On a heuristic point of view concerning the citation distribution: introducing the Wakeby distribution
 Yurij L Katchanov^{1}Email author and
 Yulia V Markova^{2}
https://doi.org/10.1186/s4006401508211
© Katchanov and Markova; licensee Springer. 2015
Received: 12 November 2014
Accepted: 14 January 2015
Published: 26 February 2015
Abstract
The paper proposes a heuristic approach to modeling the cumulative distribution of citations of papers in scientific journals by means of the Wakeby distribution. The Markov process of citation leading to the Wakeby distribution is analyzed using the terminal time formalism. The Wakeby distribution is derived in the paper from the simple and general inhomogeneous Choquet–Deny convolution equation for a nonprobability measure. We give statistical evidence that the Wakeby distribution is a reasonable approximation of the empirical citation distributions.
AMS Subject Classification: 91D30; 91D99
Keywords
Bibliometrics Choquet – Deny convolution equation Citation distributions Wakeby distributionIntroduction
The number N(z) of scientific papers that has been cited a total of z times is one of the most widely used and strong scientometric indicators. Alternatively, one may consider more sophisticated indicators (see, e.g., (Glänzel and Moed 2013; Leydesdorff et al. 2011, 2013; Radicchi and Castellano 2011; Waltman and van Eck 2013)), but we limit ourselves here to the case in which the underlying variables are defined as the nonnegative real numbers z and N(z). This approach has a formal defect that can be easily recognized. As a matter of fact, z and N(z) assume only nonnegative integer values. Yet a substantial amount of previous works on the statistical distribution of citations of scientific papers treated z and N(z) as continuous variables in the longtime limit of the observation period, and we pursue the same approach with this paper.
where z _{min} means a threshold value.
One of the classic results of scientometrics is the derivation of a model in which the probability distribution (PD for short) of z, in its asymptotic tail, is equivalent to a powerlaw PD (2) (Haitun 1982; Yablonsky 1985). A possible mechanism to explain the powerlaw distribution is a stochastic growth process in which the citation rate of a paper is defined by the total number of received citations and the time after publication (Albert and Barabási 2002; de Solla Price 1976; Dorogovtsev et al. 2000; Golosovsky and Solomon 2013; Krapivsky et al. 2000).
The other main result is the justification of the powerlaw approximation of the statistical distribution of citations of scientific papers (Albarrán and RuizCastillo 2011; Albarrán et al. 2011; Ausloos 2014; Brzezinski 2014; Egghe 2007, Eom and Fortunato 2011; Peterson et al. 2010; Radicchi and Castellano 2012; Redner 1998; Stringer et al 2010 ; Waltman et al. 2012 ; Zhao and Ye 2013 ). However, the powerlaw distribution is possessed of a number of characteristics limiting its application (Clauset et al. 2009 ; Golosovsky and Solomon 2012 ; Newman 2005).
Power laws (see detail in (Clauset et al. 2009; Newman 2005; Virkar and Clauset 2005; 2014)) are widely used to represent scientometric distributions. In reality, however, certain studies of citation distributions have used various other functional forms to provide best approximations to as wide a variety of bibliometric data as possible (see, e.g., (Burrell 2014; Davies 2002; Golosovsky and Solomon 2012; Gupta et al. 2005; Laherrère and Sornette 1998; Radicchi et al. 2008; Redner 2005; Sangwal 2013; van Raan 2001)). Nevertheless, all the same, power laws had and still have a crucial part to play in scientometrics, not only because they are established but also because they are theoretically wellfounded, for reasons arising from the generalized central limit theorem (Uchaikin and Zolotarev 2011), which has very considerable importance in probability theory.
The practice of citations evolves over time. We can conceive of the process of citation as a way of tracking discrete social acts. Time lends citations direction and meaning (see (Bouabid 2011; Burrell 2002, 2014; Eom and Fortunato 2011; Glänzel 2007; Hsu and Huang 2011; Radicchi et al. 2012; Redner 2005; Simkin and Roychowdhury 2012; Wang et al. 2013) for more details). However, when we analyze bibliometric data sets, we may interpret citations not as a series of discrete acts but rather as a statistical regularity which can be expressed in the language of timeindependent PDs. While the very meaning of the RV Z is difficult to represent in terms of the PD, it acquires a direct intuitive sense in terms of the terminal time formalism, which is developed in a systematic way (a nice general reference book for Markov processes is (Sharpe 1988)). The formal solution may consist in making the terminal time, or the lifetime, the main source of information of the RV Z. For a given process of citation, the terminal time is random. To define a realization (of the Markov process of citation) we must describe the corresponding conditional probability W. There is a natural way to associate with the terminal time problem the conditional probability W that the Markov process of citation does not stop during the fixed time interval, given that all phenomena, connected with this process during the same time interval, are known. It is proved (cf. (Sharpe 1988, [Chap. VII])) that W is connected with (nonnegative and rightcontinuous with respect to time) additive functionals of the initial Markov process of citation. We recall that an additive functional of a Markov process X is a map which associates with each interval of time [s,t] a RV \({a_{t}^{s}}\), where \({a_{t}^{s}}\) depends only on the evolution of X during the time [s,t], and also the condition \({a_{t}^{s}} + a_{\tau }^{t}=a_{\tau }^{s}\) holds for arbitrary t∈[s,τ].
The approach proposed in this paper consists in letting the probability W play a crucial part by summarizing enough information about social citation system. As a rough guide, we suppose that the RV Z depends on time through the probability W.
which we have yet to treat. We shall adopt an “asymptotic” point of view. We shall only be interested in the relation φ(·):W→Z that holds between W and Z at large times.
The proposed approach is based on the concept of the approximate invariance of the function (w∈[0,1]):w↦φ(w) by a translation of w, i.e., we claim that φ(w+·)≈ φ(w)φ(·). The considered heuristic model for the Markov process of citation is formulated as the inhomogeneous Choquet–Deny convolution equation (we shall use the abbreviated notation ICDCE) whose form is apparently determined by the approximate translation invariance. The solution of this equation gives the Wakeby distribution (WD) for citations of scientific papers. Until now, the WD has not been among the distributions employed to model observed bibliometric data.
The rest of this paper is organized as follows. The main result regarding our proposed model and its analytical solution is presented in the 2nd section. The empirical verification is provided in the 3rd section. Finally, concluding remarks are presented in the 4th section. The Appendix 1 introduces certain necessary definitions and reviews results that are needed in the rest of the paper.
Model of citation distribution

