = A
(z0 − z)γ
,
(z0 + z)β
(7)
in the domain z ∈ [−(z0 − 1), (z0 − 1)]. The logarithmic
derivative of c(z) expresses the relative change in c(z),
d ln c(z)
dc(z)/dz
=
dz
c(z)
γ
1 + θx β +
= −
= −m
, (8)
z0 − z
z0 + z
1 − x2
γ+β
. The extreme
where x = z/z0 , θ = γ−β
γ+β , and m =
z0
χ(z) ≡
values of
The function χ(z) takes on the value of χ twice at the values
z ± = z0 x± corresponding to the solutions to the quadratic
equation,
d ln c(z)
dz
for z0 1 are given by
d ln c(z) ≈ −β
dz
z=−(z0 −1)
d ln c(z) ≈ −γ
dz
z=z0 −1
(9)
(10)
which has the solution
1 + θx 1 − x2
,
p
(ln N )2 − 2 ln N + θ2
θ
±
ln N
ln N
p
θ
± 1 − 2/ ln N
≈ −
ln N
(12)
x± = −
(13)
for θ2 / ln2 N 1. Converting back to rank, then
r± ≈
p
N
θ
1−
± 1 − 2/ ln N ,
2
ln N
(14)
and so the value r∗ ≡ r− is the special rank value which distinguishes the set of excellent papers of each given author. The
c-star value ci (r∗ ) is thus a characteristic value arising from
the special analytic properties of ci (r). This method for determining the crossover value r∗ can be applied to any general
Zipf profile which can be modeled by the DGBD.
Furthermore, the crossover zx between the β scaling regime
and the γ scaling regime is calculated from the inflection
points of ln c(z),
d2 ln c(z)
−γ
β
|z=zx =
+
(15)
2
2
dz
(z0 − zx )
(z0 + zx )2
p
1±ζ
which has 2 solutions zx± = z0 1∓ζ
, where ζ ≡ γ/β.
0=
Only |zx− | < z0 is a physical solution. Transforming back to
2
+1
rank values, we find rx = z0 + zx− = z0 1+ζ
= N1+ζ
. We
illustrate these special z values in Fig. 5.
and the average value χ is calculated by,
d ln c(z)
i
dz
Z (1−1/z0 )
(1 + θx)
−m
dx
=
(1 − 1/z0 ) − (1/z0 − 1) −(1−1/z0 )
1 − x2
−m
=
ln N
(11)
2
χ ≡ h
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indicated by the solid vertical green lines, are defined as the intersection of χ with χ(z) given by Eq. [13]. The regime z < z − corresponds to
the best papers of a given author. The hatched blue line corresponds to zx− which marks the crossover between the β and γ scaling regimes.
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10
Supplementary Information
Statistical regularities in the rank-citation profile of scientists
Alexander M. Petersen,1,2 H. Eugene Stanley2 , Sauro Succi 3,4
1
2
IMT Lucca Institute for Advanced Studies, Lucca 55100, Italy
Center for Polymer Studies and Department of Physics, Boston University, Boston, Massachusetts 02215, USA
3
Istituto Applicazioni Calcolo C.N.R., Rome, IT
4
Freiburg Institute for Advanced Studies, Albertstrasse, 19, D-79104, Freiburg, Germany
(2011)
I.
SIMPLE METHOD FOR ESTIMATING THE β SCALING OF ci (r) USING TWO hp VALUES
We analyze the citation profiles of 300 prolific scientists who published Physical Review Letters, and find statistical regularity
in the functional form of ci (r) of each individual scientist i. Here we further quantify ci (r) and discuss the information contained
in the “generalized” h-index hp , defined by the relation:
c(hp ) = php ,
(S1)
with p > 1 a positive integer. In analogy to the h-index, hp is the number of papers which are cited at least php times. By
definition, h/p < hp < h. Also, the index hp can be viewed as a functional transform in p−space of the citation profile ci (r).
This transform exhibits a number of characteristic values, namely h1 ≡ h, the standard h index, h0 = N , the total number of
papers, and h∞ = c(1), a scientist’s top-cited paper. Therefore, by changing p over the entire interval [0, ∞[, one gains spectral
information of the entire citation profile ci (r) for a given scientist.
Also, for the high-rank power-law regime c(r) ∼ r−β , there is a useful relation between hp and h1/p . Since hp = h p−µ then
the ratio for complementary p-values h1/p /hp = p2µ , where µ = 1/(1 + β). Small values of µ ≈ 1 indicate slowly-decaying
ci (r) corresponding to productive authors with potentially high mobility of the h-index. Hence, if ci (r) is power-law, then the
relation
Ip ≡
hp h1/p
=1
h2
(S2)
should hold independent of the value of β. For all scientists analyzed, we calculate I2 and find I2 = 0.97 ± 0.07. This implies
that ci (r) obeys a power law around r = h. This is visually confirmed by inspecting ci (r/h) in Figs. 2(b), S1(b) and S2(b) for
r/h ≈ 1. Hence, we define two complementary mobility indices as
m2 ≡
h2
h
(S3)
h1/2
h
(S4)
and
m1/2 ≡
By definition, 0 < m2 < 1 and 1 < m1/2 < N . The potential for high mobility of the h-index is associated with m2 close
to 1 (low barrier on the high-cite side) and m1/2 >> 1 (high propensity to change in the low-cite side). We show the relation
between m1/2 and m2 in Fig. S7 along with the expected relation m2 = 1/m1/2 for visual reference.
To test the small-r scaling for each ci (r), we estimate β using two methods:
(i) We define an approximation to β by assuming c(r) ∼ r−βpq for r < h. Hence, two intersection values, hp and hq are
sufficient to calculate βpq using the relationship for power-law ci (r),
βpq =
[1] Corresponding author: Alexander M. Petersen
E-mail: [email protected]
ln(q/p)
−1.
ln(hp /hq )
(S5)
11
We use the values p ≡ 80 and q ≡ 2 since these values generally enclose the scaling regime in the ci (r) profiles (see Fig. 2).
(ii) We calculate β using a multilinear least-squares regression of ln ci (r) for 1 ≤ r ≤ r1 using the DGBD model defined in
Eq. [3]. To properly weight the data points for better regression fit over the entire range, we use only 20 values of ci (r) data
points that are equally spaced on the logarithmic scale in the range r ∈ [1, r1 ]. We plot four empirical ci (r) along with their
corresponding best-fit DGBD functions in Fig. 1 to demonstrate the goodness of fit for the entire range of r.
We compare βpq and β values calculated using methods (i) and (ii) in Tables S1 -S6 and find good agreement. Furthermore,
the average scaling exponent hβi is approximately equal to the value of β calculated for the average c(r) profile in Fig. 2. For
the scientists analyzed in dataset [A] we find hβi = 0.83 ± 0.24 as compared to β = 0.92 ± 0.01. For the scientists analyzed in
dataset [B] we find hβi = 0.70 ± 0.17 as compared to β = 0.78 ± 0.01. We plot the histograms of β and γ for datasets [A], [B],
and [C] in Fig. S8.
II.
DECOMPOSING THE HIRSCH a FACTOR TO BETTER UNDERSTAND EFFICIENCY
Many alternative single-value indicators have been proposed to address the various criticisms of the h-index. The g-index
[15] differs from the h-index in that it lends more weight to the more highly-cited papers. However, as with the h-index, the
g-index does not immediately convey much more information than the total number of citations or the productivity coefficient
a = C/h2 introduced by Hirsch. In Fig. S9 we plot the histogram of both h and a values for dataset [A] with hai = 6.0 ± 4.1
and for dataset [B] with hai = 4.2 ± 1.3, indicating that most researchers do not fall into the ‘step-function’ pathology β ≈ 0
of scientist 2 above for which a ≈ 1. Instead, most scientists have a significant number of citations arising from both their
high-cited (r < h) and low-cited (r > h) regions of ci (r).
An interesting decomposition is to write a as the product of two factors,
a=
N hci h
h
,
(S6)
where hci ≡ C/N .
(i) The first factor, by definition, is always greater than 1, and represents the number of papers in units of h. Small values of
N/h correspond to scientists who are very efficient (or less productive), while large values correspond to scientists who are very
productive (or less efficient). Highly productive authors, who may have a substantial number of papers without a single citation,
nevertheless can still have a large h-index.
(ii) The second factor is the average number of citations per paper hci in units of h. Relatively large values of hci/h > 1 signal
the presence of outstanding highly cited papers, i.e. papers with c(r) h. Many brilliant careers result from a combination of
moderate N/h ≈ 3 and hci/h > 1. Fig. S10, which plots N/h versus hci/h for the set of authors examined in this paper, shows
a tendency for the two factors to occupy a narrow band of hyperbolic curves hci/h = a/(N/h) with a ∈ {3, 7}.
III.
MOBILITY OF THE H-INDEX
The h-index is taken seriously by many research organizations, affecting important decisions such as tenure, promotions,
honors. For instance, Hirsch noted that h ∼ 12 seems to be appropriate for associate professor, h ∼ 18 might be suitable for
advancement to full professor, while h ∼ 45 is the average for NAS election [13]. However, little work has been done to measure
the “upward mobility” of the h-index with time. Here we address the question: given that a scientist has index h, what is her/his
most likely h-index value ∆t years in the future? This question is fundamentally related to the growth rate of ci (r) for r & h.
For productive scientists, Hirsch noted that h grows at a rate of about one unit a year [13]. However, a single-value indicator such
as h cannot quantify the probability of “growth spurts”, which should also enter into evaluation criteria (based on the h-index,
citation counts, etc.).
As a measure of upward mobility, we propose the gap index G(∆h), defined as a proxy for the total number of citations a
scientist needs to reach a target value h + ∆h, which is similar to the w(q)-index proposed in [18]. The merit of G(∆h) is to
detect the potential for fast growth by quantifying ci (r) around h. For ci (r) with index value h at time t, we define G(∆h) as
the minimum number of citations, distributed to papers r = {1, ...h + ∆h}, so that the h-index value at time t + ∆t becomes
h + ∆h.
