Statistical regularities in the rank-citation profile of scientists

Abstract

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 production and impact using the rank-citation profile c_i (r) of 200 distinguished professors and 100 assistant professors. For the entire range of paper rank r, we fit each c_i (r) to a common distribution function. Since two scientists with equivalent Hirsch h-index can have significantly different c_i (r) profiles, our results demonstrate the utility of the β_i scaling parameter in conjunction with h_i for quantifying individual publication impact. We show that the total number of citations C_i tallied from a scientist's N_i papers scales as . Such statistical regularities in the input-output patterns of scientists can be used as benchmarks for theoretical models of career progress.

Introduction

A scientist's career path is subject to a myriad of decisions and unforeseen events, such as Nobel Prize worthy discoveries¹, 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 (individual, mentor-apprentice, institutional, coauthorship, field)^{2,3,4,5,6,7,8,9} to account for in the statistical analysis of scientific panel data.

The rank-citation profile, c_i (r), represents the number of citations of individual i to his/her paper r, ranked in decreasing order c_i (1) ≥ c_i (2) ≥ …c_i (N) and provides a quantitative synopsis of a given scientist's publication career. Here, we analyze the rank-ordered citation distribution c_i (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 c_i (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^10,11,12,13.

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 〈h〉 = 61 ± 21. These scientists were selected using the citation shares metric⁹ to quantify cumulative career impact in the journal Physical Review Letters (PRL).

• [B] 100 additional “control” scientists with average h-index 〈h〉 = 44 ± 15.

• [C] 100 current Assistant professors with average h-index 〈h〉 = 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 in detail the selection procedure for datasets A, B and C and in tables S1-S6 we provide summary statistics for each career.

There are many conceivable ways to quantify the impact of a scientist's N_i publications. The h-index¹⁴ is a widely acknowledged single-number measure that serves as a proxy for production and impact simultaneously. The h-index h_i of scientist i is defined by a single point on the rank-citation profile c_i (r) satisfying the condition

To address the shortcomings of the h-index, numerous remedies have been proposed in the bibliometric sciences¹⁵. For example, Egghe proposed the g-index, where the most cited g papers cumulate g² citations overall¹⁶ and Zhang proposed the e-index which complements the h and g indices quantitatively¹⁷.

To justify the importance of analyzing the entire profile c_i (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 β_i is necessary in order to distinguish these two example careers. Specifically, the β_i parameter quantifies the scaling slope in c_i (r) for the high-rank papers corresponding to small r values. In this simple illustration, β ₁ ≈ 1 while β ₂ ≈ 0.

In Fig. 1 we plot c_i (r) for 5 extremely high-impact scientists. The individuals EW, ACG, MLC and PWA are physicists with the largest h_i 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. However, citation and h-index metrics should not be compared across discipline since baseline publication and citation rates can vary significantly between research fields Refs[8, 9]. To demonstrate how the singe point c_i (h_i ) is an arbitrary point along the c_i (r) curve, we also plot the lines H_p (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 H_p (r) with c_i (r) corresponds to the “generalized h-index” h_p ,

proposed in¹⁸ and further analyzed in¹⁹, with the relation h_p ≤ h_q for p > q. Since the value p ≡ 1 is chosen somewhat arbitrarily, we take an alternative approach which is to quantify the entire c_i (r) profile at once (which is also equivalent to knowing the entire h_p spectrum). Surprisingly, because we find regularity in the functional form c_i (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 c_i (r) profile for the entire set of papers ranging from rank r = 1…N_i . Using a much smaller parameter space than the h_p spectrum, we can begin to analyze the statistical regularities in the career accomplishments of scientists.

Figure 1

The citation distribution of individual scientists is heavy-tailed.

