Given a weighted directed graph $G=(V,E, w)$, suppose we generate a new graph $G'=(V,E,w')$ with the same vertices and edges, but now letting the weight of edge $(i,j)$ be
an exponential random variable with mean $w_{ij}$. My question is: what is the
expected diameter of $G'$?

Why I'm interested in this: I was intrigued by the observation that the expected diameter of $G'$ can be
quite different from the diameter of $G$. Indeed, consider the following example: define $G$ by taking the complete graph $K_n$, picking an arbitrary vertex $a$, and assigning weight $n$  to any edge incident on $a$, and weight $1$ to 
every other edge. Then, the diameter of $G$ is $n$. On the other hand, the expected diameter
of $G'$ is O(1) since we can expect one of the edges incident on $a$ to have small weight.