On S-curves and tipping points

In his discussion in this journal of Kurzweil's The Singularity is Near, Modis criticizes Kurzweil's loose characterization of the “knee” of a growth curve. Likewise, the “tipping points” described by Gladwell (The Tipping Point) are clearly relevant to forecasting systems, but Gladwell did not mathematically identify such points. Both concepts refer to a point on the curve where growth accelerates dramatically and sustains itself. What can be said in a rigorous way about knees and tipping points in growth systems?The answer has to do with the number of parameters of the growth curve, and the (probabilistic) model underlying the growth curve. Using probability theory and computational experiments, this paper clarifies these points for the logistic and Bass curves (identifying an unambiguous tipping point for the latter), and explores the merits of a 3-parameter model of innovation adoption. It concludes that if forecasters are to deal scientifically with the now-established management notion of “tipping points,” a 3-parameter model is needed. The paper also resolves four minor but annoying paradoxes in the growth curve literature.