How Popular is Your Paper ? An Empirical Study of the Citation Distribution

Numerical data for the distribution of citations are examined for: (i) papers published in 1981 in journals which are catalogued by the Institute for Scientific Information (783,339 papers) and (ii) 20 years of publications in Physical Review D, vols. 11-50 (24,296 papers). A Zipf plot of the number of citations to a given paper versus its citation rank appears to be consistent with a power-law dependence for leading rank papers, with exponent close to −1/2. This, in turn, suggests that the number of papers with x citations, N(x), has a large-x power law decay N(x) ∼ x, with α ≈ 3. PACS. 02.50.-r Probability theory, stochastic processes, and statistics – 01.75.+m Science and society – 89.90.+n Other areas of general interest to physicists In this article, I consider a question which is of relevance to those for whom scientific publication is a primary means of scholarly communication. Namely, how often is a paper cited? While the average or total number of citations are often quoted anecdotally and tabulations of highly-cited papers exist [1,2], the focus of this work is on the more fundamental distribution of citations, namely, the number of papers which have been cited a total of x times,N(x). In spite of the fact that many academics are obliged to document their citations for merit-based considerations, there have been only a few scientific investigations on quantifying citations or related measures of scientific productivity. In a 1957 study based on the publication record of the scientific research staff at Brookhaven National Laboratory, Shockley [3] claimed that the scientific publication rate is described by a log-normal distribution. Much more recently, Laherrere and Sornette [4] have presented numerical evidence, based on data of the 1120 most-cited physicists from 1981 through June 1997, that the citation distribution of individual authors has a stretched exponential form, N(x) ∝ exp[−(x/x0) ] with β ≈ 0.3. Both papers give qualitative justifications for their assertions which are based on plausible general principles; however, these arguments do not provide specific numerical predictions. Here, the citation distribution of scientific publications based on two relatively large data sets is investigated [5]. One (ISI) is the citation distribution of 783,339 papers (with 6,716,198 citations) published in 1981 and cited bea redner@sid.bu.edu tween 1981 – June 1997 that have been cataloged by the Institute for Scientific Information. The second (PRD) is the citation distribution, as of June 1997, of the 24,296 papers cited at least once (with 351,872 citations) which were published in volumes 11 through 50 of Physical Review D, 1975–1994. Unlike reference [4], the focus here is on citations of publications rather than citations of specific authors. A primary reason for this emphasis is that the publication citation count reflects on the publication itself, while the author citation count reflects ancillary features, such as the total number of author publications, the quality of each of these publications, and co-author attributes. Additionally, only most-cited author data is currently available; this permits reconstruction of just the large-citation tail of the citation distribution. The main result of this study is that the asymptotic tail of the citation distribution appears to be described by a power law, N(x) ∼ x−α, with α ≈ 3. This conclusion is reached indirectly by means of a Zipf plot (to be defined and discussed below), however, because Figure 1 indicates that the citation distribution is not described by a single function over the whole range of x. Since the distribution curves downward on a double logarithmic scale and upward on a semi-logarithmic scale (Figs. 1a and b respectively), a natural first hypothesis is that this distribution is a stretched exponential, N(x) ∝ exp[−(x/x0) ]. Visually, the numerical data fit this form fairly well for x ≤ 200 (PRD) and x ≤ 500 (ISI) as indicated in Figure 1b, with best fit values β ≈ 0.39 (PRD) and β ≈ 0.44 (ISI). However, the stretched exponential is unsuitable to describe the large-x data. Here, 132 The European Physical Journal B 10 0 10 1 10 2 10 3 10 4 x 10 0 10 1 10 2 10 3 10 4 10 5