## Abstract

Universality of scaled citation distributions was claimed a decade ago but its theoretical justification has been lacking so far. Here, we study citation distributions for three disciplines—Physics, Economics, and Mathematics—and assess them using our explanatory model of citation dynamics. The model posits that the citation count of a paper is determined by its fitness: the attribute, which, for most papers, is set at the moment of publication. In addition, the papers’ citation count is related to the process by which the knowledge about this paper propagates in the scientific community. Our measurements indicate that the fitness distribution for different disciplines is nearly identical and can be approximated by the log-normal distribution, while the viral propagation process is discipline specific. The model explains which sets of citation distributions can be scaled and which cannot. In particular, we show that the near-universal shape of the citation distributions for different disciplines and for different citation years traces its origin to the nearly universal fitness distribution, while deviations from this shape are associated with the discipline-specific citation dynamics of papers.

## PEER REVIEW

## 1. INTRODUCTION

Science is an evolving complex network of researchers, research projects, and publications. Citations of scientific publications are the most important links that glue this network together. Analysis of citations was initially focused on journal-based citation distributions. Although these vary from discipline to discipline and from journal to journal, Seglen (1992) noticed that, after proper scaling, citation distributions for different journals collapse onto one universal curve. Radicchi, Fortunato, and Castellano (2008) studied this issue by considering different disciplines/publication years and came forward with the claim of universality of citation distributions. This claim provided a stimulus to look for universality in other complex networks, and, indeed, several dynamic universalities were found there as well (Barzel & Barabasi, 2013; Candia, Jara-Figueroa et al., 2019; Gao, Barzel, & Barabasi, 2016). While significant progress in understanding these universalities has been achieved, the origin of the universality of citation distributions remained elusive. In the context of science of science (Fortunato, Bergstrom et al., 2018; Sugimoto & Larivière, 2018; Zeng, Shen et al., 2017), the striking observation of Radicchi et al. (2008) implies that different research topics develop along similar paths, thus rendering possible such generalizations as Kuhn’s paradigm shift theory (Kuhn, 1970). Understanding universality of citation distributions may provide a solid base for this theory, which has been considered as being more like a philosophical idea rather than a quantitative scientific hypothesis.

The work of Radicchi et al. (2008) paved the way for a flurry of empirical studies (Bornmann & Daniel, 2009; Chatterjee, Ghosh, & Chakrabarti, 2016; Evans, Hopkins, & Kaube, 2012; Waltman, van Eck, & van Raan, 2011) aspiring to find a fair indicator that allows quantitative comparison of the performance of papers belonging to different scientific disciplines. In the language of information science, the main achievement of Radicchi et al. (2008) was the demonstration that the variability of citation distributions for different fields is significantly reduced after going to scaled citation distributions, the scaling parameter being the mean of the distribution. The encompassing studies of Bornmann and Daniel (2009) and Waltman et al. (2011) showed deviations from such scaling, especially for research fields with a low mean number of citations. Thus, the limits to the claim of universality of citation distributions have been established. Subsequent studies (Chatterjee et al., 2016; Evans et al., 2012) extended the scaling conjecture of Radicchi et al. (2008) to sets of publications belonging to different journals, institutions, and even to Mendeley readerships (D’Angelo & Di Russo, 2019). In general, these works supported the purported scaling/universality but with some limitations; namely, research fields with a high number of uncited papers showed significant deviations from the universal distribution. To account for these deviations, two-parameter scaling (Radicchi & Castellano, 2011, 2012) was considered as well.

*K*is the number of citations of a paper,

*μ*characterizes the mean of the distribution, and

*σ*is the shape parameter.

In the past, citation distributions were empirically fitted using several functional shapes: stretched exponential, negative binomial, gamma, Weibull distribution, etc., the most popular being the log-normal and power-law distributions. Although the same citation distribution can be fitted by different functions, we followed the works of Radicchi et al. (2008), Stringer et al. (2008), and Thelwall (2016b) and chose the log-normal functional shape due to its simplicity. Under this choice, the universality claim of Radicchi et al. (2008) is more specific than the ubiquity of log-normal distributions in complex networks; it reduces to the statement that citation distributions for different natural science disciplines and journals can be modeled by a log-normal dependence with the *same shape parameter σ*. Indeed, extensive study of citation distributions for different journals by Thelwall (2016b) indicated that they can be described by the log-normal distribution with nearly the same *σ* = 1 to 1.2. This is in line with the earlier study of Stringer et al. (2008), who reported the log-normal citation distribution with *σ* ∼ 1 for hundreds of journals^{1}. D’Angelo and Di Russo (2019) reported a log-normal distribution with *σ* ∼ 1 for Mendeley readerships, Evans et al. (2012) and Chatterjee et al. (2016) reported a log-normal distribution with *σ* = 1.14 to 1.18 for many journal-based and institution-based publications, and Clough, Gollings et al. (2014) claimed a log-normal distribution with *σ* = 1.1 for U.S. patent citations. Thus, a log-normal fit of citation distributions for different journals, fields, and institutions yields more or less the same shape parameter *σ* = 1 to 1.2, indicating that although citation distributions may have very different mean numbers of citations, they have nearly the same shape.