We show 5 empirical rank-citation c_i (r) profiles , each belonging to an extremely high-impact scientists (E. Witten, A. C. Gossard, M. L. Cohen, P. W. Anderson and B. Vogelstein) whose initials and h-index as of Jan. 2010 are listed in the figure legend. The hierarchical scaling pattern in c_i (r) for small r values indicate that the pillar contributions of top scientists are “off-the-charts” since they have no characteristic scale. Put in the framework of the citation distribution, consider the probability distribution P_i (c) of the citation impact c calculated for an individual's N_i papers. If P_i (c) is heavy tailed with asymptotic power-law scaling , then ζ_i = 1 + 1/β_i . Ref. [9] calculates ζ ≈ 3, corresponding to β = 1/2, using the entire set of citations for papers from six individual journals. Hence, the citation impact of stellar scientists can be significantly more skewed than the aggregate population. This statistical regularity demonstrates the utility of the β_i 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 c_i (1) pillar paper. The c_i (r) distributions show significant variability in both the high-rank (β) and low-rank (γ) regimes. Moreover, for c_i (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 c_i (r) over the entire rank range in each case. The intersection of c_i (r) with the line H_p (r) corresponds to the generalized h-index h_p , which together uniquely quantify the c_i (r) profile. Five H_p (r) lines are provided for reference, with p = {1, 2, 5, 20, 80}.

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 scientific careers are like “startup” companies that need appropriate venture funding to support the career trajectory through lows as well as highs¹³.

Results

A Quantitative Model for c_i (r)

For each scientist i, we find that c_i (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)

has been proposed as a model for rank profiles in the social and natural sciences that exhibit such truncated scaling behavior^20,21. The parameters A_i , β_i , γ_i and N_i are each defined for a given c_i (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 β_i and γ_i using Mathematica software to perform a multiple linear regression of ln c_i (r) = ln A_i − β_i ln r + γ_i ln(N_i + 1 − r) in the base functions ln r and ln(N_i + 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/N_i ≈ 0.84 ± 0.01 for careers in datasets A and B). Figs. 1 and 2 demonstrate the utility of the DGBD to represent c_i (r), for both large and small r. The regression correlation coefficient R_i > 0.97 for all ln c_i (r) profiles analyzed.

Figure 2

Data collapse of each c_i (r) along a universal curve.

A comparison of 100 rank-citation profiles c_i (r) demonstrates the statistical regularity in career publication output. Each scientist produces a cascade of papers of varying impact between the c_i (1) pillar paper down to the least-known paper c_i (N_i ). (a) Zipf rank-citation profiles c_i (r) for 100 scientists listed in dataset [A]. For reference, we plot the average of these 100 curves and find with β = 0.92 ± 0.01. The solid green line is a least-squares fit to 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 c_i (r′) ≡ c_i (r)/A(r₁ + 1 − r)^γ, where we use the best-fit γ_i and A_i parameter values for each individual c_i (r) profile. Using the rescaled rank value , we show excellent data collapse onto the expected curve c(r′) = 1/r′. (see Figs. S1 and S2 for analogous plots for dataset [B] and [C] scientists). Green data points correspond to the average c(r′) value with 1σ error bars calculated using all 100 c_i (r′) curves separated into logarithmically spaced bins.

The DGBD proposed in²⁰ is an improvement over the Zipf law (also called the generalized power-law or Lotka-law²²) model and the stretched exponential model¹⁴ since it reproduces the varying curvature in c_i (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 γ_i which controls the curvature in c_i (r) for large r values. The DGBD has been successfully used to model numerous rank-ordering profiles analyzed in^20,21 which arise in the natural and socio-economic sciences. The relative values of the β_i and γ_i exponents are thought to capture two distinct mechanisms that contribute to the evolution of c_i (r)^20,21. Due to the data limitations in this study, we are not able to study the dynamics in c_i (r) through time. Each c_i (r) is a “snapshot” in time and so we can only conjecture on the evolution of c_i (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 “heavy-weight” 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 c_i (r) suggests that the reinforcement dynamics can be quantified by the scale-free parameters β and γ.

