Towards a simple mathematical theory of citation distributions
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Abstract
The paper is written with the assumption that the purpose of a mathematical theory of citation is to explain bibliometric regularities at the level of mathematical formalism. A mathematical formalism is proposed for the appearance of power law distributions in social citation systems. The principal contributions of this paper are an axiomatic characterization of citation distributions in terms of the Ekeland variational principle and a mathematical exploration of the power law nature of citation distributions. Apart from its inherent value in providing a better understanding of the mathematical underpinnings of bibliometric models, such an approach can be used to derive a citation distribution from first principles.
Keywords
Bibliometrics Citation distributions Power law distribution Wakeby distributionMathematics Subject Classification
91D30 91D99Background
Scholars have been investigating their own citation practice for too long. Bibliometrics already forced considerable changes in citation practice Michels and Schmoch (2014). Because the overwhelming majority of bibliometric studies focus on the citation statistics of scientific papers (see, e.g., Adler et al. 2009; Albarrán and RuizCastillo 2011; De Battisti and Salini 2013; Nicolaisen 2007; Yang and Han 2015), special attention is devoted to citation distributions (see, inter alia, Radicchi and Castellano 2012; RuizCastillo 2012; Sangwal 2014; Thelwall and Wilson 2014; Vieira and Gomes 2010). However, the fundamentals of the citation distribution (or CD for convenience) are far from being well established and the universal law of CD is still unknown (we do not go into details, and refer the reader to Bornmann and Daniel 2009; Eom and Fortunato 2011; Peterson et al. 2010; Radicchi et al. 2008; RuizCastillo 2013; Waltman et al. 2012). Furthermore, existing bibliometric models of CDs place little or no emphasis on characteristics of the mathematical formalism itself (cf. Egghe 1998; Simkin and Roychowdhury 2007; Zhang 2013).
A mathematical theory of the CDs does not considers social citation system in its actuality. (We prefer to abbreviate social citation system to SCS; for the definition of SCS the reader is referred to, e.g., Fujigaki 1998; RodríguezRuiz 2009; Rousseau and Ye 2012.) This task is completely left to scientometricists. The mathematical theory of CDs is used to investigate a mathematical substitute instead of a real process. For this mathematical substitute, the term mathematical structure has been introduced.
The objective of scientometrics is to bridge a gap between our insights of science and our knowledge of science Mingers and Leydesdorff (2015). A mathematical theory of citation can appear as an attempt to understand the structures that constitute the bases of scientometric models. To “understand” here means to bring a bibliometric structure into congruence with a mathematical structure. The purpose of a mathematical theory is fulfilled if it provides a structure of thought objects that allows us to relate bibliometric data sets and interpret the state of affairs in science by making mathematical deductions. A scientometric model attempts to create a heuristic explanation of an empirical data set. In contrast, a mathematical theory of citation is not concerned with bibliometric data per se and strives to construct a clear and coherent framework that accurately expresses some scientometric propositions in mathematical language. In this way, opportunities emerge for applying sophisticated mathematical concepts to bibliometric phenomena. The difference between a bibliometric model and a mathematical theory of citation is more apparent than real because, although the concepts of bibliometrics can be analyzed in terms of mathematics, they cannot be eliminated in favor of the latter without losing the understanding gained by bibliometrics. In particular, a firm foundation for a mathematical theory of citation can be obtained only phenomenologically by comparing the consequences of basic mathematical statements to bibliometric data.
Motivation
We will study the axioms on which a mathematical description of SCS can be based. The author risks asserting that a mathematical theory appears to be a systematic reformulation of the problem of cumulative CDs on a purely mathematical basis. That is the main intent of this paper. Before we proceed with the analysis, we remark that there are no strong arguments leading from the bibliometric facts to the axioms. However, as we hope to show below, one can obtain additional conceptual information (relating to SCS) that is not readily available from a conventional bibliometric model by means of the axioms.
Purpose
The purpose of the research reported in this article is to provide a simple and coherent presentation of CDs based on the Ekeland variational principle. We stress the elementary variational principle governing the state of SCS and have also attempted to provide enough technical detail to create a basis for potential future studies.
Methodology
The paper addresses the construction of structural hypotheses for “how SCS works” rather than statistical inferences from bibliometric data. We accept that the continuous reproduction of a scientific inequality is a conceptual basis for almost all SCSs (cf. Bourdieu 2004). An emphasis is placed on the role of the variational principle as a valid approach for describing the local behavior of an continuous SCS. We consider an SCS to obey the following scheme. Suppose an SCS is a sufficiently smooth “motion” to ensure the consistency and the integrity of citations. In phase space, this condition is equivalent to a variational principle that produces the Euler equation for the weak form of a CD. This variational principle asserts that, for an appropriate functional, one can add a small perturbation to make it attain a minimum.
