Universality of citation distributions: toward an objective measure of scientific impact

Abstract

We study the distributions of citations received by a single publication within several disciplines, spanning broad areas of science. We show that the probability that an article is cited c times has large variations between different disciplines, but all distributions are rescaled on a universal curve when the relative indicator c(f) = c/c(0) is considered, where c(0) is the average number of citations per article for the discipline. In addition we show that the same universal behavior occurs when citation distributions of articles published in the same field, but in different years, are compared. These findings provide a strong validation of c(f) as an unbiased indicator for citation performance across disciplines and years. Based on this indicator, we introduce a generalization of the h index suitable for comparing scientists working in different fields.

Fig. 1.

Normalized histogram of the number of articles P(c,c₀) published in 1999 and having received c citations. We plot P(c,c₀) for several scientific disciplines with different average number c₀ of citations per article.

Fig. 2.

Rescaled probability distribution c₀ P(c,c₀) of the relative indicator c_f = c/c₀, showing that the universal scaling holds for all scientific disciplines considered (see Table 1). The dashed line is a lognormal fit with σ² = 1.3.

Fig. 3.

We rank all articles according to the bare number of citations c and the relative indicator c_f. We then plot the percentage of articles of a particular discipline present in the top z% of the general ranking, for the rank based on the number of citations (A and C) and based on the relative indicator c_f (B and D). Different values of z (different graphs) lead to a very similar pattern of results. The average values and the standard deviations of the bin heights shown are also reported in Table 2. The numbers identify the disciplines as they are indicated in Table 1.

Fig. 4.

Rescaled probability distribution c₀ P(c,c₀) of the relative indicator c_f = c/c₀ for 3 disciplines (Hematology, Neuroimaging, and Nuclear Physics) for articles published in different years (1990, 1999, and 2004). Despite the natural variation of c₀ (c₀ grows as a function of the elapsed time), the universal scaling observed over different disciplines naturally holds also for articles published in different time periods. The dashed line is a lognormal fit with σ² = 1.3.

Fig. 5.

The same distributions as in Inset rescaled by the average number N₀ of publications per author in 1999 in the different disciplines. The dashed line is a power law with exponent −3.5. (Inset) Distributions of the number of articles, N, published by an author during 1999 in several disciplines.