The invariances of power law size distributions.

Abstract

Size varies. Small things are typically more frequent than large things. The logarithmof frequency often declines linearly with the logarithm of size. That power law relationforms one of the common patterns of nature. Why does the complexity of nature reduceto such a simple pattern? Why do things as different as tree size and enzymerate follow similarly simple patterns? Here I analyze such patterns by their invariantproperties. For example, a common pattern should not change when adding a constantvalue to all observations. That shift is essentially the renumbering of the pointson a ruler without changing the metric information provided by the ruler. A ruler isshift invariant only when its scale is properly calibrated to the pattern being measured.Stretch invariance corresponds to the conservation of the total amount of something,such as the total biomass and consequently the average size. Rotational invariancecorresponds to pattern that does not depend on the order in which underlying processesoccur, for example, a scale that additively combines the component processesleading to observed values. I use tree size as an example to illustrate how the keyinvariances shape pattern. A simple interpretation of common pattern follows. Thatsimple interpretation connects the normal distribution to a wide variety of other commonpatterns through the transformations of scale set by the fundamental invariances.