Preliminaries
In the language of \(\mathsf {P}(Z \in B)\), the PDF \(f(\cdot )\) is (almost everywhere) given by formal differentiating; as a result of this, a rather simple interpretation of \(f(\cdot )\) can be given in the framework of Sobolev spaces \(H^{k}(\mathbb {R})\). (For the definitions and properties of Sobolev spaces, see Maz’ya 2011.)
Results
Because \(\varphi (\cdot )\) and \(\zeta\) are so fundamental in this paper, it may seem strange that we have not explicitly defined them in formal mathematical terms. As with other primitive objects of the mathematical theory, the most one can do is to give the implicit definitions by postulating the properties that hold for \(\varphi (\cdot )\) and \(\zeta\).
 \(\mathbf {A_{1}}\)
 A function \(\varphi (\cdot )\) for \(\forall v \in \bigl ( V, \Vert \cdot \Vert \bigr )\) is a proper (\(\mathrm {dom}\,\varphi \not = \emptyset\)), lower semicontinuous (\(\varphi (v)\le \liminf _{n\rightarrow \infty }\varphi (v_{n})\) if \(v_{n}\rightarrow v\)), convex, and bounded below (\(\inf _{V}\varphi > \infty\)) function from \(\bigl (V, \Vert \cdot \Vert \bigr )\) to \(\mathbb {R}_{+}\), satisfying the following condition:$$\begin{aligned}(\forall v\in V)(\forall c\in \mathbb {R}):\varphi (u+v)=\varphi (u)+\varphi (v)\Leftrightarrow v= cu. \end{aligned}$$
 \(\mathbf {A_{2}}\)

Among all admissible \(\zeta\), the quantity \(\zeta _{*}\) which actually describes a given CD, is assigned in such a way that the function \(\varphi (\cdot )\) reaches its minimum.
 \(S_{1}\)
 There exists a unique \(\zeta \in E\) such thatThe formula (5) reads as the “weak” Euler equation in the current setting.$$\begin{aligned} \bigl (\forall v\in E \bigr ):\bigl ( \zeta , v \bigr )_{E} = 0. \end{aligned}$$(5)
 \(S_{2}\)
 \(\zeta\) is obtained by$$\begin{aligned} \min _{v\in {E}}\left\{ \frac{1}{2}\,\varphi (v) \right\} . \end{aligned}$$(6)
Goodness of fit—summary
Distribution  Kolmogorov–Smirnov \({\alpha = 0.1} ,\) Crit. val. 0.0013 Statistic D  Anderson–Darling \({\alpha = 0.1} ,\) Crit. val. 1.929 Statistic A 

WD  0.0406  1449.0 
Lognormal  0.1062  \(1{.}258\mathrm {E}+5\) 
Weibull  0.1262  \(1{.}319\mathrm {E}+5\) 
Discussion
One of the most exciting and fruitful applications of mathematical methods in the natural sciences is the variational principle. The substantive aim of the present paper is the derivation of a variational principle, which makes it possible to interpret the empirical regularities of the CDs as a logical necessity. Starting from the famous Ekeland variational principle, we show that the derivation of the CDs given in this paper might be considered a step in indicated above direction. Using the variational principle (6) in the energetic space E together with empirical evidences about the existence of the slowly varying functions representing the right tail of the CDs allows us to introduce the WD (and the GPD) naturally.
Let us stress that modest mathematical means concerning some simple facts of functional analysis yield a simple mathematical theory of CDs from which, as its consequence, concrete CDs are immediate derived. It is remarkable that a firstprinciples derivation of the CDs (e.g., GPD) in a bibliometric model is possible at the price of uncontrollable assumptions, which are justified a posteriori. On the contrary, in our derivation it is only assumed that Eq. (8) is relevant. This is, of course, more satisfactory. However, note that there are no proper bibliometric reasons for which the Sobolev spaces are preferred over any other, and, therefore there are also no reasons to give the vague bibliometric meaning of the consistency and the integrity of citations the mathematical form of Ekeland’s variational principle.
One must bear in mind that our result refers to properties of some “pure mathematical structure”. Like any mathematical result, the Eq. (13) cannot give a completely accurate description of a empirical CD. Moreover, in the mathematical theory of CDs, “by construction”, we have no direct knowledge of the statistical parameters. Thus, we can only measure the parameters that index the CDs, not compute them from the axioms.
Conclusions
In summary, the approach suggested here allows an interpretation of the Ekeland variational principle in terms of the standard uniform RV, which may have some interest. It is shown that in a sufficiently “smooth” SCS a powerlaw tail of the static CD can appear. However, there are no grounds to consider this a mathematical model underlying bibliometric theory. At the same time, the present study may be instructive beyond the specific research site and can contribute to a mathematical theory of CDs building.
Notes
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.
Competing interests
The author declare that he have no competing interests.
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