  Reinforcement schemes are a class of non-Markovian stochastic processes.
Their non-Markovian nature allows them to model some kind of memory of the
past. One subclass of such models are those in which the past is exponentially
discounted or forgotten. Often, models in this subclass have the property of
becoming trapped with probability~1 in some degenerate state. While previous
work has concentrated on such limit results, we concentrate here on a contrary
effect, namely that the time to become trapped may increase exponentially in
1/x as the discount rate, 1-x, approaches~1. As a result, the time to become
trapped may easily exceed the lifetime of the simulation or of the physical
data being modeled. In such a case, the quasi-stationary behavior is more
germane. We apply our results to a model of social network formation based on
ternary (three-person) interactions with uniform positive reinforcement.