After the shape of citation distributions has been empirically established for many scientific disciplines (with some caveats), the quest for the explanatory model of this shape starts. Although there have been many insightful models of citation dynamics of scientific publications, they do not address the nearly universal shape of citation distributions, in such a way that the latter remained an empirical observation lacking theoretical foundation. Our goal is to find an explanatory model accounting for this observation. We have recently developed a quantitative model of citation dynamics (Golosovsky, 2019; Golosovsky & Solomon, 2017) based on our measurements with physics papers. This model can be a good platform for understanding the shape of citation distributions. In this study, we measure and analyze the citation distributions and citation dynamics of physics, economics, and mathematics papers using the same measurement protocol for three disciplines, so that the measurements could be easily compared to the model. Our goal was to find microscopic parameters of citation dynamics for each discipline, to compare them, and to find which of them are discipline specific and which are not. To our surprise, we found that several of these parameters are the same for three disciplines. We trace the near-universal shape of citation distributions to the universality of these parameters.

## 2. UNIVERSALITY OF CITATION DISTRIBUTIONS AND DEVIATIONS THEREFROM

We illustrate here what is usually meant by the universality of citation distributions. Consider a set of papers published in the same year *t*_{0}, and denote by *K*_{j}(*t*) the number of citations garnered by a paper *j* from this set from the moment of its publication until the year *t*_{0} + *t*. Next, we consider the citation distribution *p*(*K*; *t*), namely, the number of papers having *K* citations after *t* years. Radicchi et al. (2008) introduced the scaled citation distribution *p*(*x*; *t*), where *x* = $KtMt$ and *M*(*t*) = $\u222b0\u221e$*Kp*(*K*; *t*)*dK* is the mean number of citations garnered by a paper during *t* years after publication. Although citation distributions *p*(*K* ; *t*) strongly depend on the number of years after publication, Radicchi et al. (2008) showed that the scaled distributions *p*(*x*; *t*) are very similar and hardly depend on *t*. The same study found that the scaled citation distributions for different disciplines are very similar as well. In other words, the work of Radicchi et al. (2008) implied that the scaled citation distributions collapse onto one curve, which depends on neither the discipline nor the publication year.

In what follows we present our measurements, which demonstrate only limited support for this claim. Namely, these measurements illustrate the universal shape of citation distributions for some sets of papers, and deviations from this shape for others. In particular, Figure 1 (left panel) shows cumulative citation distributions for all 40,195 physics papers published in 1984. The distributions for different citation years are markedly different. After dividing each distribution by its mean we obtain scaled distributions. While early scaled distributions are very similar and collapse onto a single curve, the scaled distributions in later years do not collapse. Thus, one-parameter scaling, suggested by Radicchi et al. (2008) for papers in one discipline published in 1 year, is valid only for early citation distributions. As time passes, the one-parameter scaling becomes unsatisfactory.

Figure 2 shows citation distributions for the papers belonging to three different disciplines and published in 1 year. After division of each distribution by its mean, they all collapse onto a single curve. Again, this scaling works well for the early citation distributions and breaks for the late distributions (not shown here).

Figure 3 plots citation distributions for economics papers for two publication years, 1984 and 1995. We compare here two citation distributions after the same long 17-year period elapsed after publication. Although the papers published later have more citations due to the explosion in the number of publications in economics, which have been covered in the Web of Science since 2000, the scaled citation distributions again collapse onto a single curve.

Figure 4 compares citation distributions for the physics papers published in three different journals during the same period. Obviously, these distributions are very different and even after division by the mean number of citations they do not collapse onto one curve. Here, we have intentionally focused on journals with very different mean numbers of citations. If we were considering the scaled citation distributions for the journals with more or less the same mean number of citations (Stringer et al., 2008; Thelwall, 2016a), they would collapse onto one curve.

Thus, in some cases the scaled citation distributions collapse onto one curve, while in other cases they do not. In what follows we explain these observations using our model of citation dynamics (Golosovsky, 2019; Golosovsky & Solomon, 2017).

