Statistical distributions and self-organizing phenomena: what conclusions should be drawn?

Some salient properties of the inverse power law distribution, the exponential distribution, catastrophe distributions, and the relationships among them were explored and compared. Self-organizing events may display any of these distributions. Catastrophe functions and their distributions do not display fractional (fractal) dimensions. Thus it is possible to have self-organization without the fractal. An empirical example from leadership emergence research illustrated a situation where a power law distribution provided a poor characterization of the data, but a swallowtail catastrophe model did so quite well. The results call into question some simplistic assumptions about the relationships among fractals, inverse power laws, self-organization and so-called pink noise.