(Pokemon Black only)
So, according to Serebii, the value which keeps the trainers there at Black City is some number, where when A Day Passes: the value decreases by 5 Entering the Area: increases by 3 Defeating Trainers: increases by 7* Talking to those in White Forest increases by 10
(+) Here's the website by the way: > http://serebii.net/blackwhite/blackcitywhiteforest.shtml
The points only raise for the individual interacted with. Once you battle a trainer, their point goes up by one.
source