Consider the citation gap g(r) = h+ − c(r) of each paper r with c(r) < h+ . Then G(∆h) is given by the exact relation which
can be easily verified graphically,
+
G(∆h) ≡
h
X
r=h−
g(r) = h+ (h+ − h− + 1) − C(h− , h+ ) ,
(S7)
12
Pn
where h+ = h + ∆h, h− is the smallest r value for which c(r) < h+ , and C(m, n) = r=m c(r) is the number of citations
from paper m up to paper n. Hence, G(∆h) quantifies the minimum number of citations, assuming perfect assignment, required
to bring papers r = h− . . . h . . . h+ up to citation level h+ .
The gap index G(∆h) establishes a characteristic time scale ∆t for the dynamics of h(t). The estimated amount of time for
the transition is ∆t = M ax{tr ≡ g(r)/ċ(r)} for h− ≤ r ≤ h+ , where ċ(r) is its average citation rate (citations/year). This
estimate does not take into account ‘rampant papers’, papers with relatively large r and ċ(r), which are either new or rejuvenated
after a lengthy period with very few citations [46]. In practical terms, the short-term utility of G(∆h) is for moderate values of
∆h, say in multiples of 5 or 10. In other words, for a scientist with h = 12, a plausible target could be h+ = 17 and a longer
term target h+ = 22.
In Fig. S11 we plot the histograms for G(5) and G(10). The common distributions between authors in dataset [A] and dataset
[B] indicate that the growth potential of h does not depend very strongly on prestige, but rather on the publication patterns of
individual authors. Indeed, the average annual growth rate h/L are larger for dataset [A] physicists than dataset [B] physicists,
with a significant number of exceptional “outliers” with h/L > 3.
For young careers corresponding to small h-values, there will be a correlation between G(∆h) and h because most new
citations will contribute to the increase of h. However, for an advanced career, not all incoming citations will contribute to
an increase in h. Hence, to test the dependence of G(∆h) on h, we perform a linear regression Gi (∆h) = g0 + g1 hi for
both datasets [A] and [B] and for ∆h = 5, 10. In each of the four regressions we calculate correlation values R2 < 0.05 and
ANOVA (analysis of variance) F-statistics F < 2.3 for each case, indicating that we accept the null hypothesis that the linear
regression coefficient g1 = 0. Thus, for significantly large h, the gap-index G(∆h) is not dependent on h. A similar regression
analysis between G(∆h) and β results in the same conclusion, that the gap index G(∆h) is not dependent on β for profiles with
sufficiently large h. Hence, the gap index can be used to estimate the mobility of h and as a comparison index between ci (r).
IV.
CHARACTERIZING THE RANK-CITATION ci (r) PROFILE
As many previous studies have shown, and further demonstrated here, there are many conceivable ways to quantify ci (r). In
Tables S1-S4 we list 16 values derived from ci (r) which can serve as quantitative indicators of a scientific career:
[1] the author’s total number of papers N ,
[2] the author’s total number of citations C ≡
PN
r=1
c(r),
[3] the author’s most-cited paper ci (1),
[4] the author’s c-star paper c(r∗ ), which distinguishes the minimum citation tally of his/her stellar papers in the range
r ∈ [1, r∗ ].
[5] the author’s r∗ value calculated from his/her DGBD parameter values according to Eq. [14]
[6] the author’s original h-index h1 ≡ h and the generalized h-index hp for p = 2, 80,
[7] the author’s scaling exponent βpq calculated using the values p = 2 and q = 80 corresponding to h2 and h80 ,
[8] the author’s scaling exponents β and γ calculated using multilinear least-squares regression fit to the DGBD model ci (r)
in Eq. [3],
[9] the author’s peak-value Λ given by the c-star value c∗ in units of the h-index, Λ ≡ c(r∗ )/h
[10] the author’s number of papers in units of the h-index N 0 ≡ N/h,
[11] the author’s average number of citations per paper in units of the h-index hci0 ≡ hci/h,
[12] the author’s “productivity” value proposed by Hirsch, a ≡ C/h2 ,
[13] the author’s mobility estimator G(∆h) quantifying the minimum number of citations needed to increase an individual’s
h-value by ∆h units for ∆h = 5 and ∆h = 10,
[14] the author’s mobility indices m1/2 and m2 where m1/2 m2 ≈ 1,
[15] the author’s peak number P =
1
h2
Ph
r=1
c(r),
[16] the author’s average h-index growth rate h/L over the L-year time period between an author’s first and most recent paper.
13
The h-index conveys a very informative one-number picture of productivity, however it does not tell the whole story, since it
does not fully capture the impact an author’s most cited papers. Instead, we show the utility of the c-star value c(r∗ ), which is
a better representative of an author’s most cited papers. Thus, we introduce the peak-value indicator Λ ≡ c(r∗ )/h in order to
characterize the most distinguished papers (1 ≤ r ≤ r∗ ) of each given author. The probability distribution of Λ values is given
in Fig. [? ].
We use the Discrete Generalized Beta Distribution (DGBD) to quantify ci (r) for the whole range of r. However, typically
a scientist is mostly evaluated by his/her highest ranked papers, say for r ≤ r∗ . Hence, in this regime, we show that ci (r) can
be parameterized by only two variables, β and h1 , in order to comprehensively capture a publication career. The emergence
of a such a compact (two-parameter) and general parametrization highlights an amazing statistical regularity in the scientific
productivity of single individuals. Without endorsing the extreme viewpoint ”you are what you publish” or “publish or perish”,
such statistical regularity, nevertheless, highlights an outstanding question on the role of social factors in ironing out individual
details of human productivity. We believe that such question bears a great relevance to most fields of economic, natural and
social sciences, where productivity data are available.
14
a 10
4
Average c(r)
3
citations, c(r)
10
10
2
H80
1
H2
10
0
10
0
10
10
1
2
paper rank, r
3
10
10
c(r') = c(r)/A (N+1-r)
"
b
10
0
1
-2
10
0
10
10
1
2
3
10
scaled paper rank, r' = r
10
!
FIG. S1: Zipf rank-citation curves plotted in panel (a) correspond to the the 100 dataset [B] scientists. For reference, we plot the average
c(r) of these 100 curves and find c(r) ∼ r−β with β = 0.78 ± 0.01. The solid green line is a least-squares fit to c(r) over the range (a)
1 ≤ r ≤ 30. We also plot the Hp (r) lines corresponding to p = 2 and p = 80 for reference. (b) We re-scale the curves in panel (a), plotting
c(r0 ) ≡ c(r)/A(N + 1 − r)γ , where we use the multilinear least-squares regression values for each individual ci (r) profile. Using the rescaled
rank value r0 ≡ rβ , we show excellent data collapse along the expected curve c(r0 ) = 1/r0 . Green data points correspond to the average c(r0 )
value with 1σ error bars calculated using all 100 ci (r0 ) curves separated into logarithmically spaced bins.
15
a 10
4
H20
citations, c(r)
3
10
10
2
1
10
H2
0
10 0
10
2
10
paper rank, r
c(r') = c(r)/A (N+1-r)
"
b
1
10
10
0
1
-1
10
-2
10
0
10
1
2
10
scaled paper rank, r' = r
10
!
FIG. S2: Zipf rank-citation curves plotted in panel (a) correspond to the the 100 dataset [C] assistant professor scientists. For reference, we
plot the Hp (r) lines corresponding to p = 1.. (b) We re-scale the curves in panel (a), plotting c(r0 ) ≡ c(r)/A(N + 1 − r)γ , where we use
the multilinear least-squares regression values for each individual ci (r) profile. Using the rescaled rank value r0 ≡ rβ , we show excellent data
collapse along the expected curve c(r0 ) = 1/r0 even for young careers. Green data points correspond to the average c(r0 ) value with 1σ error
bars calculated using all 100 ci (r0 ) curves separated into logarithmically spaced bins. We note that young careers might possibly be analogous
to advanced careers in other disciplines where overall publication rates are lower.
16
Set A100
Set B100
Set C100
P( h1 / L)
1.2
1
0.8
0.6
0.4
0.2
00
1
2
h1 / L
3
0
1
2
h1 / L
3
0
1
2
h1 / L
3
4
FIG. S3: Probability distribution of h/L values calculated for scientists in datasets [A], [B] and [C].
Set A100
Set B100
Set C100
0.01
P(c(r*))
0.01
0.008
0.006
0.004
0.002
0
0
150 300 450
c(r )
*
0
100
200
c(r )
*
0
100
200
c(r )
*
FIG. S4: Probability distribution of c(r∗ ) values calculated for scientists in datasets [A], [B] and [C].
300
17
Set A100
Set B100
Set C100
0.05
P( r* )
0.04
0.03
0.02
0.01
0
0
20
40
r
*
60
80 0
20
40
r
*
60
0
2
4
6
r
*
8 10 12
FIG. S5: Probability distribution of r∗ for scientists in datasets [A], [B] and [C].
Set A100
Set B100
Set C100
P( Λ )
0.5
0.4
0.3
0.2
0.1
0
0
2
4
8 10 0
6
Λ = c(r ) / h1
*
2
4
Λ = c(r ) / h1
*
6 0
4
8
12
Λ = c(r ) / h1
*
FIG. S6: Probability distribution of peak-values Λ ≡ c(r∗ )/h for scientists in datasets [A], [B] and [C].
18
Set A100
Set B100
Set C100
1.1
1
m2
0.9
0.8
0.7
0.6
0.5
0.4
1
1.1
1.2
1.3
1.4
m1/2
1.5
1.6
FIG. S7: Scatter plot of m1/2 = h1/2 /h and m2 = h2 /h for for scientists datasets [A], [B] and [C]. The quantity I2 ≡ m1/2 m2 ≈ 0.97±0.07
for all scientists analyzed. We plot 3 green curves corresponding to I2 = 0.9, I2 = 1.0, and I2 = 1.1 for comparison.
