I am interested in any known results/empirical studies done on the average height of a poset with $N$ elements. Obviously this would depend on how that poset relation was randomly defined, however, at this point, I'll take any reasonable result on the topic regardless of how the poset was formed.

In my case $N$ is a very large number (billions of elements), so I am interested whether the height $h \ll N$ asymptotically.