Power-law distributions from exponential processes: an explanation for the occurrence of long-tailed distributions in biology and elsewhere

A possible explanation for the frequent occurrence of power-law distributions in biology and elsewhere comes from an analysis of the interplay between random time evolution and random observation or killing time. If the system population or its topological parameters grow exponentially with time, and observations on the system correspond to stopping the evolution at an exponentially distributed random time, power-law behaviour in one or both tails of the distribution of observed quantities may result. We pursue this theme for two specific models. The first model is a randomly killed birth-and-death process, with applications to the numbers of genes per gene family and proteins per protein family, the distribution of taxonomic elements in live taxa, and other areas. The second model is a randomly growing network, with the state of a random node (which thus has a random age) observed. For the growing network, we consider both tree-like networks, appropriate in biological applications, and networks in which closed loops can appear, which model communication networks and networks of human sexual interactions.