19
3.5
3.5
Set A100
Set A100
3
3
2.5
2
2
1.5
1.5
1
P( " )
P( ! )
2.5
1
0.5
0.5
0 0.4
0.6 0.8
1
1.2 1.4
0.4
0.6
0.8
1
"
!
0
1.2
3.5
4.5
Set B100
Set B100
4
3
3.5
3
2
2.5
2
1.5
P( " )
P( ! )
2.5
1.5
1
1
0.5
0.5
0 0.4
0.6 0.8
1
1.2 1.4
0.4
!
0.6
0.8
"
1
1.2
2
2
Set C100
Set C100
1.5
1
1
0.5
P( " )
1.5
P( ! )
0
0.5
00 0.4 0.8 1.2
1.6 2 2.4
!
0
0.3 0.6 0.9 1.2 1.5
0
"
FIG. S8: Probability distribution of β and γ values calculated using multilinear least-squares regression of ln ci (r) ≡ ln A−β ln r +γ ln[r1 +
1 − r] for scientists in datasets [A], [B] and [C].
20
Set A100
Set A100
0.03
0.5
0.02
0.3
P( a )
P( h1 )
0.4
0.2
0.01
0.1
0
25
50
75
h1
100
125
2
4
Set B100
6
a
0
8
10
Set B100
0.03
0.5
0.02
0.3
P( a )
P( h1 )
0.4
0.2
0.01
0.1
25
50
75
h1
100
125
2
6
a
0
8
10
0.3
Set C100
0.08
4
Set C100
0.25
0.2
P( h1 )
0.06
0.15
0.04
P( a )
0
0.1
0.02
00
0.05
10
20
h1
30
40
50
0
10
a
20
0
FIG. S9: Probability distribution of h and a values calculated for scientists in datasets [A], [B] and [C].
21
<c>/h
10
10
Set A 100
Set B 100
Set C 100
1
0
a=7
a=3
+ impact
+ efficiency
10
+ productivity
-1
10
0
10
1
N/h
FIG. S10: Statistical regularities in the impact-productivity space of scientists. Scatter plot of N and hci, in units of h, for the 300 scientists
analyzed. The quantity hci represents the impact per paper, while the quantity N/h is inversely related to efficiency, since small values
correspond to scientists with many citations arising from a relatively small number of papers. Because each quantity is measured in units of
h, only individuals with similar h can be compared. Dotted green lines correspond to hyperbolic curves hyperbolic curves hci/h = a/(N/h)
with values a = {3, 4, 5, 6, 7} (bottom to top).
22
0.05
0.025
0.04
0.02
0.03
0.015
0.02
0.01
0.01
0.005
020
40
80
60
G(5)
0
100
150 200 250 300 350
G(10)
0.05
0.025
Set B100
P( G5 )
P( G10 )
Set A100
Set B100
0.04
0.02
0.03
0.015
0.02
0.01
0.01
0.005
020 30 40
50 60 70 80
G(5)
P( G10 )
P( G5 )
Set A100
0
125 150 175 200 225 250 275
G(10)
0.05
0.015
Set C100
Set C100
0.04
P( G10 )
P( G5 )
0.01
0.03
0.02
0.005
0.01
00
20
40
G(5)
60
80
0
0
50 100 150 200 250 300
G(10)
FIG. S11: Probability distribution of G(5) and G(10) values calculated for scientists datasets [A], [B] and [C].
β
23
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
Set A100
Set B100
Set C100
0.2 0.4 0.6 0.8
1
γ
1.2 1.4 1.6 1.8
FIG. S12: Scatter plot of best-fit β and γ values.
2
24
c(1) c(r∗ ) r∗
N
C
A, E
A, I
A, A
A, BJ
A, PW
A, A
B, P
B, J
B, CP
B, CWJ
B, CH
B, G
B, K
B, M
C, N
C, R
C, DM
C, DJ
C, SW
C, JI
C, ML
C, PB
D, S
D, SD
E, DE
E, JH
E, VJ
E, M
F, RP
F, ME
F, MPA
F, DS
G, H
G, C
G, SL
G, AC
G, DJ
H, FDM
H, BI
H, DR
H, TW
H, H
H, SE
H, JE
I, F
I, Y
J, R
J, S
K, HJ
K, G
116
205
363
185
344
111
173
141
60
265
78
91
190
218
140
267
162
194
469
295
752
220
594
108
235
347
129
58
69
362
145
137
193
128
157
1064
217
89
272
200
398
78
200
186
249
241
229
185
241
211
13848
15073
14996
18658
65575
12514
16709
25350
10960
14474
17715
15707
21887
16279
9022
18716
16307
13801
25808
19894
50269
14257
19992
8339
13741
15475
11496
16166
21058
33076
16913
16532
23540
19273
20303
44312
24264
13658
32647
18673
21854
4287
13256
10380
14235
12384
26017
12356
16011
11298
hxi
σ
275 20368 2686
190 11381 2436
Name
3339
783
708
6267
4661
1689
3037
5636
2486
1835
3849
6280
12906
1445
2261
5147
6264
1572
1444
1514
1588
1878
2119
744
780
891
1630
12906
1715
1490
2260
1834
2425
6250
2548
1602
1722
1823
2978
2482
1889
535
1423
535
1231
1598
1742
3836
1228
1949
145
181
92
133
379
306
154
305
264
101
400
240
82
148
117
128
136
166
115
136
153
125
65
189
170
109
194
55
1288
231
190
275
241
197
203
103
300
274
241
210
125
121
157
128
131
102
286
75
215
89
16
18.3
33.9
22.1
32.5
9.9
20.9
16.1
8.3
28.4
9.8
11.5
23.8
22.7
15.4
25.1
19.4
18.2
44.4
30.4
63.8
23.4
57.6
10.8
15.9
29.6
13.8
11.9
3.8
29.5
18.1
14
20.7
16.4
19.4
80.1
17.3
11.8
26.9
19.1
35.1
9
18.7
18.2
22.7
24.6
19.9
25.1
16.7
25.1
h1 h2 h80 βxy
41
57
63
51
103
48
53
57
28
58
39
46
45
55
43
66
52
56
82
71
107
55
65
45
65
65
46
22
38
93
59
61
77
51
61
108
67
44
78
64
70
33
56
52
52
52
74
43
62
47
β
γ
30 6 1.29 1.17 0.64
45 6 0.83 0.68 0.76
44 4 0.54 0.66 0.55
38 7 1.18 0.94 0.59
78 13 1.06 0.91 0.73
38 6
1 0.75 0.98
39 5 0.8 0.96 0.63
44 9 1.32 0.94 0.81
21 6 1.94 1.35 0.96
39 4 0.62 0.8 0.55
33 8 1.6 1.01 1.02
33 5 0.95 1.01 0.83
31 5 1.02 1.17 0.44
39 6 0.97 0.84 0.65
32 3 0.56 0.77 0.69
49 4 0.47 0.74 0.62
36 5 0.87 0.91 0.62
39 5 0.8 0.74 0.71
56 6 0.65 0.77 0.53
50 6 0.74 0.75 0.6
69 9 0.81 0.7 0.54
37 5 0.84 0.84 0.61
44 4 0.54 0.69 0.43
33 4 0.75 0.67 0.87
44 4 0.54 0.48 0.79
45 4 0.52 0.7 0.59
34 5 0.92 0.81 0.79
17 3 1.13 1.69 0.37
35 9 1.72 0.39 1.64
66 8 0.75 0.63 0.71
42 7 1.06 0.96 0.7
43 6 0.87 0.72 0.85
52 7 0.84 0.8 0.74
40 5 0.77 1.05 0.7
44 6 0.85 1 0.69
71 6 0.49 0.69 0.47
51 8 0.99 0.75 0.85
34 6 1.13 0.93 0.88
59 10 1.08 0.84 0.7
46 5 0.66 0.83 0.71
50 5 0.6 0.69 0.59
25 3 0.74 0.7 0.84
40 5 0.77 0.78 0.69
38 4 0.64 0.63 0.7
36 6 1.06 0.85 0.64
38 4 0.64 0.87 0.57
58 8 0.86 0.75 0.8
33 5 0.95 1.04 0.48
48 6 0.77 0.66 0.79
34 4 0.72 0.81 0.55
183 26 61 44
158 14.4 20 14
6
2
0.85 0.83 0.67
0.31 0.23 0.19
c(r∗ )/h1 N/h1 hci/h1
a
Gap(5) Gap(10) m1/2 m2
P
h1 /L
0.71
1.9
1.58
0.86
1.72
0.81
1.61
1.08
0.58
2.07
0.95
1.28
1.1
1.67
0.86
1.83
1.53
1.44
3.57
3.38
1.73
1.53
1.71
0.87
1.71
1.41
0.88
1.22
0.76
1.79
2.11
1.97
2.03
1.55
1.2
2.12
1.52
1.29
1.73
1.31
1.59
1.18
1.19
1.53
1.13
1.16
1.72
0.98
1.68
1.57
2.83
3.6
5.76
3.63
3.34
2.31
3.26
2.47
2.14
4.57
2
1.98
4.22
3.96
3.26
4.05
3.12
3.46
5.72
4.15
7.03
4
9.14
2.4
3.62
5.34
2.8
2.64
1.82
3.89
2.46
2.25
2.51
2.51
2.57
9.85
3.24
2.02
3.49
3.13
5.69
2.36
3.57
3.58
4.79
4.63
3.09
4.3
3.89
4.49
2.91
1.29
0.66
1.98
1.85
2.35
1.82
3.15
6.52
0.94
5.82
3.75
2.56
1.36
1.5
1.06
1.94
1.27
0.67
0.95
0.62
1.18
0.52
1.72
0.9
0.69
1.94
12.7
8.03
0.98
1.98
1.98
1.58
2.95
2.12
0.39
1.67
3.49
1.54
1.46
0.78
1.67
1.18
1.07
1.1
0.99
1.54
1.55
1.07
1.14
8.24
4.64
3.78
7.17
6.18
5.43
5.95
7.8
14
4.3
11.6
7.42
10.8
5.38
4.88
4.3
6.03
4.4
3.84
3.95
4.39
4.71
4.73
4.12
3.25
3.66
5.43
33.4
14.6
3.82
4.86
4.44
3.97
7.41
5.46
3.8
5.41
7.05
5.37
4.56
4.46
3.94
4.23
3.84
5.26
4.58
4.75
6.68
4.17
5.11
37
44
51
45
66
71
57
55
73
47
66
89
46
48
38
52
52
54
50
58
40
50
46
60
49
47
37
51
100
56
74
46
59
47
45
58
36
40
48
46
51
54
47
44
55
36
46
39
45
34
165
168
206
176
228
257
207
190
262
182
229
292
186
203
168
214
187
210
197
205
193
174
191
221
217
197
160
199
325
216
238
189
245
189
202
243
171
177
201
173
179
221
195
177
198
163
188
172
166
154
1.37
1.37
1.4
1.37
1.27
1.19
1.34
1.32
1.14
1.38
1.23
1.15
1.38
1.4
1.4
1.33
1.38
1.34
1.32
1.38
1.42
1.45
1.46
1.29
1.38
1.37
1.35
1.23
1.11
1.33
1.2
1.28
1.23
1.29
1.25
1.39
1.33
1.32
1.27
1.31
1.43
1.21
1.27
1.38
1.38
1.38
1.22
1.37
1.35
1.47
3.37
3.87
4.23
1.9
1.88
2
6.04
4.09
51
11
201
30
1.34 0.72 5.18 1.58
0.09 0.05 4.27 0.59
TABLE S1: Career citation statistics for 100 dataset [A] scientists: 1-50.