## 3. MODELING CITATION DYNAMICS AND CITATION DISTRIBUTIONS

### 3.1. Recursive Search Model

We present here a short summary of our model. Consider a discipline, namely, a network of papers which are densely connected through citations and only loosely connected to the outside world of science (other disciplines)—a community, in the language of complex networks. Consider some paper *j* published in year *t*_{0}. The author of a new paper that belongs to the same discipline and was published *t* years later may cite paper *j* after picking it up from the databases, scientific journals, or following the recommendations of colleagues or news portals. We name this a direct citation. An author of another new paper can pick up paper *j* from the references of the papers already included in his or her reference list. Such a strategy is known as *copying* or *redirection*, although we call it *indirect citation*. (Our definitions of direct and indirect citations are different from those of Peterson, Presse, and Dill (2010) and Milojevic (2020), who base their models on the preferential attachment mechanism.)

*k*

_{j}(

*t*) citations in year

*t*is captured by a Poisson distribution, $\lambda jkjkj!$

*e*

^{−λj}, where

*λ*

_{j}(

*t*) is the latent citation rate, which, in contradistinction to the rate of the conventional Poisson process, depends on the papers’ citation history. It is given by the following expression:

The first and the second addends in Eq. 2 capture, correspondingly, the direct and indirect citations, and the time *t* is counted from the moment of publication. *Ã*(*t*) is the aging function for citations and *R*_{0} is the average reference list length of papers belonging to this discipline and published in the same year. The main individual property of the paper is its fitness *η*_{j}—a real number that captures the appeal that this paper makes to readers, in other words, its citation potential. This definition of fitness can be traced to Caldarelli, Capocci et al. (2002) and is very different from that of Bianconi and Barabasi (2001). To determine *η*_{j} quantitatively, the papers’ citation trajectory must be compared to the model prediction given by Eq. 2. The best proxy to *η*_{j} is the initial citation rate (i.e., the number of citations that the paper garners during the first 2–3 years after publication, namely, *η*_{j} ∝ $Ktt$, where *t* = 2–3 years).

Each past citation of a paper triggers a cascade of indirect citations. These are captured by the integral in Eq. 2, where *k*_{j}(*τ*) is the number of citations garnered in year *τ* (it is also equal to the number of first-generation citing papers published in year *τ*), *m*(*t* − *τ*) is the average number of second-generation citing papers garnered by a first-generation citing paper in year *t*, and $TR0$*e*^{−γ(t−τ)} is the probability of a second-generation citing paper to cite paper *j*.

*j*with fitness

*η*

_{j}and citation history

*k*

_{j}(

*τ*). The parameters of the model, which are common for all papers in one discipline and one publication year, are

*R*

_{0},

*γ*,

*T*, and the functions

*Ã*(

*t*) and

*m*(

*t*). The latter one is not an independent function. Indeed, by averaging Eq. 2 over a collection of all papers in one discipline published in 1 year, we obtain

*η*

_{0}is an average fitness of the papers in this collection. Equation 3 implicitly defines a single-valued function

*m*(

*t*) and relates it to

*Ã*(

*t*),

*R*

_{0},

*γ*,

*T*, and

*η*

_{0}.

*Ã*(

*t*). Indeed, one paper’s citation is another paper’s reference. This translates into the symmetry between synchronous (retrospective) and diachronous (prospective) citation distributions (Nakamoto, 1988; Roth, Wu, & Lozano, 2012). Theoretical understanding of this symmetry yields a reference-citation duality (Glanzel, 2004; Golosovsky & Solomon, 2017; Yin & Wang, 2017), which relates the dynamic of the mean number of citations to the age distribution of references in the reference lists of papers belonging to one discipline. As the growth of the number of publications and of the average reference list length may be crudely approximated by exponentials (Evans et al., 2012; Milojevic, 2012; Sugimoto & Larivière, 2018), namely,

*N*(

*t*

_{0}) ∝

*e*

^{αt0}and

*R*

_{0}(

*t*

_{0}) ∝

*e*

^{βt0}the reference-citation duality is captured by the following equation,

*t*

_{0}is the publication year,

*r*(

*t*

_{0},

*t*

_{0}−

*t*) is the average fraction of references of age

*t*in the reference list of papers published in year

*t*

_{0}, and $\u222b0\u221e$

*r*(

*t*

_{0},

*t*

_{0}−

*t*)

*dt*= 1, by definition. Equation 4 and the requirement of the finite average reference list length yield a useful relation

*r*(

*t*

_{0},

*t*

_{0}−

*t*) only weakly depends on the publication year

*t*

_{0}. The latter study also showed that the function

*r*(

*t*

_{0},

*t*

_{0}−

*t*), where

*t*is the argument and

*t*

_{0}is the parameter, varies with

*t*

_{0}only on a long time scale, on the order of 10–20 years. We are interested here in the shorter time scales, hence we assume that

*r*(

*t*

_{0},

*t*

_{0}−

*t*) does not depend on

*t*

_{0}. This allows us to drop

*t*

_{0}from our notation, for clarity. Then Eqs. 3 and 4 yield

Under this condition, *η*_{0} is equal to the average fraction of direct references in the reference list of papers belonging to this discipline^{2}.

