A general expression based on the concepts of the progressive nucleation mechanism is proposed in the form to describe the growth behavior of items in an individual system and a collective of systems. In the above relation, α(t) is the ratio of items N(t) at time t to the maximum number C of possible items for the system, Θ is the corresponding time constant and q is the exponent. The above relation is then used to analyze: (1) the growth behavior of cumulative number N(t) of papers published by individual authors and cumulative citations L(t) of N(t) papers of an author as a function of citation duration t, and (2) the relationship between cumulative citations L(t) of papers and cumulative number N(t) of papers. The proposed approach predicts that: (1) the fraction of items produced by successive systems is additive, (2) the cumulative fraction αsum(t) of maximum number of sites is the sum of contributions of fractions of maximum number of items produced by different systems, and (3) the values of time constants Θ and exponent q increase with the addition of fraction of items produced by subsequent systems, but their values are the lowest for individual systems. The approach is applied to explain the growth behavior of cumulative N(t) papers and L(t) citations of four selected Polish professors.
The basic concepts and equations of the progressive nucleation mechanism (PNM) are presented first for the growth and decay of items. The mechanism is then applied to describe the cumulative citations L and citations ΔL per year of the individual most-cited papers i of four selected Polish professors as a function of citation duration t. It was found that the PNM satisfactorily describes the time dependence of cumulative citations L of the papers published by different authors with sufficiently high citations ΔL, as represented by the highest yearly citations ΔLmax during the entire citation period t (normal citation behavior). The citation period for these papers is less than 15 years and it is even 6–8 years in several cases. However, for papers with citation periods exceeding about 15 years, the growth behavior of citations does not follow the PNM in the entire citation period (anomalous citation behavior), and there are regions of citations in which the citation data may be described by the PNM. Normal and anomalous citation behaviors are attributed, respectively, to the occurrence and nonoccurrence of stationary nucleation of citations for the papers. The PNM also explains the growth and decay of citations ΔL per year of papers exhibiting normal citation behavior.
The nature of the empirical proportionality constant A in the relation L = Ah2 between total number of citations L of the publication output of an author and his/her Hirsch index h is analyzed using data of the publication output and citations for six scientists elected to the membership of the Royal Society in 2006 and 199 professors working in different institutions in Poland. The main problem with the h index of different authors calculated by using the above relation is that it underestimates the ranking of scientists publishing papers receiving very high citations and results in high values of A. It was found that the value of the Hirsch constant A for different scientists is associated with the discreteness of h and is related to the tapered Hirsch index hT by A1/2 ≈ 1.21hT. To overcome the drawback of a wide range of A associated with the discreteness of h for different authors, a simple index, the radius R of circular citation area, defined as R = (L/π)1/2 ≈ h, is suggested. This circular citation area radius R is easy to calculate and improves the ranking of scientists publishing high-impact papers. Finally, after introducing the concept of citation acceleration a = L/t2 = π(R/t)2 (t is publication duration of a scientist), some general features of citations of publication output of Polish professors are described in terms of their citability. Analysis of the data of Polish professors in terms of citation acceleration a shows that: (1) the citability of the papers of a majority of physics and chemistry professors is much higher than that of technical sciences professors, and (2) increasing fraction of conference papers as well as non-English papers and engagement in administrative functions of professors result in decreasing citability of their overall publication output.
It is shown that the age-independent index based on h-type index per decade, called hereafter an α index instead of the a index, suggested by Kosmulski (Journal of Informetrics 3, 341–347, ) and Abt (Scientometrics ) is related to the square-root of the ratio of citation acceleration a to the Hirsch constant A.