0.73
0.79
0.7
0.75
0.76
0.79
0.74
0.77
0.75
0.67
0.85
0.72
0.69
0.71
0.74
0.74
0.69
0.7
0.68
0.7
0.64
0.67
0.68
0.73
0.68
0.69
0.74
0.77
0.92
0.71
0.71
0.7
0.68
0.78
0.72
0.66
0.76
0.77
0.76
0.72
0.71
0.76
0.71
0.73
0.69
0.73
0.78
0.77
0.77
0.72
7.67
3.84
2.49
6.25
5.6
5.14
5.23
7.32
13.8
3.22
11.3
7.17
9.94
4.49
4.11
3.51
5.27
3.64
2.78
3
2.79
3.75
2.54
3.63
2.47
2.66
4.81
32.9
14.5
2.94
4.44
3.96
3.53
6.93
5
2.31
4.86
6.58
4.71
3.92
3.17
3.49
3.55
2.95
4.42
3.66
4.21
5.74
3.49
3.84
3.55
3.19
1.46
2.63
3.69
6.38
2.91
5.36
9.45
1.75
10.3
5.24
1.84
2.7
2.74
1.95
2.62
2.98
1.4
1.93
1.44
2.29
1
4.22
2.63
1.68
4.24
2.51
33.9
2.49
3.23
4.51
3.14
3.88
3.33
0.96
4.49
6.25
3.1
3.3
1.79
3.69
2.8
2.47
2.52
1.96
3.87
1.76
3.48
1.9
25
Name
N
C
L, RB
L, PA
L, EH
L, SG
L, MD
M, AH
M, ND
M, RN
N, DR
O, E
O, SR
P, G
P, SSP
P, M
P, JB
P, JP
P, A
P, LN
P, JC
P, HD
R, L
R, TM
S, JJ
S, LM
S, GA
S, DJ
S, M
S, JR
S, MO
S, YR
S, DJ
S, HE
S, PJ
S, R
S, RH
T, J
T, M
T, DC
V, CM
W, S
W, DA
W, KW
W, SR
W, F
W, E
W, WK
Y, E
Y, CN
Z, P
Z, A
hxi
σ
79
344
234
379
151
455
216
371
191
438
146
529
330
435
298
250
169
784
620
71
105
345
265
154
335
333
415
174
573
637
242
909
173
46
127
181
262
493
253
208
330
742
124
263
264
49
172
194
331
581
275
190
7751
32668
20139
27530
11231
17708
10409
18413
21742
22310
5051
29994
19184
29719
26621
62338
10053
24901
23513
8721
12124
25117
7662
9510
21292
17958
19276
24689
19269
26458
15414
41505
19462
8952
9186
22501
15755
17649
14935
42287
16955
24655
9821
26549
65014
13348
17852
23798
22263
36151
20368
11381
c(1) c(r∗ ) r∗
2271
3228
1862
1355
876
641
2741
1919
1371
2973
1236
2768
1760
5147
2719
12906
3849
460
1330
1807
3491
2112
868
3062
1328
589
580
5636
1456
1038
7118
892
1700
3491
1526
1782
1687
1602
2466
5094
610
458
1511
1722
2034
3815
6022
1537
1514
7861
2686
2436
147
208
167
160
178
93
83
87
265
101
59
108
108
123
170
173
50
89
75
333
100
193
52
71
155
129
115
208
74
114
71
115
201
82
96
275
133
82
112
488
140
90
144
254
860
495
153
318
148
109
183
158
10.8
30.8
25.1
34.9
14.5
37.9
20.5
40.6
18.3
41.6
14.8
51.1
34.7
43.6
31.5
31.4
25.9
52.4
59.4
6.7
16.6
27.2
29.2
21.1
28.3
28.8
33.2
18.9
49.2
44.1
27.8
68.9
19.9
9.8
16.3
18.1
24.3
41.6
25.3
17.9
25.5
51.6
15.2
22.2
14.6
6.5
19.6
16.2
30.8
54.3
26
14.4
h1 h2 h80 βxy
32
80
62
84
50
67
41
62
73
76
26
81
58
85
75
62
37
82
71
35
37
81
43
37
77
71
74
52
68
86
49
100
58
21
35
70
60
70
58
91
68
81
48
81
121
27
49
67
77
85
61
20
β
γ
24 5 1.35 1.04 0.76
59 9 0.96 0.8 0.67
46 6 0.81 0.82 0.65
62 6 0.58 0.66 0.62
37 5 0.84 0.73 0.8
46 4 0.51 0.58 0.56
29 5 1.1 1.07 0.53
42 5 0.73 0.84 0.47
52 8 0.97 0.72 0.81
49 5 0.62 0.77 0.51
21 3 0.9 1.14 0.53
55 7 0.79 0.82 0.49
41 7 1.09 0.84 0.53
57 5 0.52 0.76 0.53
48 7 0.92 0.83 0.61
46 9 1.26 1.38 0.49
25 3 0.74 1.1 0.38
56 3 0.26 0.53 0.53
46 6 0.81 0.78 0.43
27 4 0.93 0.7 1.11
26 4 0.97 1.21 0.55
58 6 0.63 0.69 0.68
29 3 0.63 0.82 0.46
27 5 1.19 1.08 0.46
53 5 0.56 0.61 0.66
49 5 0.62 0.59 0.64
48 4 0.48 0.54 0.61
41 8 1.26 1.16 0.69
48 5 0.63 0.75 0.46
59 4 0.37 0.54 0.57
32 4 0.77 0.94 0.47
68 7 0.62 0.61 0.52
44 7 1.01 0.87 0.73
16 3 1.2 1.72 0.5
28 4 0.9 1.22 0.56
51 8 0.99 0.72 0.82
43 4 0.55 0.72 0.64
46 4 0.51 0.66 0.51
39 5 0.8 0.92 0.57
71 10 0.88 0.68 0.92
48 5 0.63 0.57 0.67
54 3 0.28 0.53 0.53
34 4 0.72 0.85 0.7
63 7 0.68 0.67 0.78
92 13 0.89 0.45 1.06
21 6 1.94 1.17 1.18
36 6 1.06 0.92 0.65
54 8 0.93 0.71 0.89
52 6 0.71 0.68 0.62
59 4 0.37 0.69 0.5
44 6 0.85 0.83 0.67
14 2 0.31 0.23 0.19
c(r∗ )/h1 N/h1 hci/h1
4.62
2.61
2.7
1.91
3.58
1.4
2.03
1.41
3.63
1.34
2.31
1.34
1.87
1.45
2.27
2.8
1.37
1.09
1.07
9.53
2.7
2.39
1.22
1.93
2.02
1.82
1.56
4.01
1.1
1.33
1.45
1.15
3.48
3.95
2.76
3.94
2.23
1.18
1.94
5.37
2.07
1.12
3
3.14
7.11
18.4
3.14
4.76
1.93
1.29
3.37
3.87
2.47
4.3
3.77
4.51
3.02
6.79
5.27
5.98
2.62
5.76
5.62
6.53
5.69
5.12
3.97
4.03
4.57
9.56
8.73
2.03
2.84
4.26
6.16
4.16
4.35
4.69
5.61
3.35
8.43
7.41
4.94
9.09
2.98
2.19
3.63
2.59
4.37
7.04
4.36
2.29
4.85
9.16
2.58
3.25
2.18
1.81
3.51
2.9
4.3
6.84
4.23
1.9
3.07
1.19
1.39
0.86
1.49
0.58
1.18
0.8
1.56
0.67
1.33
0.7
1
0.8
1.19
4.02
1.61
0.39
0.53
3.51
3.12
0.9
0.67
1.67
0.83
0.76
0.63
2.73
0.49
0.48
1.3
0.46
1.94
9.27
2.07
1.78
1
0.51
1.02
2.23
0.76
0.41
1.65
1.25
2.04
10.1
2.12
1.83
0.87
0.73
1.88
2
a
7.57
5.1
5.24
3.9
4.49
3.94
6.19
4.79
4.08
3.86
7.47
4.57
5.7
4.11
4.73
16.2
7.34
3.7
4.66
7.12
8.86
3.83
4.14
6.95
3.59
3.56
3.52
9.13
4.17
3.58
6.42
4.15
5.79
20.3
7.5
4.59
4.38
3.6
4.44
5.11
3.67
3.76
4.26
4.05
4.44
18.3
7.44
5.3
3.75
5
6.04
4.09
Gap(5) Gap(10) m1/2 m2
43
40
57
53
50
43
62
36
57
43
55
53
40
48
54
63
57
45
57
51
64
45
55
38
49
46
36
57
53
54
58
53
58
45
49
64
40
38
60
57
41
54
80
52
37
53
40
53
65
47
51
11
190
166
197
177
211
175
216
173
215
199
183
194
157
196
218
224
203
174
243
202
212
190
202
165
195
183
161
230
205
197
239
216
232
186
195
232
180
168
207
231
181
205
282
194
167
233
171
218
237
183
201
30
TABLE S2: Career citation statistics for 100 dataset [A] scientists: 51-100.