The social network (sufficiently structured subsystem) is a polycentric complex of interrelated scholars.

The scientific market (sufficiently stochastic subsystem) contains autonomous scholars who enter into the competition.

The social network is characterized by structural cohesion, while the scientific market is actually an amorphous medium for sharing information resources.

The evolution of the scientific market is of a stochastic nature.

The social network corresponds to the notion of a dynamic system.

The statistical properties of the citation distribution are partially determined by the nature of interactions between scientific market and social network.
 (1)In the event horizon where the scientific market “lives”, it can be assumed that the function φ(w) in the expression (5) is invariant under translation of w$$ \varphi(w+\cdot)=\varphi(w)\varphi(\cdot). $$(6)
 (2)In the event horizon of the social network the function φ(w) may be intuitively considered as the positive contraction semigroup τ(w) on a real onedimensional Banach space generated by −β$$ (\beta \in \mathbb{R})\colon \tau(w) = \exp( \beta w). $$(7)
 (3)The social logic of the citation distribution is such that there is a twoway influence between the scientific market and the social network (Bourdieu 1975). However, in the limit of long time, social effects of the process of citation bring to screening “longrange” interactions. As a result, the subsystems in social citation system are almost independent and we obtain approximate translation invariance$$ \varphi(w + \cdot) = \varphi(w)\varphi(\cdot) + r(w), $$(8)
where r(w) indicates a remainder term.