The β_i value determines the relative change in the c_i (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 c_i (r) for each scientist from dataset [A] as well as the average of the 100 individual curves (see Figs. S1 and S2 for analogous plots for datasets [B] and [C]). We find robust power-law scaling

for 10⁰ ≤ r ≤ 10². The scaling value calculated for other rank-size (Zipf) distributions in the social and economic sciences is typically around unity, β ≈ 1, for example in studies of word frequency²³ and city size^20,21,24. Here we calculate β_i for each individual author and observe a distribution which is centered around characteristic values 〈β〉 = 0.83 ± 0.23 [A], 〈β〉 = 0.70 ± 0.16 [B], 〈β〉 = 0.79 ± 0.38 [C].

We calculate each β_i value using a multilinear least-squares regression of ln c_i (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 c_i (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 c_i (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 DGBD model, we collapse each c_i (r) along a universal scaling function c(r′) = 1/r′, by using the rescaled rank values defined for each curve. In Figs. 2(b), S1(b) and S2(b), we plot the quantity c_i (r′) ≡ c_i (r)/A(r₁ + 1 − r) ^γ , using the best-fit γ_i and A_i parameter values for each individual c_i (r) profile. While the curves in Fig. 2(a) are jumbled and distributed over a large range of c(r) values, the rescaled c_i (r) curves in Fig. 2(b) all lie approximately along the predicted curve c(r′) = 1/r′.

Using c_i (r) to quantify career production and impact

A main advantage of the h-index is the simplicity in which it is calculated, e.g. ISI Web of Knowledge²⁵ readily provides this quantity online for distinct authors. Another strength of the h-index is its stable growth with respect to changes in c_i (r) due to time and information-dependent factors²⁶. 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 Hirsch¹⁴). Annual growth rates h/L ≈ 3 correspond to exceptional scientists (for the histogram of P(h/L) see Fig. S3 and for h/L values see the SI text (Tables S1–S6)). As a result, h/L is a good predictor for future achievement along with h²⁷.

It is truly remarkable how a single number, h_i , 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^28,29,30, nor can h_i incorporate the full information contained in the entire c_i (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 ≤ h_i , its citation record is discounted, until other less-cited papers with r > h_i eventually overcome the rank “barrier” r = h_i . Moreover, as noted in¹⁴, the papers for which r > h_i do not contribute any additional credit.

Instead of choosing an arbitrary h_p as an productivity-impact indicator, we use the analytic properties of the DGBD to calculate a crossover value . In the methods section, we derive an exact expression for which highlights the distinguished papers of a given author. To calculate , we use the logarithmic derivative χ(r) ≡ d ln c(r)/dr to quantify the relative change in c_i (r) with increasing r. We defined papers as “distinguished” if they satisfy the inequality , where 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 corresponds to a “knee” in c_i (r) when plotted on log-linear axes. The dependence of and on the three DGBD parameters β_i , γ_i and N_i are provided in the methods section.

The advantage of is that this characteristic rank value is a comprehensive representation of the stellar papers in the high-rank scaling regime since it depends on the DGBD parameter values β_i , γ_i and N_i and thus probes the entire citation profile. Fig. 3 shows a scatter plot of the “c-star” and h_i 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 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 incorporate the entire c_i (r) profile. We plot the histogram of and values in Figs. S4 and S5, respectively.

Figure 3

Limitations to the use of the h -index alone.

The h-index can be insufficient in comprehensively representing c_i (r). (a) The h-index does not contain any information about c_i (r) for r < h_i 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 , in which case the h-index cannot account for the stellar impact of the papers. (b) For a given h_i value, prolific careers are characterized by a large β_i value, as it is harder to maintain large β_i values for large h_i . As a result, the β_i vs h_i parameter space can be used to identify anomalous careers and to better compare two scientists with similar h_i indices. We find that a third career metric C_i , the total number of citations to the papers of author i, can be calculated with high accuracy by the scaling relation , which we illustrate in Fig. 4(b).

To further contrast the values of and the h-index, we propose the “peak indicator” ratio , which corrects specifically for the h-index penalty on the stellar papers in the peak region of c_i (r). Thus, all papers in the peak region of c_i (r) satisfy the condition c_i (r) ≥ h_i Λ _i . 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 the histogram of Λ _i values, with typical values for dataset [A] scientists 〈Λ〉 ≈ 3.4 ± 3.9 and for dataset [B] scientists 〈Λ〉 ≈ 2.2 ± 1.1. This indicator can only be used to compare scientists with similar h values, since a small h_i can result in a large Λ _i .