1.31
1.38
1.42
1.43
1.24
1.49
1.22
1.48
1.26
1.36
1.31
1.41
1.55
1.41
1.36
1.34
1.46
1.44
1.49
1.2
1.35
1.38
1.35
1.35
1.36
1.41
1.46
1.19
1.4
1.4
1.33
1.44
1.24
1.33
1.26
1.24
1.38
1.44
1.34
1.19
1.41
1.43
1.21
1.31
1.23
1.19
1.43
1.27
1.45
1.51
1.34
0.09
0.75
0.74
0.74
0.74
0.74
0.69
0.71
0.68
0.71
0.64
0.81
0.68
0.71
0.67
0.64
0.74
0.68
0.68
0.65
0.77
0.7
0.72
0.67
0.73
0.69
0.69
0.65
0.79
0.71
0.69
0.65
0.68
0.76
0.76
0.8
0.73
0.72
0.66
0.67
0.78
0.71
0.67
0.71
0.78
0.76
0.78
0.73
0.81
0.68
0.69
0.72
0.05
P
h1 /L
7.06
4.26
4.29
2.74
3.97
2.34
5.52
3.29
3.62
2.75
6.7
3.23
4.18
2.94
3.77
15.4
6.23
2.04
2.82
6.81
8.23
3.03
2.95
6.02
2.64
2.43
2.1
8.7
2.72
2.1
5.48
2.36
5.15
19.9
6.92
4.06
3.37
2.23
3.73
4.76
2.53
2.05
3.76
3.44
4.09
18.1
6.6
4.84
2.7
3.19
5.18
4.27
1.07
1.82
1.19
2.33
3.13
1.91
0.8
1.77
2.09
1.85
0.65
1.84
2
2.36
1.79
1.63
0.79
1.86
1.31
1.06
1.54
1.84
0.84
0.95
1.75
2.03
2.24
0.98
1.55
1.87
2.23
2.22
1.66
0.88
0.92
2.26
1.05
1.59
1.38
1.47
2.06
1.93
1.66
2.19
3.56
0.84
1.44
1.03
2.33
2.24
1.58
0.59
26
Name
N
C
A, P
A, DE
B, RZ
B, BB
B, WF
B, AL
B, RH
B, L
B, K
B, KI
B, RW
B, AJ
B, JH
B, SJ
B, RA
C, EM
C, NJ
C, NS
C, G
D, C
D, TJ
D, G
E, JP
E, RW
F, JC
F, AR
F, PA
F, KJ
F, ED
F, D
G, GW
G, DC
G, W
G, SC
G, AM
G, P
H, P
H, S
H, SW
H, HJ
H, F
H, JJ
H, MS
H, CE
I, J
J, PK
K, LP
K, E
K, W
K, DV
hxi
σ
125
469
143
252
73
170
87
112
763
64
311
240
334
275
384
108
107
140
125
208
361
507
101
188
113
385
99
135
240
347
210
75
322
132
284
255
491
527
146
383
236
163
279
151
147
423
188
231
172
78
217
121
5167
18982
4946
6928
2723
25048
2589
1841
35274
1199
7063
9685
8108
19230
9774
6069
2898
2953
10245
19421
15040
26718
5833
12092
1854
17615
5286
8154
5711
15664
13358
2057
7594
4725
6316
16210
20518
18490
25088
8042
12176
27512
3331
6326
5831
7661
17057
8413
18752
1112
9230
6860
c(1) c(r∗ ) r∗ h1 h2 h80 βxy
668
1819
200
520
227
4461
298
107
2726
124
282
1384
733
1696
442
1306
255
166
5600
2693
872
1352
383
2535
174
4267
413
406
532
518
1199
342
375
407
376
2645
2568
1145
3444
232
821
5232
292
403
1028
212
2007
839
2763
106
1024
1158
94
89
102
71
96
203
63
41
100
49
61
81
51
161
62
103
57
59
74
165
94
123
151
139
37
74
158
176
45
108
131
69
54
89
55
105
71
67
387
52
124
370
25
92
71
46
218
76
228
29
96
64
13
40.9
9.3
20.6
6.9
21.5
10.4
10.5
66.1
5.9
25
24.4
32.2
25
32
12.2
12
11.2
16.9
22.1
32.4
43.5
9.1
16.6
11.4
39.6
8.4
10.4
26.5
29.8
22.1
7.9
28.8
12.8
24.7
28
50.4
52.4
14.6
31.7
20.8
15.8
27.8
15.7
17.4
32.3
16.9
23.6
17.3
9.8
20.7
10.6
36
66
41
45
29
61
25
25
89
21
44
48
44
74
49
34
28
30
34
58
64
75
36
40
26
59
37
46
37
65
61
21
47
38
42
55
63
64
69
47
57
68
27
42
40
42
54
47
63
19
44
14
26
46
28
33
20
45
18
16
58
13
31
33
31
50
32
25
18
20
27
43
41
52
29
32
16
41
28
35
24
45
41
17
30
26
28
39
41
42
50
32
41
52
19
30
27
27
44
32
46
12
31
10
3
6
2
2
2
8
2
1
5
1
3
3
3
5
3
4
2
1
3
7
4
7
3
6
1
4
4
4
2
4
3
2
2
2
2
5
4
5
8
2
4
8
1
3
3
2
7
3
6
1
3
2
0.71
0.81
0.4
0.32
0.6
1.14
0.68
0.33
0.51
0.44
0.58
0.54
0.58
0.6
0.56
1.01
0.68
0.23
0.68
1.03
0.59
0.84
0.63
1.2
0.33
0.59
0.9
0.7
0.48
0.52
0.41
0.72
0.36
0.44
0.4
0.8
0.59
0.73
1.01
0.33
0.59
0.97
0.25
0.6
0.68
0.42
1.01
0.56
0.81
0.48
0.62
0.25
β
γ
0.84
0.82
0.38
0.62
0.5
1.08
0.76
0.5
0.68
0.45
0.62
0.66
0.77
0.67
0.56
1.02
0.62
0.48
0.99
1
0.58
0.78
0.6
0.96
0.55
0.85
0.6
0.52
0.63
0.56
0.73
0.86
0.58
0.68
0.6
0.97
0.83
0.8
0.73
0.54
0.59
0.87
0.7
0.63
0.85
0.48
0.87
0.72
0.74
0.68
0.7
0.16
0.68
0.49
0.83
0.6
0.89
0.65
0.67
0.64
0.48
0.79
0.54
0.57
0.45
0.67
0.52
0.67
0.64
0.67
0.54
0.64
0.56
0.53
0.92
0.68
0.6
0.46
0.9
0.9
0.46
0.61
0.65
0.74
0.51
0.7
0.53
0.52
0.42
0.43
0.9
0.51
0.68
0.84
0.38
0.67
0.57
0.49
0.77
0.55
0.79
0.56
0.62
0.14
c(r∗ )/h1 N/h1 hci/h1
2.64
1.36
2.51
1.59
3.34
3.34
2.54
1.65
1.13
2.35
1.39
1.69
1.18
2.18
1.27
3.04
2.05
1.99
2.2
2.86
1.47
1.64
4.2
3.5
1.46
1.27
4.29
3.84
1.22
1.67
2.16
3.32
1.17
2.36
1.32
1.92
1.14
1.06
5.62
1.11
2.19
5.45
0.94
2.2
1.78
1.11
4.04
1.63
3.63
1.54
2.19
1.08
3.47
7.11
3.49
5.6
2.52
2.79
3.48
4.48
8.57
3.05
7.07
5
7.59
3.72
7.84
3.18
3.82
4.67
3.68
3.59
5.64
6.76
2.81
4.7
4.35
6.53
2.68
2.93
6.49
5.34
3.44
3.57
6.85
3.47
6.76
4.64
7.79
8.23
2.12
8.15
4.14
2.4
10.3
3.6
3.68
10.1
3.48
4.91
2.73
4.11
4.92
2.01
1.15
0.61
0.84
0.61
1.29
2.42
1.19
0.66
0.52
0.89
0.52
0.84
0.55
0.94
0.52
1.65
0.97
0.7
2.41
1.61
0.65
0.7
1.6
1.61
0.63
0.78
1.44
1.31
0.64
0.69
1.04
1.31
0.5
0.94
0.53
1.16
0.66
0.55
2.49
0.45
0.91
2.48
0.44
1
0.99
0.43
1.68
0.77
1.73
0.75
1.01
0.54
a
3.99
4.36
2.94
3.42
3.24
6.73
4.14
2.95
4.45
2.72
3.65
4.2
4.19
3.51
4.07
5.25
3.7
3.28
8.86
5.77
3.67
4.75
4.5
7.56
2.74
5.06
3.86
3.85
4.17
3.71
3.59
4.66
3.44
3.27
3.58
5.36
5.17
4.51
5.27
3.64
3.75
5.95
4.57
3.59
3.64
4.34
5.85
3.81
4.72
3.08
4.23
1.23
Gap(5) Gap(10) m1/2 m2
63
44
41
64
52
47
61
51
78
53
31
42
44
45
38
57
49
48
48
51
47
36
57
49
59
61
35
46
66
61
43
54
42
56
56
51
42
72
62
53
44
48
30
33
79
52
61
56
61
49
52
11
TABLE S3: Career citation statistics for 100 dataset [B] scientists: 1-50.