The simplest and most intuitive general approach to translate invariance is via convolution. Let T _{ a } be the translation operator defined by T _{ a } φ(w)=φ(w+a). Translation invariance of the convolution (φ∗χ) means that the convolution with a fixed function χ commutes with T _{ a }, i.e.,$$ T_{a}\left(\varphi \ast \chi\right) = \left(T_{a}\varphi\right) \ast \chi = \varphi \ast \left(T_{a}\chi\right). $$
It can involve explicitly the wellknown Choquet–Deny convolution equation (CDCE for short, see (20)).

By virtue of formula (7), whatever the precise form of r(w) may be it will give to (8) a contribution of the form$${\lim}_{w\uparrow{1}} r(w) = O\left(\exp( \beta w)\right). $$
This proposition corresponds to a functional equation that can be rewritten as the ICDCE (see (21)).
The translation invariance is an important concept, so it should be understood in a thorough manner. The probability W, of course, corresponds to terminal time, while the RV Z occurs at random in time. Since the RV Z in the scientific market should be independent of an arbitrary translation a, the constancy of termination rate of the process of citation take place in the scientific market. This is what we mean when we say that φ(w) has the translation invariant property (6) in the scientific market.
The motivation of the approximate translation invariance is to take the relation between the scientific market and the social network into consideration. In rough approximation, the scientific market and the social network can be considered as relatively independent. Consequently, their contributions to φ(w) are additive. Summing (6), and (7), we obtain (8), i.e., approximate translation invariance. The Eq. 8 therefore expresses some kind of linear superposition of the effect of the scientific market and the social network. This superposition is not valid in the general case.
The construction of such a subprocess is minutely described in (Sharpe 1988 [p. 65–74]).
where, as in Appendix 1, φ(w) is locally integrable (with respect to the Lebesgue measure Λ). However, the function φ(w) is not yet completely defined. In fact, the general problem of studying the form of φ(w) can be reduced to the case in which this function satisfies certain extra conditions. One can attempt to define φ(w) implicitly by some functional equation rather than by direct definitions. In particular, the general form of φ(w) may be derived uniquely from its invariance.
This expression is the relation we were seeking between the quantities we were interested in, Z and W. As could be expected, the RV Z contains two parts: one corresponds to the incident stream of citations, the other to the scattered stream of citations.
The formula (17) defines the distribution, which is called the WD (Johnson et al. 2010, [p. 44–46]). This distribution was established by H. A. Thomas (Houghton 1978) (who lived on Wakeby pond on Cape Cod, Massachusetts) for hydrological data case studies (Griffiths 1989; Hosking and Wallis 2005). We stress that the explicit formula for the PDF of Z is not generally available.
The WD in (18), when α=0 or γ=0 reduces to the GPD. The Eq. 18 is not very tractable for analysis but can yield efficient algorithms for the numerical simulation of the WD.
Nearly all the papers that deal with inference for the WD are based on the theory of Lmoments (Hosking 1990, 2006; Hosking and Wallis 2005). The free software statistical environment R contains functions to estimate the parameters of the WD from the data (see, e.g., (Asquith 2011), and packages ‘lmom’, ‘lmomco’).
Illustration
To demonstrate the applicability of the proposed heuristic model, we evaluate the goodnessoffit of the WD to two bibliometric datasets.
Data sets
This study is based on the citation distribution of papers published by the American Physical Society (APS), the American Mathematical Society (AMS), the European Mathematical Society (EMS), and the Institute of Physics (IOP) (see the list of journals in Appendix 2) in the years 1980 —2008 and indexed in Thomson Reuters Journal Citation Reports, Science Edition 2012. The data on citations was obtained from the Thomson Reuters Web of Science Core Collection. The data on citations of papers of APS, AMS and EMS were obtained in December 2013. The data for IOP were received in April 2014. The number of citations z is counted as the total number of times a paper appears as a reference of a more recently published paper indexed in the Web of Science Core Collection.