Equation 6 is the counterpart of Eq. 3 for citations. The scenario of the referencing process, which underlies Eq. 6, is as follows. An author, composing the reference list of a new paper, selects several direct references. The probability of choosing a paper *j* as a direct reference is given by the product *η*_{j}*A*(*t*), where *η*_{j} is the paper’s fitness and the aging function *A*(*t*) is given by Eq. 7. From the reference list of each preselected paper of age *τ* the author randomly chooses *Te*^{−γτ} (indirect) references, copies them, and proceeds recursively. By averaging over all papers in one discipline published in 1 year, we come to Eq. 6.

Our model of citation dynamics and its verification has been described in our previous publication (Golosovsky & Solomon, 2017). This model makes a probabilistic prediction of the citation trajectory of a paper which is based on its citation history. Our model was developed several years after the model of Wang, Song, and Barabasi (2013) and is complementary to it, in the sense that the latter is predictive whereas our model is explanatory and is based on a realistic scenario of the citation process. It should also be noted that the model of Wang et al. (2013) yields citation trajectories of papers using three parameters for each paper, and all three are paper specific. This should be compared to our model, which operates with only one individual parameter (the paper’s fitness), while five other empirical parameters and two empirical functions are the same for all papers that were published in 1 year and belong to one discipline.

### 3.2. Citation Distributions

*ρ*(

*η*). To analyze dynamic factors appearing in the model, for pedagogical reasons we replace the actual citation rate

*k*

_{j}(

*t*) in Eq. 2 with the latent citation citation rate

*λ*

_{j}(

*η*

_{j},

*t*). Next, we assume temporarily that

*T*=

*const*(we will revise this assumption later). After making corresponding substitutions into Eq. 2, it reduces to a linear integral Volterra equation of the second kind,

*η*

_{j}and

*R*

_{0}, namely,

*λ*

_{j}(

*t*) ∝

*η*

_{j}

*R*

_{0}. Next, we introduce the integrated latent citation rate,

*B*(

*t*) is the same for all papers in one discipline that were published in 1 year.

*ρ*(

*η*) is the fitness distribution for this collection and Λ(

*η*) is given by Eq. 10. For

*K*>> 1, the Poisson factor $e\u2212\Lambda \Lambda KK!$ reduces to the delta-function

*δ*(Λ −

*K*), in such a way that

*η*

_{0}is the average fitness. Because

*K*≈ Λ for

*K*>> 1, Eqs. 10 and 13 yield that the ratio $KjtMt$ ≈ $\eta j\eta 0$ does not depend on time. This prompts us to introduce the reduced fitness $\eta \u02dc$

*= $\eta j\eta 0$. Next, we note that for*

_{j}*K*>> 1 the scaled citation distribution is nothing else but the reduced fitness distribution

It should be noted that the conclusion about time-independence of the scaled citation distribution *p*(*x*; *t*) relies on the assumption *T* = *const* (Eq. 2). We will revise this assumption later and demonstrate that Eq. 14 has a very limited range of applicability.

We are now in a position to assess the conjecture of Radicchi et al. (2008). It consists of two separate statements:

The scaled citation distributions for the same set of papers and for different citation windows collapse onto one curve; namely, the scaled citation distributions do not depend on the time after publication. This statement follows naturally from our model and is captured by Eq. 14.

The scaled citation distributions for different sets of papers and for the same citation window collapse onto one curve. In the framework of our model, this statement is equivalent to the assertion that different collections of papers are characterized by the same reduced fitness distribution

*ρ*($\eta \u02dc$). This assertion is beyond our model, because the latter does not presuppose any particular shape of the fitness distribution.

The model presented above is a synthesis of the fitness model of Caldarelli et al. (2002) and the recursive search/copying/redirection models of Krapivsky and Redner (2005), Vazquez (2001), and Simkin and Roychowdhury (2007). The new ingredient is a realistic rather than cartoon-like representation of the citation habits of authors. Our model is based on two important assumptions:

A paper’s fitness

*η*_{j}does not change during the paper’s lifetime; in other words, the aging function*Ã*(*t*) is the same for all papers in one discipline published in 1 year.The kernel $TR0$

*e*^{−γ(t−τ)}, which characterizes indirect citations in Eq. 2, is the same for all papers in one discipline published in 1 year.