An alternative “single number” indicator is C_i , an author's total number of citations

which incorporates the entire c_i (r) profile. However, it has been shown that correlates well with h_i³¹, a result which we will demonstrate in Eq. [6] to follow directly from a c_i (r) with β_i ≈ 1.

We test the aggregate properties of c_i (r) by calculating the aggregate number of citations C_β,h for a given profile,

where H_N′,β is the generalized harmonic number and is of order O(1) for β ≈ 1. We neglect the γ_i scaling regime since the low-rank papers do not significantly contribute to an author's C_i tally. We approximate the coefficient A in Eq. [6] using the definition c(h) ≡ h, which implies that A/h^β ≈ h. We use the value N′ ≡ 3 h, so that C_β,h can be approximated by only the two parameters h_i and β_i for any given author. We justify this choice of N′ by examining the rescaled c_i (r/h), which we consider to be negligible beyond rank r = 3 h_i for most scientists. In Fig. 4(a), we plot for each scientist the predicted C_β,h value versus the empirical C_i value and we find excellent agreement with our theoretical prediction given by Eq. [6]. In Fig. 4(b), we plot for each scientist the total number of citations using the best-fit DGBD model c_m (r) ≡ c_i (r; β_i , γ_i , A_i , r₁) to approximate c_i (r). The excellent agreement demonstrates that the fluctuations in the residual difference c_m (r) − c_i (r) cancel out on the aggregate level. Furthermore, a comparison of the quality of agreement between the theoretical C_i values and the empirical C_i values in Fig. 4(a) and (b) shows the importance of the additional γ_i scaling regime in the DGBD model.

Figure 4

Aggregate publication impact C.

The total number of citations C_i is also comprehensive productivity-impact measure.For most best-fit DGBD model curves, the C_i value is preserved with high precision. This shows that the difference between a given c_i (r) and the corresponding best-fit DGBD model function are negligible on the macroscopic scale. (a) The exact aggregate number of citations C_i , calculated from c_i (r) using Eq. [5], can be analytically approximated by using Eq. [6] which depends only on the scientist's β_i and h_i values. (b) We justify the use of the DGBD model defined in Eq. [3] for the approximation of c_i (r) by comparing the aggregate citations C_i with the expected aggregate citations calculated from the best-fit DGBD model c_m (r). Including the extra scaling-parameter, as in the DGBD model, improves the agreement between the theoretical and empirical C_i values in (a) and (b). We plot the line y = x (dashed-green line) for visual reference.

Discussion

We use the DGBD model to provide an analytic description of c_i (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 β_i , which for many scientists analyzed, can be approximated using only two values of the generalized h-index h_p (see SI text). In particular, we show that for a given h-value, a larger β_i value corresponds to a more prolific publication career, since .

Many studies analyze only the high rank values of generic Zipf ranking profiles c(r), e.g. computing the scaling regime for r < r_c below some some rank cutoff r_c . 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 , r_× and , 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 quantifying the Zipf rank profiles in various phenomena²¹.

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 Δt years in the future? In consideration of the bulk of c_i (r) and following from the regularity of c_i (r) for r ≈ h, we propose a model-free gap-index G(Δh) as both an estimate and a target for future achievement which can be used in the review of career advancement. The gap index G(Δh), defined as a proxy for the total number of citations a scientist needs to reach a target value h+Δh, can detect the potential for fast h-index growth by quantifying c_i (r) around h. This estimator differs from other estimators for the time-dependent h-index^33,34,35 in that G(Δh) is model independent.

Even though the productivity of scientists can vary substantially^{9,36,37,38,39} and despite the complexity of success in academia, we find remarkable statistical regularity in the functional form of c_i (r) for the scientists analyzed here from the physics community. Recent work in^8,9,40 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^2,3,9 versus (ii) the intrinsic role of the “sacred spark” in combination with intellectual genius³⁷ manifest in the parameters of the DGBD model.