The first set contains papers published by APS, AMS, and EMS. There are 10,043,731 citations among 356,287 papers.

The second set consists of 233,570 papers published by IOP. This dataset includes 5,885,458 citations.
Empirical results
Goodness of fit — Summary for Dataset 1
Distribution  Kolmogorov – Smirnov  Anderson – Darling  

( p =0 . 00205, α =0 . 1)  ( p =1 . 9286, α =0 . 1)  
Statistic  Rank  Statistic  Rank  
WD  0.05036  1  785.96  1 
GPD  0.05818  2  878.3  2 
Gen. Extreme Value  0.0861  3  2359.3  3 
Pareto 2  0.09083  4  47317.0  6 
Phased BiExponential  0.09143  5  53274.0  8 
Goodness of fit — Summary for Dataset 2
Distribution  Kolmogorov – Smirnov  Anderson – Darling  

( p =0 . 00253, α =0 . 1)  ( p =1 . 9286, α =0 . 1)  
Statistic  Rank  Statistic  Rank  
WD  0.05845  1  655.6  1 
GPD  0.07127  2  796.09  2 
Gen. Extreme Value  0.09249  3  1817.5  3 
Gen. Logistic  0.09848  4  2022.5  4 
Phased BiExponential  0.10584  5  37917.0  10 
Comparing the obtained values and goodnessoffit statistics given in the Tables, it will be seen that the WD offers a greater level of accuracy than the other PDs considered.
Discussion
We conclude that the WD is in some sense the best PD to adequately fit the examined bibliometric data sets.
It should be clear that the proposed heuristic approach is only a phenomenological model of the citation distribution. The Eq. 11 has not been derived yet but has rather been injected into the model. The vector of parameters (α,β,γ,δ,ξ), which fixes the WD, is assumed to be given. We can say that the formula (17) does not reproduce the exact citation distribution. We should rather view the expression (17) as an approximate representation, in which the fine details of the citation distribution have been rounded up for clarity. Nevertheless, discrepancies with observation may be caused by errors in data collection or by random influences, which will be explained later. Also, there may be many still unknown secondary effects that could change the shape of the citation distribution. But it does not detract from the consistency or the cognitive value of the mathematical model. The proposed heuristic model of the citation distribution may be considered as a potentially useful amalgamation of mathematical abstraction and scientometric intuition.
Appendixes
Appendix 1. Mathematical preliminaries to model development
In the context of this paper we are interested in mathematical formulations. Therefore, we briefly indicate here how the function φ(·) can be treated mathematically.

\(\psi \colon G \rightarrow \mathbb {R}_{+}\): the realvalued nonnegative function;

\(C(G,\mathbb {R}_{+})\): the space of continuous functions from G to \(\mathbb {R}_{+}\);

μ: the Radon measure on the Borel σfield \(\mathcal {B}(G)\) that is generated by G;

Λ: the Lebesgue measure;

Ψ: the space of all realvalued nonnegative functions \(\psi (\cdot)\colon G \rightarrow \mathbb {R}_{+}\) such that$$ (\forall x \in G)\left(\psi(\cdot)\in C(G,\mathbb{R}_{+})\right)\colon \psi(x + y) = \psi(x)\psi(y). $$(19)
From the definition, Ψ _{ μ } is a Borel subset of Ψ. In addition, let G itself be the smallest closed subgroup of G that contains supp(μ).
For an extensive discussion of the whole problem, the reader is referred to (Lukec̆s et al. 2010, [Chap. 3]). The CDCE (20) and its ramifications occupy a central place in our study.
It should be noted that, according to (Deny 1959), if μ is a probability measure, then every bounded solution of (20) reduces to a constant.
In the case \(G = \mathbb {R}_{+}\), μ is assumed to be nonarithmetic such that μ(∅)<1, and ϕ(·) is assumed to be nonnegative, realvalued and locally integrable with respect to the Λ function (ignoring the trivial case of ϕ(·)=0 a.e. (mod Λ)) such that it satisfies a.a. (mod Λ) to the CDCE (20).
The proof of this theorem can be found in (Lau and Rao 1982).
where r(x)≤κ exp(−β x) is an “error term”, is a generalization of the Eq. 20 given by Shimizu. The solutions of the ICDCE on \(\mathbb {R}_{+}\) were considered by (Shimizu 1980)and by (Gu and Lau 1984).
Appendix 2. List of journals