*T*has a weak logarithmic dependence on the number of accumulated citations

*K*,

*T*

_{0}and

*b*are empirical parameters. The

*T*(

*K*) dependence introduces nonlinearity into Eq. 2 and its derivative, Eq. 9

^{3}. Although this nonlinearity is weak, it is important. Indeed, because Eq. 9 is weakly nonlinear, its solution, strictly speaking, cannot be factorized. Thus, Eqs. 10, 13, and 14 are only approximately valid. The nonlinearity, which is captured by Eq. 15, results in deviations from the universality of the scaled citation distributions, the magnitude of these deviations being proportional to the nonlinear coefficient

*b*.

In summary, our model (Eqs. 2–8, and the empirical Eq. 15) provide a framework for the assessment of the purported universality of citation distributions. The model reduces this assessment to the analysis of certain parameters and functions. To find these parameters and functions, we need to perform dedicated measurements of citation dynamics and citation distributions for different collections of papers. We have already reported such measurements for one collection of papers—physics (Golosovsky & Solomon, 2017). Here, we report similar measurements and analysis for two additional collections of papers—mathematics and economics. By analyzing and comparing citation distributions and the corresponding model parameters for three disciplines, we assess the validity of the universality hypothesis of Radicchi et al. (2008).

## 4. ANALYSIS OF CITATION DISTRIBUTIONS AND COMPARISON TO THE MODEL

### 4.1. Measurements

Using Clarivate’s Web of Science, we pinpointed all physics and pure mathematics papers published in 1984, and all economics papers published in 1984 and in 1995. We considered research papers, letters, notes, and uncited papers, while editorial material and reviews were excluded. We measured the citation dynamics of the papers belonging to these collections during a long period after publication and using a citation window of 1 year. The measurements for physics papers were partially described in our previous publication (Golosovsky & Solomon, 2017), and we compare them here with our new measurements for mathematics and economics papers.

### 4.2. Fitting Procedure

Figure 5 shows citation distributions for two disciplines and for several citation years. To model these distributions, for each discipline and publication year we considered a synthetic set containing the same number of papers and characterized by a certain fitness distribution. On these synthetic sets, we ran the stochastic numerical simulation based on Eqs. 2 and 15 and tuned the parameters of the simulation to achieve close correspondence between citation dynamics of the real and synthetic sets of papers.

Besides citation distributions, we considered the citation lifetime *τ*_{0}, which was defined from the exponential approximation of the paper’s citation trajectory, *K*(*t*) = *K*_{∞}(1 − *e*^{−Γt}), where Γ = 1/*τ*_{0} is the obsolescence rate (see Figure 6). We tuned the parameters of the numerical simulation to fit not only citation distributions but citation lifetime as well. While the fitting of citation distributions using many parameters leaves some ambiguity, the simultaneous fitting of the citation distributions (snapshot measurements) and citation lifetime (longitudinal measurements), using the same parameters, pinpoints these parameters unambiguously.

To perform numerical simulation of the citation distributions, we have to choose some trial fitness distribution. While Eq. 2 contains papers’ fitness *η*_{j}, our analysis of the scaled citation distributions (Eqs. 13 and 14) focuses on reduced fitness, $\eta \u02dc$_{j} = $\eta j\eta 0$, where *η*_{0} is the average fitness. Therefore, in our simulations we used the reduced fitness, which we assumed to follow a log-normal distribution, *ρ*($\eta \u02dc$) = $1\eta \u02dc\sigma 2\pi e\u2212ln\eta \u02dc+\sigma 2222\sigma 2$. The mean of this distribution is unity and it is fully defined by its shape parameter *σ*.

Next, we recast the first term in Eq. 2 as $\eta \u02dc$* _{j}*[

*η*

_{0}

*R*

_{0}

*Ã*(

*t*)] and considered the expression

*η*

_{0}

*R*

_{0}

*Ã*(

*t*) as one composite fitting function. Thus, the fitting parameters and functions used in our simulation were

*σ*, the shape parameter of the reduced fitness distribution; the composite function

*η*

_{0}

*R*

_{0}

*Ã*(

*t*) which characterizes direct citations; and the parameters $T0R0$ and

*γ*,

*b*which characterize indirect citations.

The fitting procedure was as follows. For each discipline, we found some initial combination of the parameters *σ*, $T0R0$, *γ*, *b*, and *η*_{0}*R*_{0}*Ã*(*t*), which satisfactorily fit citation distributions for different citation years. Of these, the parameters *σ*, $T0R0$, *γ*, and *b* were the same for all citation years, while *η*_{0}*R*_{0}*Ã*(*t*) was the only parameter that was specific for each citation year. After achieving a reasonable fit for citation distributions, we focused on citation lifetime of papers and its dependence on the number of citations. Because citation lifetime *τ*_{0} = 1/Γ does not depend on the fitness distribution, we used only two parameters, $T0R0$ and *b*, for fitting while the remaining parameter *γ* was determined independently from the analysis of the Pearson correlation coefficient for citation fluctuations in subsequent years (not shown here). Figure 6 shows that Γ decreases logarithmically with *K* (*τ*_{0} increases) and this is a direct consequence of the *T*(*K*) dependence captured by Eq. 15. The slope of the Γ(*K*) dependence yields the nonlinear coefficient *b*. After fitting citation lifetimes, we came back to citation distributions and ran our simulation with the previously found parameters $T0R0$, *γ*, and *b*, while fine-tuning *σ* and *η*_{0}*R*_{0}*Ã*(*t*). Then we came back to citation lifetime and fine-tuned the parameters $T0R0$, *γ*, and *b*. After several loops of fitting we achieved *simultaneously* a good correspondence between the measured and simulated citation distributions, on the one hand, and the measured and simulated citation lifetimes, on the other.