236
184
165
215
212
226
207
200
251
209
169
172
166
179
171
215
184
186
174
199
207
156
205
176
212
225
153
185
223
210
196
190
192
213
212
201
176
240
217
198
170
184
136
152
244
198
239
201
226
189
196
27
1.31
1.41
1.34
1.38
1.24
1.25
1.32
1.36
1.48
1.33
1.41
1.44
1.45
1.35
1.59
1.21
1.57
1.47
1.41
1.31
1.48
1.44
1.31
1.4
1.38
1.46
1.3
1.35
1.51
1.48
1.3
1.24
1.47
1.32
1.43
1.35
1.46
1.41
1.28
1.45
1.4
1.24
1.56
1.43
1.3
1.57
1.22
1.47
1.3
1.37
1.39
0.09
0.72
0.7
0.68
0.73
0.69
0.74
0.72
0.64
0.65
0.62
0.7
0.69
0.7
0.68
0.65
0.74
0.64
0.67
0.79
0.74
0.64
0.69
0.81
0.8
0.62
0.69
0.76
0.76
0.65
0.69
0.67
0.81
0.64
0.68
0.67
0.71
0.65
0.66
0.72
0.68
0.72
0.76
0.7
0.71
0.68
0.64
0.81
0.68
0.73
0.63
0.7
0.05
P
h1 /L
3.41
3.12
2.22
2.41
2.75
6.21
3.38
1.98
2.49
2.08
2.33
2.85
2.74
2.72
2.07
4.8
2.6
2.11
7.98
5.13
2.11
3.34
4.01
6.77
1.86
3.72
3.38
3.3
2.27
2.34
2.84
4.16
1.92
2.65
2.17
4.45
3.45
2.92
4.84
1.97
2.7
5.53
2.44
2.7
2.95
1.81
5.33
2.62
4.16
2.15
3.19
1.36
0.86
1.5
1.24
1.13
0.57
2.9
0.83
1.19
2.07
0.68
1.19
1.3
1.13
1.68
1.11
1.1
1.47
0.71
0.83
2.32
1.31
1.32
1.16
1.08
0.84
1.28
1.19
1.02
1.48
1.71
1.56
0.84
1.07
1.73
0.89
1.28
1.75
1.68
1.53
1.52
1.16
1.33
0.77
1.56
1.25
1
1.06
1.88
2.33
0.76
1.28
0.5
27
Name
N
C
K, TR
K, L
K, W
K, WR
L, RB
L, P
L, MJ
L, M
L, AJ
L, RA
L, H
L, MS
M, L
M, BT
M, P
M, DE
M, JE
M, GE
N, AHC
O, V
O, SA
P, VM
P, CJ
P, PM
P, VL
P, CY
R, AR
S, BEA
S, RD
S, F
S, WD
S, J
S, L
S, GF
S, D
S, KR
S, EA
S, S
S, A
S, S
T, MA
T, LJ
T, D
T, MS
V, JJM
W, IA
W, RE
W, RB
W, H
W, JA
hxi
σ
161
268
395
111
157
255
180
240
152
190
234
143
264
244
398
107
176
420
158
104
150
83
184
204
137
118
113
284
121
266
45
77
108
202
363
211
272
220
158
220
123
138
315
246
131
209
261
185
240
120
217
121
5394
10661
4433
15302
6244
6264
2110
7535
14577
5481
6277
2379
13179
9633
5915
6011
8053
10571
3509
6588
9554
2089
8877
8569
2932
3214
5257
4937
4585
10047
1330
3254
4026
26489
7894
7371
12743
9322
16325
4178
1304
3494
11639
17261
5524
4156
12111
5611
11408
2722
9230
6860
c(1) c(r∗ ) h1 h2 h80 βpq
549
3016
193
2535
571
300
298
535
2261
489
279
319
863
686
372
865
572
862
431
663
538
254
522
432
433
548
295
337
449
636
154
643
440
3501
238
482
882
2280
1760
577
119
228
569
2888
491
383
880
375
657
196
1024
1158
66
61
27
301
98
57
40
59
113
69
60
41
125
91
36
139
91
52
44
164
176
50
115
109
41
42
101
43
109
82
78
78
86
150
53
81
103
67
230
35
28
61
95
134
99
49
103
73
127
60
96
64
18.5
29.7
33.5
11.3
14.6
23.4
12.4
27.1
19.8
17.9
22
13.6
22.9
22.3
34.3
10.2
19.4
40.6
17.9
9.6
11.7
10.4
17.7
17.2
15.5
16.2
12.3
25
9.6
25.6
4.2
10
11.3
24.8
30.5
20.2
26.1
25.5
16
23.6
11.4
13.3
25.5
25.5
12.8
19.1
24.6
17.5
18.9
10.8
20.7
10.6
35
49
35
45
39
41
21
43
42
36
42
24
57
52
38
40
44
52
30
40
53
24
49
50
27
25
36
38
37
53
21
28
30
52
45
48
50
45
59
31
21
33
59
63
42
35
54
40
60
31
44
14
26
31
23
37
30
30
16
28
29
27
27
17
40
37
25
29
30
33
20
31
39
18
33
34
20
17
25
23
27
36
15
20
22
37
30
32
36
32
45
21
14
22
40
42
29
23
39
27
41
21
31
10
3
2
1
6
3
2
2
3
7
3
2
2
5
3
2
3
4
3
2
3
4
2
4
3
1
3
3
2
2
3
1
3
2
7
2
3
5
3
5
3
1
2
3
4
2
2
4
3
4
1
3
2
β
γ
0.71
0.35
0.18
1.03
0.6
0.36
0.77
0.65
1.6
0.68
0.42
0.72
0.77
0.47
0.46
0.63
0.83
0.54
0.6
0.58
0.62
0.68
0.75
0.52
0.23
1.13
0.74
0.51
0.42
0.48
0.36
0.94
0.54
1.22
0.36
0.56
0.87
0.56
0.68
0.9
0.4
0.54
0.42
0.57
0.38
0.51
0.62
0.68
0.59
0.21
0.62
0.25
0.75
0.84
0.63
0.8
0.76
0.59
0.82
0.74
1.23
0.78
0.55
0.89
0.74
0.59
0.72
0.63
0.8
0.64
0.78
0.58
0.5
0.68
0.64
0.58
0.74
0.87
0.63
0.59
0.53
0.67
0.4
0.89
0.78
1.23
0.57
0.68
0.78
0.8
0.72
0.91
0.65
0.56
0.59
0.76
0.57
0.7
0.72
0.63
0.57
0.54
0.7
0.16
c(r∗ )/h1 N/h1 hci/h1
0.56
0.45
0.39
0.95
0.7
0.54
0.56
0.49
0.56
0.57
0.57
0.51
0.65
0.62
0.4
0.83
0.61
0.44
0.49
0.88
0.87
0.63
0.68
0.7
0.53
0.45
0.75
0.48
0.8
0.57
0.93
0.69
0.71
0.56
0.5
0.6
0.58
0.49
0.81
0.39
0.54
0.64
0.61
0.61
0.72
0.52
0.6
0.61
0.71
0.68
0.62
0.14
1.89
1.25
0.8
6.69
2.54
1.4
1.94
1.39
2.71
1.94
1.45
1.71
2.2
1.76
0.96
3.48
2.09
1.02
1.47
4.12
3.34
2.09
2.36
2.19
1.54
1.7
2.83
1.13
2.97
1.56
3.75
2.8
2.89
2.9
1.2
1.7
2.07
1.51
3.91
1.14
1.35
1.85
1.61
2.14
2.37
1.4
1.92
1.84
2.13
1.94
2.19
1.08
4.6
5.47
11.3
2.47
4.03
6.22
8.57
5.58
3.62
5.28
5.57
5.96
4.63
4.69
10.5
2.68
4
8.08
5.27
2.6
2.83
3.46
3.76
4.08
5.07
4.72
3.14
7.47
3.27
5.02
2.14
2.75
3.6
3.88
8.07
4.4
5.44
4.89
2.68
7.1
5.86
4.18
5.34
3.9
3.12
5.97
4.83
4.63
4
3.87
4.92
2.01
a
0.96
0.81
0.32
3.06
1.02
0.6
0.56
0.73
2.28
0.8
0.64
0.69
0.88
0.76
0.39
1.4
1.04
0.48
0.74
1.58
1.2
1.05
0.98
0.84
0.79
1.09
1.29
0.46
1.02
0.71
1.41
1.51
1.24
2.52
0.48
0.73
0.94
0.94
1.75
0.61
0.5
0.77
0.63
1.11
1
0.57
0.86
0.76
0.79
0.73
1.01
0.54
Gap(5) Gap(10) m1/2 m2
4.4
4.44
3.62
7.56
4.11
3.73
4.78
4.08
8.26
4.23
3.56
4.13
4.06
3.56
4.1
3.76
4.16
3.91
3.9
4.12
3.4
3.63
3.7
3.43
4.02
5.14
4.06
3.42
3.35
3.58
3.02
4.15
4.47
9.8
3.9
3.2
5.1
4.6
4.69
4.35
2.96
3.21
3.34
4.35
3.13
3.39
4.15
3.51
3.17
2.83
4.23
1.23
39
59
64
56
42
42
43
63
56
47
59
48
62
55
46
44
49
68
62
34
55
43
46
60
46
41
51
69
50
41
33
76
49
72
55
58
34
38
51
53
63
47
83
51
58
53
43
64
64
69
52
11
TABLE S4: Career citation statistics for 100 dataset [B] scientists: 51-100.
