In the graph where nodes are persons and edges follow authorship (that is, having co-authored a peer-reviewed scientific paper), the Erdős number is the number of steps you need to reach mathematician Paul Erdős.
Whenever you progress in a scientific career, your Erdős number necessarily decreases from an infinite value (whenever you published your first paper alone for instance) to a more respectable one. However, Paul Erdős passed away in 1996 and since its last publication, nobody on earth would get a number of 1. As time passes, the same will be true for number 2 and so on... Looking overall at the whole demographics, the counts of Erdős numbers is therefore stabilizing for the lowest numbers.
So my question is, is there any way of predicting the shape of this count as time would reach eternity?