Our next step was to determine each one of the three multipliers in the composite function *η*_{0}*R*_{0}*Ã*(*t*). To this end, we analyzed the mean number of citations, *M*(*t*) = $\u222b0t$*m*(*τ*)*dτ*, in particular, we considered *M*_{detrended}(*t*) = $\u222b0t$*m*(*τ*)*e*^{−(α+β)τ}*dτ* (Figure 7). It follows from Eq. 5 that $MdetrendedtR0$ = $\u222b0t$*r*(*τ*)*dτ*, where *r*(*t*) is the reduced age composition of the average reference list. From the requirement of the convergence of this integral to unity, in the long time limit, we found the sum of the growth exponents (*α* + *β*) and the reference list length *R*_{0}. From known *R*_{0} and $T0R0$ we found *T*_{0} (see Eq. 15). At the next step, we recast the composite function *η*_{0}*R*_{0}*Ã*(*t*), found from the fitting procedure for citation distributions, as *η*_{0}*R*_{0}*A*(*t*)*e*^{(α+β)t}, substituted there *R*_{0} and (*α* + *β*) found from the analysis of *M*(*t*), and determined the remaining parameter *η*_{0} (average fitness) from the normalization condition $\u222b0\u221e$*A*(*t*)*dt* = 1.

### 4.3. Fitting Parameters and Functions

Table 1 summarizes the parameters of citation dynamics for three disciplines, as determined through the above fitting procedure. Our most important finding is that all three disciplines are characterized by the same reduced fitness distribution, namely, a log-normal distribution with the shape parameter *σ* = 1.13. The aging function *A*(*t*) for the three disciplines turned out to be nearly identical, as well. Indeed, Figure 8a shows that while the aging functions *Ã*(*t*) differ from discipline to discipline, the detrended aging functions, *A*(*t*) = *Ã*(*t*)*e*^{−(α+β)t} (the aging function for references, which captures the proportion of recent references versus old references in the average reference list of papers—see Eq. 6) collapse onto one curve. It should be noted that our model does not presuppose the universality of *ρ*($\eta \u02dc$) and *A*(*t*): This unexpected result follows from our measurements for three widely different disciplines.

**Table 1.**

Disciplines
. | Publication year
. T_{publ} | Number of papers
. N | Reference list length
. R_{0} | Growth exponents (
. α + β), yr^{−1} | Average fitness
. η_{0} | Shape parameter
. σ | Copying exponent
. γ, yr^{−1} | Indirect citations
. T_{0} | Nonlinear coefficient
. b |
---|---|---|---|---|---|---|---|---|---|

Physics | 1984 | 40,195 | 19.2 | 0.045 | 0.48 | 1.13 | 1.2 ± 0.2 | 3.2 | 0.42 |

Economics | 1995 | 4,782 | 8.8 | 0.12 | 0.49 | 1.13 | 1 ± 0.2 | 7.2 | 0.30 |

Economics | 1984 | 3,043 | 8.4 | 0.09 | 0.45 | 1.13 | 1 ± 0.2 | 6.6 | 0.29 |

Mathematics | 1984 | 6,313 | 3.9 | 0.092 | 0.46 | 1.13 | 0.8 ± 0.2 | 5.5 | 0.25 |

Disciplines
. | Publication year
. T_{publ} | Number of papers
. N | Reference list length
. R_{0} | Growth exponents (
. α + β), yr^{−1} | Average fitness
. η_{0} | Shape parameter
. σ | Copying exponent
. γ, yr^{−1} | Indirect citations
. T_{0} | Nonlinear coefficient
. b |
---|---|---|---|---|---|---|---|---|---|

Physics | 1984 | 40,195 | 19.2 | 0.045 | 0.48 | 1.13 | 1.2 ± 0.2 | 3.2 | 0.42 |

Economics | 1995 | 4,782 | 8.8 | 0.12 | 0.49 | 1.13 | 1 ± 0.2 | 7.2 | 0.30 |

Economics | 1984 | 3,043 | 8.4 | 0.09 | 0.45 | 1.13 | 1 ± 0.2 | 6.6 | 0.29 |

Mathematics | 1984 | 6,313 | 3.9 | 0.092 | 0.46 | 1.13 | 0.8 ± 0.2 | 5.5 | 0.25 |

While the average fitness *η*_{0}, which captures the proportion of the direct references in the average reference list, has little variation from discipline to discipline^{4}, the remaining parameters (the average reference list length *R*_{0}^{5}, the growth exponents *α*, *β*^{6}, and the parameters *T*_{0}, *γ*, and *b*, which characterize the indirect citations) differ from discipline to discipline. Thus, they are nonuniversal.

