So question, does the fact that they can achieve a ranking with 90% of unreciprocated edges directed from lower to higher status people WHEN THE DATA IS RANDOMLY GENERATED worry you... I mean, its good that they can increase this to 98% when the use the ME, but it seems like they have a lot of flexibility. I hypothesize that in schools where people fill out their survey more thoroughly (fewer people with very few edges) that they end up with more edges directed the wrong way. But that the rankings are in some sense truer. Do you think I'm right?
Congratulations, you have rediscovered the phenomenon of overfitting! Everything you said is correct and important. In general, too-flexible statistical models are bad news.
The only way to really know whether or not you're doing it is to have some data held out for testing/cross-validation. For example, you could erase a random 10% of the edges, do their procedure on the remaining 90%, and then see how many rank violations you get in the invisible 10% of edges.