American Physical Society
 1.
Physical Review A
 2.
Physical Review B
 3.
Physical Review B
 4.
Physical Review C
 5.
Physical Review D
 6.
Physical Review E
 7.
Physical Review Letters
 8.
Physical Review Special Topics Accelerators And Beams
 9.
Physical Review Special Topics Physics Education Research
 10.
Physical Review X
 11.
Reviews of Modern Physics
 1.

American Mathematical Society
 1.
Bulletin of American Mathematical Society
 2.
Journal of the American Mathematical Society
 3.
Mathematics of Computation
 4.
Memoirs of the American Mathematical Society
 5.
Proceedings of the American Mathematical Society
 6.
St. Petersburg Mathematical Journal
 7.
Transactions of the American Mathematical Society
 1.

European Mathematical Society
 1.
Commentarii Mathematici Helvetici
 2.
Groups Geometry and Dynamics
 3.
Interfaces and Free Boundaries
 4.
Journal of Noncommutative Geometry
 5.
Journal of the European Mathematical Society
 6.
Portugaliae Mathematica
 7.
Rendiconti Lincei —Matematica e Applicazioni
 8.
Revista Matematica Iberoamericana
 9.
Zeitschrift für Analysis und Ihre Anwendungen
 1.

Institute of Physics
 1.
Astronomical Journal
 2.
Astrophysical Journal
 3.
Astrophysical Journal Letters
 4.
Astrophysical Journal Supplement Series
 5.
Bioinspiration Biomimetics
 6.
Biomedical Materials
 7.
Chinese Physics B
 8.
Chinese Physics Letters
 9.
Classical and Quantum Gravity
 10.
Communications in Theoretical Physics
 11.
Environmental Research Letters
 12.
European Journal of Physics
 13.
Fluid Dynamics Research
 14.
Inverse Problems
 15.
Journal of Breath Research
 16.
Journal of Cosmology and Astroparticle Physics
 17.
Journal of Geophysics and Engineering
 18.
Journal of Instrumentation
 19.
Journal of Micromechanics and Microengineering
 20.
Journal of Neural Engineering
 21.
Journal of Physics A Mathematical and Theoretical
 22.
Journal of Physics B Atomic Molecular and Optical Physics
 23.
Journal of Physics: Condensed Matter
 24.
Journal of Physics D Applied Physics
 25.
Journal of Physics G Nuclear and Particle Physics
 26.
Journal of Radiological Protection
 27.
Journal of Statistical Mechanics Theory and Experiment
 28.
Laser Physics
 29.
Laser Physics Letters
 30.
Measurement Science Technology
 31.
Metrologia
 32.
Modelling and Simulation in Materials Science and Engineering
 33.
Nanotechnology
 34.
New Journal of Physics
 35.
Nonlinearity
 36.
Physica Scripta
 37.
Physical Biology
 38.
Physics in Medicine and Biology
 39.
Physics World
 40.
Physiological Measurement
 41.
Plasma Physics and Controlled Fusion
 42.
Plasma Science Technology
 43.
Plasma Sources Science Technology
 44.
Reports on Progress in Physics
 45.
Semiconductor Science and Technology
 46.
Smart Materials and Structures
 47.
Smart Materials Structures
 48.
Superconductor Science Technology
 1.
Declarations
Acknowledgements
The financial support from the Government of the Russian Federation within the framework of the Basic Research Program at the National Research University Higher School of Economics and within the framework of implementation of the 5100 Programme Roadmap of the National Research University Higher School of Economics is acknowledged.
Authors’ Affiliations
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