153
215
213
211
165
165
159
215
216
202
203
178
225
195
190
169
189
238
217
147
201
150
187
212
183
147
196
244
190
173
148
234
180
227
190
195
138
168
197
200
211
182
254
206
203
203
159
233
255
226
196
27
1.49
1.43
1.4
1.24
1.31
1.51
1.43
1.51
1.24
1.28
1.48
1.33
1.35
1.46
1.45
1.35
1.41
1.38
1.37
1.38
1.3
1.5
1.39
1.34
1.3
1.6
1.36
1.37
1.35
1.4
1.43
1.21
1.33
1.35
1.49
1.33
1.5
1.47
1.32
1.29
1.24
1.45
1.39
1.49
1.43
1.4
1.41
1.35
1.28
1.32
1.39
0.09
P
h1 /L
0.74
0.63
0.66
0.82
0.77
0.73
0.76
0.65
0.69
0.75
0.64
0.71
0.7
0.71
0.66
0.73
0.68
0.63
0.67
0.78
0.74
0.75
0.67
0.68
0.74
0.68
0.69
0.61
0.73
0.68
0.71
0.71
0.73
0.71
0.67
0.67
0.72
0.71
0.76
0.68
0.67
0.67
0.68
0.67
0.69
0.66
0.72
0.68
0.68
0.68
0.7
0.05
3.18
3.26
1.9
7.11
3.41
2.2
3.65
2.57
7.66
3.24
2.08
3.25
3.22
2.37
2.31
3.25
3.28
2.09
2.74
3.54
2.86
2.69
2.8
2.6
3.02
3.67
3.27
1.86
2.68
2.48
2.51
3.68
3.82
9.09
2.08
2.38
3.8
3.37
4.09
3.12
2.08
2.14
2.23
3.3
2.33
2.3
3.05
2.45
2.43
2.07
3.19
1.36
1.17
0.98
0.63
1.41
0.87
0.93
0.88
1.02
0.91
0.86
1.83
0.5
1.3
1.41
0.9
1.08
1.22
1.33
1.5
3.64
1.2
1.41
1.09
0.98
0.6
0.56
1.24
1.03
1.32
1.89
0.68
2.8
0.97
1.27
1.55
1.3
0.86
1.32
1.84
1.15
0.68
1.5
1.97
2.03
1.4
1.3
1.08
1.03
1.4
1.29
1.28
0.5
28
Name
N
C
A, AG
A, MW
A, A
A, J
A, BP
A, NP
B, A
B, DR
B, M
B, BA
B, MD
B, BB
B, SK
B, D
B, M
B, J
B, R
C, I
C, AL
C, NJ
D, AJ
D, C
D, M
D, RD
D, R
D, MVG
E, DA
E, H
F, A
F, F
F, GA
F, DP
G, VM
G, ML
G, M
G, GH
H, H
H, F
H, M
H, ER
I, A
I, MF
J, P
J, E
J, AN
K, E
K, HG
K, J
K, EA
K, I
hxi
σ
64
32
50
17
18
39
51
55
71
55
17
35
35
13
17
19
24
27
81
64
27
37
32
15
31
11
24
24
56
33
36
74
23
23
14
53
50
39
12
15
16
30
22
22
29
21
50
28
27
19
33
19
1009
268
1169
250
2472
1370
2202
1245
10032
1345
162
646
729
894
578
252
644
1492
4249
1432
620
712
1452
1940
987
623
631
793
738
1042
592
17020
458
1029
1576
1720
2499
1013
54
413
289
1442
1075
1469
466
1934
655
1711
336
384
1334
2022
c(1) c(r∗ )
135
53
112
83
825
208
440
208
1153
260
25
216
198
422
225
43
428
600
833
343
200
124
431
1036
142
152
146
151
89
216
87
5611
175
244
965
318
364
208
15
91
64
586
244
332
79
377
73
495
39
88
326
596
r∗
h1 h2 h80 βxy
28
9.4 16 12
9
7.07 10 7
61 6.06 18 13
26
3
8 7
157 4.07 13 12
89 4.73 17 14
68 7.86 21 15
39 8.28 18 13
323 8.27 40 33
40 8.54 16 13
11 5.33 8 5
39 4.83 14 10
35
5.9 12 10
78 3.48 7 7
59 3.26 11 8
33 2.25 10 6
17 5.63 9 6
43 6.32 12 9
111 9.12 33 23
29 10.54 17 13
26 6.07 11 8
39 5.38 14 12
68 5.84 16 12
36 5.66 10 9
74 3.84 16 14
160 1.23 9 7
45 4.41 15 9
72 3.55 14 11
29 7.08 16 11
49 6.09 16 12
37 4.74 15 10
473 8.3 47 37
26 5.09 12 8
130 2.64 14 12
13
6.1 7 6
70 6.72 21 17
89 7.43 21 17
44 6.49 16 11
3
5.24 5 3
71 1.91 10 8
37 2.77 10 9
98
4.3 15 13
121 3.09 13 11
165 2.58 17 14
38 2.93 15 10
149 4.22 15 12
29 6.17 15 9
79 5.84 16 12
29 2.86 12 8
37 3.59 10 8
71
5.1 14 11
70 2.04 7 5
1
0
1
1
3
1
3
1
6
2
0
1
1
2
1
0
1
2
2
1
1
1
2
3
1
1
1
1
1
1
1
6
1
1
2
2
3
1
0
1
0
1
1
2
0
3
0
2
0
1
1
1
0.66
0.18
0.95
0.18
1.63
1.24
1.1
0.57
1.69
0.95
0.43
0.43
0.91
1.72
1.35
0.29
1.1
2.92
0.71
0.95
1.35
0.66
1.1
1.1
0.49
3.11
1.1
0.77
0.35
1.1
0.43
1.37
0.11
1.1
2.32
0.59
0.88
0.77
0
0.66
0.05
0.95
1.92
0.84
0.43
1.63
0.53
1.63
0.11
0.66
0.99
0.67
β
γ
0.81
1.12
0.7
0.54
1.17
0.69
0.97
0.71
0.73
0.84
0.93
0.76
0.92
1.12
0.67
0.27
1.4
1.5
0.62
1.13
1.03
0.67
0.94
1.9
0.48
0.05
0.61
0.57
0.67
0.84
0.59
0.79
0.75
0.5
2.5
0.73
0.98
0.77
1.2
0.3
0.56
0.8
0.54
0.39
0.32
0.84
0.59
1.02
0.31
0.68
0.79
0.38
0.59
0.44
0.86
1.06
1.28
1.12
0.76
0.59
0.98
0.64
0.37
0.87
0.71
1.14
1.11
0.96
0.56
0.76
0.8
0.56
0.6
0.86
0.87
0.32
1.06
2.1
0.71
1.11
0.73
0.74
0.98
1.14
0.55
1.62
0.07
0.92
0.96
0.69
0.17
1.5
1.33
0.98
1.05
1.37
0.98
0.9
0.81
0.71
1
0.95
0.89
0.36
c(r∗ )/h1 N/h1 hci/h1
1.78
0.98
3.42
3.27
12.14
5.25
3.25
2.2
8.08
2.51
1.45
2.82
2.92
11.23
5.45
3.31
1.98
3.62
3.38
1.76
2.38
2.79
4.28
3.68
4.64
17.79
3.04
5.21
1.83
3.1
2.47
10.07
2.17
9.3
1.98
3.37
4.28
2.79
0.77
7.15
3.76
6.59
9.31
9.75
2.58
9.97
1.95
4.94
2.47
3.76
4.91
4.24
4
3.2
2.78
2.13
1.38
2.29
2.43
3.06
1.78
3.44
2.13
2.5
2.92
1.86
1.55
1.9
2.67
2.25
2.45
3.76
2.45
2.64
2
1.5
1.94
1.22
1.6
1.71
3.5
2.06
2.4
1.57
1.92
1.64
2
2.52
2.38
2.44
2.4
1.5
1.6
2
1.69
1.29
1.93
1.4
3.33
1.75
2.25
1.9
2.26
0.86
0.99
0.84
1.3
1.84
10.56
2.07
2.06
1.26
3.53
1.53
1.19
1.32
1.74
9.82
3.09
1.33
2.98
4.6
1.59
1.32
2.09
1.37
2.84
12.93
1.99
6.29
1.75
2.36
0.82
1.97
1.1
4.89
1.66
3.2
16.08
1.55
2.38
1.62
0.9
2.75
1.81
3.2
3.76
3.93
1.07
6.14
0.87
3.82
1.04
2.02
3.35
4.57
a
3.94
2.68
3.61
3.91
14.63
4.74
4.99
3.84
6.27
5.25
2.53
3.3
5.06
18.24
4.78
2.52
7.95
10.36
3.9
4.96
5.12
3.63
5.67
19.4
3.86
7.69
2.8
4.05
2.88
4.07
2.63
7.7
3.18
5.25
32.16
3.9
5.67
3.96
2.16
4.13
2.89
6.41
6.36
5.08
2.07
8.6
2.91
6.68
2.33
3.84
6.15
6.21
Gap(5) Gap(10) m1/2 m2
50
69
34
40
65
45
53
47
38
45
55
70
42
15
57
62
53
55
49
55
47
50
49
25
49
21
75
57
52
56
67
52
59
67
43
52
48
48
35
48
69
65
63
84
79
67
51
50
53
52
51
14
TABLE S5: Career citation statistics for 100 dataset [C] scientists: 1-50.
