This is not really an answer and rather an extended comment. But there is a lot of serious research on network systems that formalize or are related to collaboration networks of authors. A good starting point may be this webpage on research on collaboration in research by the Erdős Number Project. I'm no expert, but the article seems to summarize the history very well and provides lots of links to actual serious research. For example, the abstract of one article mentioned there reads:
A.L. Barabasi, H. Jeong, Z. Neda, E. Ravasz, A. Schubert, T. Vicsek, Evolution of the social network of scientific collaborations, Physica A 311, (2002), 590-614.
The co-authorship network of scientists represents a prototype of complex evolving networks. By mapping the electronic database containing all relevant journals in mathematics and neuro-science for an eight-year period (1991-98), we infer the dynamic and the structural mechanisms that govern the evolution and topology of this complex system. First, empirical measurements allow us to uncover the topological measures that characterize the network at a given moment, as well as the time evolution of these quantities. The results indicate that the network is scale-free, and that the network evolution is governed by preferential attachment, affecting both internal and external links. However, in contrast with most model predictions the average degree increases in time, and the node separation decreases. Second, we propose a simple model that captures the network's time evolution. Third, numerical simulations are used to uncover the behavior of quantities that could not be predicted analytically.
The article's preprint is available here on arXiv.
The following is a review article on serious research on network systems:
M. E. J. Newman, The structure and function of complex networks, SIAM Review 45, (2003) 167-256.
(And its shorter version on Proc. Natl. Acad. Sci. USA 98, (2001) 404-409.)
I wish I could answer your questions by writing an interesting mathematical post. Alas, all I know is that it is an interesting and serious research topic. Since you got downvoted right away, hopefully someone familiar with this kind of research will post a nice answer before this thread gets closed...