Consequences of lotka law in the case of fractional counting of authorship and of 1st author counts

Abstract

In a recent paper, Rousseau [1] notes the fact that if we give weights of 1/m to each author in an m-authored paper, Lotka's law does not apply. However, he also notes that the function modeling the number of authors with weights j, j greater-than-or-equal-to 0, starts increasing from zero to about one and then decreases. In that paper, the present author is quoted as stating that this is not a breakdown of Lotka's law but merely a composition of two Lotka laws: one for j greater-than-or-equal-to 1 modeling papers per author, and one for 0 less-than-or-equal-to j less-than-or-equal-to 1 modeling authors per paper. The stochastic problem of how these two laws fit into each other was not solved in Rousseau's paper, however. This is done in this paper, where we will show that the weight-distribution function has indeed a maximum for the weight equal to one. We then study the same problem in the case where only the first author gets weight one and the others weight zero. We solve this case completely providing a formula for the probability of the weights. Also this function has a maximum for the weight equal to one. The main tool in these models is the technique of repeated convolution of continuous or discrete distribution functions.