Reinforcement-driven spread of innovations and fads

We investigate how social reinforcement drives the spread of permanent innovations and transient fads. We account for social reinforcement by endowing each individual with M + 1 possible awareness states 0, 1, 2,..., M, with state M corresponding to adopting an innovation. An individual with awareness k < M increases to k + 1 by interacting with an adopter. Starting with a single adopter, the time for an initially unaware population that consists of N individuals to adopt an innovation grows as lnN for M = 1, and as N1−1/M for M > 1. When individuals can abandon the innovation at rate λ, the population fraction that remains clueless about the fad undergoes a phase transition at a critical rate λc; this transition is second order for M = 1 and first order for M > 1, with macroscopic fluctuations accompanying the latter. The time for the fad to disappear has an intriguing non-monotonic dependence on λ