Let's consider a random directed acyclic graph (DAG) with $n$ vertices and $m$ edges. What is the expected length of the longest path in this graph? Is there a function $f(n,m)$ that can approximate this value? Or some kind of $O(f(n, m))$ expression?
I have run some calculations and e.g. for $n = 300, m = 400$ average length of the longest path was something around 8.2. For $n = 500, m = 1200$ it was around 13.