Nonuniversal power law scaling in the probability distribution of scientific citations

Abstract

We develop a model for the distribution of scientific citations. The model involves a dual mechanism: in the direct mechanism, the author of a new paper finds an old paper A and cites it. In the indirect mechanism, the author of a new paper finds an old paper A only via the reference list of a newer intermediary paper B, which has previously cited A. By comparison to citation databases, we find that papers having few citations are cited mainly by the direct mechanism. Papers already having many citations ("classics") are cited mainly by the indirect mechanism. The indirect mechanism gives a power-law tail. The "tipping point" at which a paper becomes a classic is about 25 citations for papers published in the Institute for Scientific Information (ISI) Web of Science database in 1981, 31 for Physical Review D papers published from 1975-1994, and 37 for all publications from a list of high h-index chemists assembled in 2007. The power-law exponent is not universal. Individuals who are highly cited have a systematically smaller exponent than individuals who are less cited.C ommonly observed in nature and in the social sciences are probability distribution functions that appear to involve dual underlying mechanisms, with a "tipping point" between them. Examples of such probability distributions include the distributions of city sizes (1, 2); fluctuations in stock market indices (3, 4); U.S. firm sizes (5, 6); degrees of Internet nodes (7, 8); numbers of followers of religions (8); gamma-ray intensities of solar flares (9); sightings of bird species (8); and citations of scientific papers (10-13). In these situations, a distribution pðkÞ may have exponential behavior for small k and a power-law tail for large k. Here we develop a generative model for one such dual-mechanism process, scientific citations, for which databases are large and readily available. Here, k represents the number of citations a paper receives, ranging from zero to hundreds or, sometimes, thousands. pðkÞ is the distribution of the relative numbers of such citations, taken over a database of papers.There have been several important studies of power-law tails of distributions, including those involving scientific citations. Price noted that highly cited scientific papers accumulate additional citations more quickly than papers that have fewer citations (14). He called this "cumulative advantage" (CA): the probability that a paper receives a citation is proportional to the number of citations it already contains. Price showed that this rule asymptotically gives a power law for large k. Power-law tails have been widely explored in various contexts and under different names -"the rich get richer," the Yule process (15, 16), the Matthew effect (17), or preferential attachment (18). Barabási and Albert noted that networks, such as the World Wide Web, often have power-law distributions of vertex connectivities, called "scalefree" behavior (18). Their model, called preferential attachment, leads to a fixed power-law exponent of −3. Because many propertie...

The evaluation of citation distributions

“…Recent contributions using a social network approach by, for example, Dorogovstev and Mendes (2001); Jackson and Rogers (2007), and Peterson et al (2010) constitute a formidable first attempt in this direction. The similarities that have been documented about citation distributions seem to indicate that a plausible working hypothesis is that the distribution of talent to achieve an international impact is, certainly skewed, but similar in all sciences.…”

Stochastic Dynamical Model of a Growing Citation Network Based on a Self-Exciting Point Process

“…We find that the cumulative citation distribution is neither stable nor stationary but develops in time. Immediately after publication the spread of initial conditions (journal circulation numbers) yields convex cumulative distribution of citations that can be fitted equally well by the (discrete) power-law [20][21][22] or log-normal [10,23,24,31] functions. Thereafter, citation dynamics of most papers is dominated by the first term in Eq.6 in such a way that the citation history of papers that managed to garner less than 50-70 citations is completed after 10-15 years.…”

Modeling nonuniversal citation distributions: the role of scientific journals

“…Here, the degree k represents the number of citations a paper receives and p(k) is the distribution of the relative numbers of such citations. To explain this feature, several mechanisms have been proposed based on the cumulative advantage in different contexts [22,23,24]. For instance, Peterson et al [23] developed a direct-indirect citing mechanism to fit the citation distribution and analyzed the tipping-point transition.…”

“…To explain this feature, several mechanisms have been proposed based on the cumulative advantage in different contexts [22,23,24]. For instance, Peterson et al [23] developed a direct-indirect citing mechanism to fit the citation distribution and analyzed the tipping-point transition. Eom and Fortunato [24] employed the linear preferential attachment with initial attractiveness to reproduce empirical distributions and account for the presence of citation bursts.…”