## 5. DISCUSSION

We are now in a position to assess the empirically established universality of citation distribution and the deviations therefrom. Radicchi et al. (2008) conjectured that properly scaled citation distributions, for the collection of papers published in 1 year and for different citation windows, collapse onto one curve. We assessed this conjecture theoretically, using our model of citation dynamics, and came to the conclusion that, if citation dynamics were linear, the scaled citation distributions would indeed collapse. However, as our measurements show, citation dynamics are nonlinear. Therefore, citation distributions do not exhibit perfect scaling, and there are deviations from one universal curve, such as those presented in Figure 1c. Because the nonlinearity of citation dynamics is associated with viral propagation, namely, with the parameter *T*(*K*) (Eq. 15), the magnitude of these deviations is determined by the term *b* ln *K*, where *b* is the nonlinear coefficient and *K* is the number of accumulated citations. Notably, deviations from the scaling which result from nonlinearity are mostly associated with *K* >> 1, namely, with the highly cited papers.

For collections of papers belonging to different disciplines and published in the same year, early citation distributions contain a very small number of highly cited papers, hence they obey scaling (Figure 1b); while late citation distributions, containing many highly cited papers, do not scale (Figure 1c). Thus, one source of the deviations from scaling is the *nonlinear* citation dynamics.

Another conjecture that follows from Radicchi et al. (2008) is that properly scaled citation distributions for different sets of papers and for the same citation window collapse onto one curve. This is a much stronger statement. When analyzed in the framework of our model of citation dynamics, this is equivalent to the assertion that different collections of papers are characterized by the same reduced fitness distribution *ρ*($\eta \u02dc$). Our measurements support this remarkable claim for collections of papers belonging to different disciplines and published in 1 year^{7}. The same reasoning predicts that the scaled citation distributions for collections of papers belonging to different journals (Figure 4) would be nonuniversal, as the corresponding fitness distribution *ρ*_{journal}($\eta \u02dc$) is journal-specific. Although a set of papers published in a journal is a subset of those belonging to the whole discipline, the sampling performed by each journal is not the same due to different acceptance criteria. For example, while *Science* and *Physical Review Letters* skim the high-fitness tail of the fitness distribution, the *Journal of Applied Physics* samples it more uniformly.

Thus, we have demonstrated that citation distributions for the sets of papers published in 1 year are determined by the fitness distribution, on the one hand, and by the citation dynamics of papers, on the other. While the latter differ from discipline to discipline, the fitness distribution is the same for physics, economics, and mathematics and it is fairly well approximated by a log-normal distribution with shape parameter *σ* ∼ 1.13. Limpert, Stahel, and Abbt (2001) reviewed the log-normal distributions occurring in nature and demonstrated that the distribution with *σ* ∼ 1 is one of the narrowest observed. In fact, Ghadge, Killingback et al. (2010) showed that this distribution is something special: It generates a citation network that is a borderline between two classes—a gel-like network and a network consisting of isolated clusters. On the other hand, the log-normal distribution belongs to the class of fat-tailed distributions and is reminiscent of self-organized criticality in sand piles. Such an analogy is not unexpected, as each new paper in the scientific enterprise causes a cascade of citations and, ideally, an avalanche of new and fruitful ideas. Why different disciplines adjust themselves to produce this specific shape of the fitness distribution—a log-normal with *σ* ∼ 1—is an intriguing question. This nearly universal fat-tailed distribution probably reveals some facet of science as a self-organizing system.

Our results can be considered from another perspective. In the framework of our recursive search model, the information about a paper propagates in the scientific community in two ways: broadcasting (the authors find this paper after reading news, searching in the Internet, reading the journals, etc.—this corresponds to direct citations) and word-of-mouth (finding this paper in the reference lists of other papers—we name these *indirect citations*). These two modes of propagation are coupled: Each direct citation gives rise to cascades of indirect citations, which can turn viral. Although direct citations are garnered in proportion to the paper’s fitness, which captures its intrinsic quality and attributes, indirect citations depend on the structure of the citation network and gauge the paper’s fame. The number of citations combines the paper’s fitness and fame (Simkin & Roychowdhury, 2013). As indirect citations originate from direct citations, the paper’s fitness is the key parameter that determines the overall number of citations. Our results imply that the fitness distributions for different disciplines are very similar whereas citation distributions are not, inasmuch as they are associated with viral propagation of information in the network of communications corresponding to each discipline. In other words, the static attributes of the citation network for each discipline are universal, while the dynamic attributes are not. This differentiation between the dynamic and static attributes can be relevant to other growing complex networks as well.