With little calculation, the β_i metric developed here, used in conjunction with the h_i , can better answer the question, “How popular are your papers?”⁴¹. Since the cumulative impact and productivity of individual scientists are also found to obey statistical laws^9,11, 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^9,11,42 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^11,43,44. Hence, whenever data are available^45,46, finding statistical regularities emerging from human endeavors is a first step towards better understanding the dynamics of human productivity.

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^9,32,40 and career progress¹¹. 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 in⁹. 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 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.

Statistical significance tests for the c(r) DGBD model

We test the statistical significance of the DGBD model fit using the χ ² test between the 3-parameter best-fit DGBD c_m (r) and the empirical c_i (r). We calculate the p-value for the χ ² distribution with r₁ − 3 degrees of freedom and find, for each data set, the number of c_i (r) with p-value [A], 19 [B], 22 [C] for p_c = 0.05 and 8 [A], 22 [B], 37 [C] for p_c = 0.01.

The significant number of c_i (r) which do not pass the χ ² test for P_c = 0.05, results from the fact that the DGBD is a scaling function over several orders of magnitude in both r and c_i (r) values and so the residual differences [c_i (r) − c_m (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 c_i (r) do pass the χ ² test at such a high significance level, provides evidence for the quality-of-fit of the DGBD model. For comparison, none of the c_i (r) pass the χ ² test using the power-law model at the P_c = 0.05 significance level. In the next section, we will also compare the macroscopic agreement in the total number of citations for each scientist and the total number of citations predicted by the DGBD model for each scientist and find excellent agreement.

Derivation of the characteristic DGBD r values

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 c_i (r) to characterize the crossover between the high-rank and the low-rank regimes of c_i (r).

We begin by considering c_i (r) under the centered rank transformation z = r − z₀, where z₀ = (N + 1)/2, then

in the domain z ∈ [− (z₀ − 1), (z₀ − 1)]. The logarithmic derivative of c(z) expresses the relative change in c(z),

where x = z/z₀, and . The extreme values of for are given by

and the average value is calculated by,

The function χ(z) takes on the value of twice at the values corresponding to the solutions to the quadratic equation,

which has the solution

for . Converting back to rank, then

and so the value is the special rank value which distinguishes the set of excellent papers of each given author. The c-star value c_i (r*) is thus a characteristic value arising from the special analytic properties of c_i (r). This method for determining the crossover value r* can be applied to any general rank order profile which can be modeled by the DGBD.

Furthermore, the crossover z_x between the β scaling regime and the γ scaling regime is calculated from the inflection points of ln c(z),

which has 2 solutions , where . only is a physical solution. Transforming back to rank values, we find . We illustrate these special z values in Fig. 5.

Figure 5

Characteristic properties of the DGBD.

We graphically illustrate the derivation of the characteristic c_i (r) crossover values that locate the two tail regimes of c_i (r), in particular, the distinguished “peak” paper regime corresponding to paper ranks r ≤ r* (shaded region). The crossover between two scaling regimes suggests a complex reinforcement relation between the impact of a scientist's most famous papers and the impact of his/her other papers. (a) The c_i (r) plotted on log-log axes with N = 278, β = 0.83 and γ = 0.67, corresponding to the average values of the Dataset [A] scientists. The hatched magenta curve is the H₁(z) line on the log-linear scale with corresponding h-index value h = 104. The r* value for c_i (r) is not visibly obvious. (b) We plot on log-linear axes the centered citation profile c_i (z) (solid black curve) given by the symmetric rank transformation z = r − z₀ in Eq. [7]. This representation better highlights the peak paper regime, but fails to highlight the power-law β scaling. (c) We plot the corresponding logarithmic derivative χ(z) of c(z) (solid black curve), which represents the relative change in c(z). The dashed red line corresponds to , where is the average value of χ(z) given by Eq. [12]. The values of , indicated by the solid vertical green lines, are defined as the intersection of with χ(z) given by Eq. [13]. The regime corresponds to the best papers of a given author. The hatched blue line corresponds to which marks the crossover between the β and γ scaling regimes.