Let $E(n,t)$ be the number of mathematicians with finite positive Erdős number $n$ at time $t$. As old mathematicians leave, and new mathematicians come, how does $E(n,t)$ change over time ?
We can consider only the cooperations between two mathematicians, and assume that every year, the number of new articles and new mathematicians are both constant. We can fix the length of career for every mathematician. Feel free to add any other assumptions.
Some obvious facts are
- for every $n>0$, there exists a time $T(n)$, such that $E(n,t)$ is constant for $t>T(n)$.
- the average Erdős number will increase over time, but is it linear ?
It would be nice to find a model for the growth of network that can fit the real data. But since we are not far from Erdős ($T(1)\leq Y1996$), no data is available for large $t$, unfortunately. The distribution seems to be converging towards low Erdős numbers at this time.