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Balls and Bins Models with Feedback

We examine generalizations of the classical balls and bins models,
where the probability a ball lands in a bin is proportional to the
number of balls already in the bin raised to some exponent p. Such
systems exhibit positive or negative feedback, depending on the
exponent p, with a phase transition occurring at p = 1. Similar models
have proven useful in economics and chemistry; for example, systems
with positive feedback (p > 1) tend naturally toward monopoly. We
provide several results and useful heuristics for these models,
including showing a bound on the time to achieve monopoly with high
probability.