Introduction

What do biological evolution, the physics of glassy materials, a superfluid near its phase transition, the brain, a slowly driven sandpile, and a fluvial network have in common? Little, it might seem. Nevertheless, they give rise to similar theoretical notions and statistical features (Bak, 1996). Central to their similitude is the power-law nature of the probability distribution that describes key geometric and topologic quantities, or the descriptors of time activity that yield signals (usually termed “ 1/f noise,” from the algebraic decay with exponent –f of the power spectrum) having components of all durations. The algebraic decay (the so-called fat tail) of the distribution ensures that extreme events will be much more likely than in any “regular” case, where the exponential decay of the probability of large events makes catastrophic events (the extremes) prohibitively unlikely. It is this feature that is responsible for the enhanced probability of individual, anomalously large events.

Power laws are also the essential ingredients of scaling arguments; that is, they describe a scale-free arrangement of the parts and the whole such that no characteristic scale is present in the growth of the structure and such a scale can arise only through the size of the system. The foregoing description postulates that in infinite domains, no embedded scale typical of the process is found. In perhaps simpler terms, this means that “events” (or values for the quantity under study) range from infinitely small to infinitely large without preference, and thus they scale with the system size.

A fundamental question among scientists in diverse disciplines is related to the dynamic reasons behind scale-free growth.