## 6. CONCLUSIONS

We explored the conjecture of Radicchi et al. (2008) who claimed that the scaled citation distributions collapse onto one curve, namely, their shape is nearly universal. We found that the scaling holds for collections of papers belonging to one discipline, published in 1 year, and measured several years after publication. We explain this observation using our recently developed model of citation dynamics, which delineates between the static and dynamic factors affecting the citation dynamics of papers. The model attributes the accumulated citations to the paper’s fitness, on the one hand, and to the viral propagation of the information about this paper in the scientific community, on the other. We believe that the underlying reason for the scaling of citation distributions is the universal fitness distribution for scientific disciplines. This claim has been verified by our measurements with physics, economics, and mathematics papers. Although extrapolation from these three disciplines to all science may be too ambitious, because the three disciplines are so different, that is still plausible.

We also found that citation distributions do not scale well when one compares collections of papers many years after publication. In this case, our model traces the deviations from the scaling to the discipline-specific viral propagation. On the other hand, we find that citation distributions for different journals also do not scale. In this case, we attribute deviations from the scaling to the journal-specific fitness distribution which can differ from the universal fitness distribution for a scientific discipline as a whole.

Thus, our model of citation dynamics explains the near-universality of the scaled citation distributions and also accounts for the deviations from this near-universality.

## ACKNOWLEDGMENTS

I am grateful to Sorin Solomon for fruitful discussions, to Magda Fontana for stimulating discussions and for the assessment of the economics journals, and to Yakov Varshavsky for the assessment of the mathematics journals.

## COMPETING INTERESTS

The author has no competing interests.

## FUNDING INFORMATION

This research was not funded.

## DATA AVAILABILITY

Citation distributions for the physics, mathematics, and economics papers are available in the supplementary material (https://doi.org/10.5281/zenodo.4558007).

## Notes

^{1}

Obviously, the values of *σ* listed in Stringer et al. (2008) should be multiplied by ln 10 = 2.3025.

^{2}

One may wonder why we introduced a discipline-dependent parameter *R*_{0} into Eq. 2. Indeed, by redefinition of *η*_{j} and *T*, this parameter could have been absorbed there, but then it would pop up in Eq. 6. Our motivation to hold it in Eq. 2 instead of Eq. 6 was driven by the observation that the function *r*(*t*), which is a solution of Eq. 6, is almost independent of *R*_{0}, as shown by Roth et al. (2012) and Yin and Wang (2017). Hence, we wished to demonstrate that Eq. 6 does not not contain *R*_{0}.

^{3}

It should be noted that Eq. 2 with a kernel containing logarithmic nonlinearity pops out not only in the context of citations; Iribarren and Moro (2011) found that the dynamics of viral marketing is captured by a very similar equation.

^{4}

Our measurements of this fraction in the reference lists of physics papers published in 2010 yielded *η*_{0} = 0.34, which is slightly lower than *η*_{0} found from the analysis of citations of the physics papers published in 1984. Probably *η*_{0} varies with time.

^{5}

We estimated here the average reference list length *R*_{0} from the measurements of citation dynamics. For physics papers published in 1984, this estimate agrees well with the direct measurements of the reference list length. However, our estimates of *R*_{0} for mathematics and economics are too small. It should be remembered, however, that *R*_{0}, as estimated from citation dynamics through Eq. 4, includes only original research papers and excludes books, conference proceedings, and interdisciplinary references. These constitute a very small proportion of physics references, while they are abundant among mathematics and economics references; hence our estimate of *R*_{0} for these disciplines is smaller than the actual reference list length.

^{6}

The data of Sugimoto and Larivière (2018) show that the exponential approximation for the growth of the number of publications is reasonable, and most disciplines exhibit the growth exponent α ∼ 0.04 in the period 1984–2010; while the exponential growth of the reference list length is a very crude approximation and it grows with time very nonuniformly, in such a way that the corresponding effective growth exponent β depends upon the time window of measurements.

^{7}

When we compare citation distributions for different disciplines, we consider here, for clarity, only early citation distributions, for which the nonlinearity associated with viral propagation has not yet developed.

## REFERENCES

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_{0}at the micro level using data from chemistry

## Author notes

Handling Editor: Ludo Waltman

DOI:https://doi.org/10.1038/nbt.2601,PMID:23851447,PMCID:PMC3740009