A scientometrics law about co-authors and their ranking. The co-author core

Rather than "measuring" a scientist impact through the number of citations which his/her published work can have generated, isn't it more appropriate to consider his/her value through his/her scientific network performance illustrated by his/her co-author role, thus focussing on his/her joint publications, - and their impact through citations? Whence, on one hand, this paper very briefly examines bibliometric laws, like the $h$-index and subsequent debate about co-authorship effects, but on the other hand, proposes a measure of collaborative work through a new index. Based on data about the publication output of a specific research group, a new bibliometric law is found. Let a co-author $C$ have written $J$ (joint) publications with one or several colleagues. Rank all the co-authors of that individual according to their number of joint publications, giving a rank $r$ to each co-author, starting with $r=1$ for the most prolific. It is empirically found that a very simple relationship holds between the number of joint publications $J$ by coauthors and their rank of importance, i.e. $J \propto 1/r$. Thereafter, in the same spirit as for the Hirsch core, one can define a "co-author core", and introduce indices operating on an author. It is emphasized that the new index has a quite different (philosophical) perspective that the $h$-index. In the present case, one focusses on "relevant" persons rather than on "relevant" publications. Although the numerical discussion is based on one case, there is little doubt that the law can be verified in many other situations. Therefore, variants and generalizations could be later produced in order to quantify co-author roles, in a temporary or long lasting stable team(s), and lead to criteria about funding, career measurements or even induce career strategies.