Suppose we have a simple directed graph with $n$ nodes and $m$ edges, and we label each edge from $1$ to $m$ (with distinct labels). Define the weighted "length" of a directed path to be the maximum of all edge labels on that path (or $0$ for a trivial path), and define the "distance" $d(a, b)$ from node $a$ to $b$ to be the minimum weight of all paths from $a$ to $b$ (or $\infty$ if $b$ is not reachable from $a$).
Define a triple $a, b, c$ of nodes to be defective if $d(a, c) < d(a, b) < \infty$ and $d(b, c) < d(a, b)$.
I have two questions:
A) What is the maximum number of defective triples possible?
B) What is the average number of defective triples if edge labels are assigned at random?
Unfortunately, I haven't been able to make much progress on either one, so I was hoping other people might have insight into the problem.