Given the fact that $s$-$t$ path enumeration is a #P-complete problem, could there be efficient methods that compute (or at least approximate) the average length of $s$-$t$ path without enumerating them? What if paths are allowed to revisit vertices?
Relevant results on special graphs could also be helpful.
calculating/estimating/approximating the average path length has been studied for some random graph models including the Erdos-Renyi model and the Barabasi-Albert scale free networks, and also the Strogatz small world graphs which may be suitable as approximations for your graphs. [it would be better if you could narrow down/detail some nature/characteristics of the graphs you're studying.]