174
207
161
124
127
176
198
170
189
155
153
227
158
69
171
184
183
192
178
192
171
166
186
82
175
56
234
195
187
192
208
249
189
144
103
183
185
180
66
108
150
186
188
214
257
210
193
197
199
170
167
45
1.31
1
1.33
1.38
1.08
1.35
1.29
1.44
1.23
1.5
1.25
1.14
1.33
1.14
1.18
1.2
1.33
1.25
1.36
1.29
1.27
1.36
1.38
1.2
1.25
1
1.2
1.14
1.31
1.19
1.2
1.15
1.25
1.14
1.14
1.24
1.19
1.38
1.2
1.1
1
1.13
1.15
1.06
1.07
1.13
1.4
1.25
1.33
1.2
1.22
0.18
0.75
0.7
0.72
0.88
0.92
0.82
0.71
0.72
0.83
0.81
0.63
0.71
0.83
1
0.73
0.6
0.67
0.75
0.7
0.76
0.73
0.86
0.75
0.9
0.88
0.78
0.6
0.79
0.69
0.75
0.67
0.79
0.67
0.86
0.86
0.81
0.81
0.69
0.6
0.8
0.9
0.87
0.85
0.82
0.67
0.8
0.6
0.75
0.67
0.8
0.77
0.09
P
h1 /L
746
230
982
221
2447
1246
2003
962
9477
1073
133
589
639
885
552
225
608
1430
3547
1222
547
589
1343
1918
877
622
583
743
552
944
493
16575
405
999
1556
1497
2302
864
47
398
279
1402
1046
1439
422
1904
464
1621
265
354
1225
1948
0.84
0.91
1.38
1
0.76
1.55
2.1
1.38
3.08
1.6
0.73
0.82
1.09
0.7
1.22
1.43
0.64
0.86
2.36
1.13
1.22
2
1.78
0.63
1.45
1
1.25
1.75
1.45
1.14
1.5
3.62
1
1.08
0.7
1.24
1.75
1.33
0.29
1.25
1.25
1.07
1.08
1.55
1.88
1.25
1.25
1.33
1.5
0.91
1.29
0.56
29
Name
N
C
K, SM
K, AA
K, IN
L, A
L, LJ
L, RL
L, BJ
L, J
L, Y
M, O
M, V
M, BA
M, L
M, P
M, D
M, B
M, E
M, OI
M, AE
N, D
N, A
N, V
N, Z
O, AL
O, SB
P, N
P, NB
P, AT
P, MG
P, A
P, F
P, S
S, T
S, TR
S, D
S, MD
S, L
S, OG
S, GT
S, M
S, AM
T, N
T, AP
T, H
V, O
V, MG
W, RH
W, M
Y, A
Z, MW
hxi
σ
15
11
42
21
18
44
19
36
14
22
80
19
18
15
24
16
38
36
44
45
25
7
54
28
33
49
26
24
14
48
21
62
121
60
19
17
22
22
19
47
51
14
29
22
22
41
57
18
21
20
33
19
395
1028
1815
289
815
879
591
881
648
151
2038
378
966
723
1029
132
374
716
1695
1427
759
1536
1175
516
505
5544
225
2094
206
1000
998
1576
3788
933
1906
548
476
444
284
1265
1431
2073
85
347
653
804
2377
469
1176
3151
1334
2022
c(1) c(r∗ )
71
383
567
69
290
91
156
93
177
32
206
202
345
180
283
60
77
90
303
231
116
1174
258
93
92
550
45
659
57
106
317
231
240
133
686
194
72
83
43
126
288
1256
26
88
123
129
364
90
536
592
326
596
r∗
h1 h2 h80 βxy
56 2.17 10 9
160 2.61 9 8
71 6.52 20 15
25 3.86 9 7
83 3.33 9 8
43 5.49 16 11
23 5.43 8 7
41 5.96 17 13
47 4.55 11 8
5
6.76 5 4
62 8.71 24 17
17 5.26 9 6
140 2.22 13 10
100 2.77 10 8
53 5.15 12 11
5
4.7 5 3
14 7.15 10 8
53 3.71 15 11
56 7.67 18 12
77 5.46 19 15
69 2.62 16 14
147 2.8 6 6
44 6.98 19 13
65 1.96 14 10
27 5.59 12 8
241 6.33 30 22
14 4.92 8 6
153 4.11 13 11
24 2.33 9 6
48 5.83 18 13
63 4.83 13 11
51 8.22 23 16
81 11.89 33 25
55 4.81 16 12
171 3.39 16 15
26 5.66 12 9
43 3.62 12 10
48 2.66 13 8
27
3.5 9 7
68 4.51 21 15
70
5.4 21 15
45 4.72 8 7
3
6.58 4 2
27 4.07 11 8
55 3.93 12 9
43 5.75 17 12
107 6.27 25 19
38 4.07 10 9
126 3.09 9 7
311 3.52 16 15
71
5.1 14 11
70 2.04 7 5
0
2
2
0
2
1
1
1
1
0
2
1
1
2
2
0
0
1
2
1
1
2
1
1
1
5
0
3
0
1
1
1
2
1
2
1
0
1
0
1
1
2
0
1
1
1
2
1
2
4
1
1
0.53
2.32
0.74
0.18
1.35
0.77
1.72
0.57
1.35
0.66
0.59
0.29
0.91
2.32
0.77
1.1
0.66
0.77
2.32
0.74
0.49
2.32
0.57
0.43
0.66
1.92
0.29
2.8
0.29
0.57
0.77
0.38
0.81
0.66
0.74
1.1
0.43
0.66
0.18
0.43
0.43
1.72
2.32
0.66
1.1
0.66
1
1.1
1.72
2.02
0.99
0.67
β
γ
0.29
0.67
0.78
0.68
0.93
0.52
1.4
0.5
0.9
1.1
0.76
1.09
0.39
0.74
1.22
1.53
0.78
0.44
1.02
0.7
0.26
0.97
0.73
0.3
0.86
0.83
0.72
1.22
0.26
0.57
0.78
0.67
0.74
0.55
0.53
1.04
0.61
0.38
0.52
0.41
0.61
2.12
0.87
0.63
0.73
0.78
0.68
0.82
1.05
0.63
0.79
0.38
1.07
1.49
0.72
0.85
1.3
0.84
0.72
0.55
0.53
0.23
0.88
0.41
1.51
1.58
1.01
0.51
0.51
1.06
0.73
1.01
0.93
2.16
0.85
1.28
0.63
1.15
0.62
1.47
0.64
0.75
0.67
0.59
0.94
0.95
0.9
0.19
1.04
1
0.75
0.92
1.07
0.8
0.16
0.82
1.02
0.89
1.12
0.9
1.53
0.95
0.89
0.36
c(r∗ )/h1 N/h1 hci/h1
5.68
17.81
3.6
2.86
9.28
2.71
2.88
2.44
4.31
1.11
2.58
1.93
10.81
10.06
4.49
1.16
1.48
3.55
3.15
4.09
4.32
24.56
2.35
4.65
2.3
8.05
1.84
11.8
2.75
2.67
4.91
2.26
2.48
3.44
10.7
2.24
3.6
3.72
3.05
3.25
3.35
5.71
0.91
2.52
4.65
2.58
4.31
3.85
14.11
19.48
4.91
4.24
1.5
1.22
2.1
2.33
2
2.75
2.38
2.12
1.27
4.4
3.33
2.11
1.38
1.5
2
3.2
3.8
2.4
2.44
2.37
1.56
1.17
2.84
2
2.75
1.63
3.25
1.85
1.56
2.67
1.62
2.7
3.67
3.75
1.19
1.42
1.83
1.69
2.11
2.24
2.43
1.75
7.25
2
1.83
2.41
2.28
1.8
2.33
1.25
2.26
0.86
2.63
10.38
2.16
1.53
5.03
1.25
3.89
1.44
4.21
1.37
1.06
2.21
4.13
4.82
3.57
1.65
0.98
1.33
2.14
1.67
1.9
36.57
1.15
1.32
1.28
3.77
1.08
6.71
1.63
1.16
3.66
1.11
0.95
0.97
6.27
2.69
1.8
1.55
1.66
1.28
1.34
18.51
0.73
1.43
2.47
1.15
1.67
2.61
6.22
9.85
3.35
4.57
a
3.95
12.69
4.54
3.57
10.06
3.43
9.23
3.05
5.36
6.04
3.54
4.67
5.72
7.23
7.15
5.28
3.74
3.18
5.23
3.95
2.96
42.67
3.25
2.63
3.51
6.16
3.52
12.39
2.54
3.09
5.91
2.98
3.48
3.64
7.45
3.81
3.31
2.63
3.51
2.87
3.24
32.39
5.31
2.87
4.53
2.78
3.8
4.69
14.52
12.31
6.15
6.21
Gap(5) Gap(10) m1/2 m2
58
49
53
48
49
40
33
45
29
36
41
53
49
68
56
46
39
34
57
61
60
10
49
79
45
82
40
75
38
41
38
51
35
36
52
49
65
69
44
55
49
54
37
54
56
70
49
54
58
16
51
14
166
152
190
152
146
166
145
167
87
128
183
118
79
124
187
106
157
165
200
204
186
15
194
198
186
270
155
198
172
170
141
207
150
156
159
160
201
218
158
197
213
88
131
195
176
237
189
136
157
31
167
45
TABLE S6: Career citation statistics for 100 dataset [C] scientists: 51-100.
1.1
1.11
1.3
1.33
1.22
1.56
1.5
1.41
1.18
1.6
1.33
1.33
1.08
1.1
1.17
1.2
1.4
1.33
1.33
1.16
1.13
1
1.32
1.07
1.25
1.1
1.38
1.08
1.22
1.33
1.38
1.22
1.36
1.38
1.13
1.17
1.17
1.15
1.33
1.29
1.24
1.13
1.5
1.18
1.17
1.24
1.28
1
1
0
1.22
0.18
0.9
0.89
0.75
0.78
0.89
0.69
0.88
0.76
0.73
0.8
0.71
0.67
0.77
0.8
0.92
0.6
0.8
0.73
0.67
0.79
0.88
1
0.68
0.71
0.67
0.73
0.75
0.85
0.67
0.72
0.85
0.7
0.76
0.75
0.94
0.75
0.83
0.62
0.78
0.71
0.71
0.88
0.5
0.73
0.75
0.71
0.76
0.9
0.78
0.94
0.77
0.09
P
h1 /L
376
1020
1638
251
794
658
547
688
627
112
1670
348
951
712
990
122
266
597
1524
1303
707
1535
977
486
440
5374
169
2072
188
808
928
1306
2950
769
1878
520
436
407
244
1047
1241
2058
47
303
606
707
2110
440
1168
3125
1225
1948
1.43
0.82
1.25
0.82
0.82
0.89
1
0.81
1
0.33
2.18
0.82
1.63
1
0.92
0.42
1.43
1.07
1.5
1.73
1.45
1
1.19
1.27
0.86
2.73
0.67
0.87
1
1.38
1
2.56
2.36
1.14
2.29
1.2
2
1.08
0.9
1.75
1.5
0.89
0.57
1.22
0.92
1.21
1.92
1.25
1.29
2
1.29
0.56