in a power law). For example, Gibrat's law about proportional growth processes produce distributions that are lognormal, although their log–log plots look linear over a limited range. An explanation of this is that although the logarithm of the lognormal density function is quadratic in , yielding a "bowed" shape in a log–log plot, if the quadratic term is small relative to the linear term then the result can appear almost linear, and the lognormal behavior is only visible when the quadratic term dominates, which may require significantly more data. Therefore, a log–log plot that is slightly "bowed" downwards can reflect a log-normal distribution – not a power law. In general, many alternative functional forms can appear to follow a power-law form for some extent. proposed plotting the empirical cumulative distribution function in the log-log domain and claimed that a candidate power-law should cover at least two orders of magnitude. Also, researchers usually have to face the problem of deciding whether or not a real-world probability distribution follows a power law. As a solution to this problem, Diaz proposed a graphical methodology based on random samples that allow visually discerning between different types of tail behavior. This methodology uses bundles of residual quantile functions, also called percentile residual life functions, which characterize many different types of distribution tails, including both heavy and non-heavy tails. However, claimed the need for both a statistical and a theoretical background in order to support a power-law in the underlying mechanism driving the data generating process. One method to validate a power-law relation tests many orthogonal predictions of a particular generative mechanism against data. Simply fitting a power-law relation to a particular kind of data is not considered a rational approach. As such, the validation of power-law claims remains a very active field of research in many areas of modern science. See also Fat-tailed distribution Heavy-tailed distributions Hyperbolic growth Lévy flight Long tail Pareto distribution Power-law fluid Simon model Stable distribution Stevens's power law References Notes Bibliography External links Zipf, Power-laws, and Pareto – a ranking tutorial Stream Morphometry and Horton's Laws "How the Finance Gurus Get Risk All Wrong" by Benoit Mandelbrot & Nassim Nicholas Taleb. Fortune, July 11, 2005. "Million-dollar Murray": power-law distributions in homelessness and other social problems; by Malcolm Gladwell. The New Yorker, February 13, 2006. Benoit Mandelbrot & Richard Hudson: The Misbehaviour of Markets (2004) Philip Ball: Critical Mass: How one