I don't know that your characteristic has been explicitly studied before, nor would I be surprised if it has, but it fits into a more general setting as follows. 

The [directed graph distance][1] $d(a,b)$ from vertex $a$ to $b$ is the minimum length of a directed path from $a$ to $b$. (Of course, $d(a,b)\ne d(b,a)$ in general.)

Then the [distance from a vertex to a set of vertices][2] is also often defined, but in the directed case there are two versions:
$$d(a,B)=\min\{d(a,b):b\in B\},$$
$$d(B,a)=\min\{d(b,a):b\in B\},$$
where again typically $d(a,B)\ne d(B,a)$.

You're looking at the case where $B$ is the set of all sources or the set of all sinks.

And then, upon picking an edge $e=(v_0,v_1)$, the random variable 
$$X(e)=d(v_0,B)-d(v_1,B)$$ tells you how much closer to $B$ you got.

  [1]: https://en.wikipedia.org/wiki/Distance_(graph_theory)
  [2]: https://math.stackexchange.com/questions/2099527/distance-between-a-point-and-a-set-or-closure-of-it