Power law in the context of "Herbert A. Simon"

Vilfredo Federico Damaso Pareto (/pəˈreɪtoʊ/; Italian: [paˈreːto]; born Wilfried Fritz Pareto; 15 July 1848 – 19 August 1923) was an Italian polymath, whose areas of interest included sociology, civil engineering, economics, political science, and philosophy. He made several important contributions to economics, particularly in the study of income distribution and in the analysis of individuals' choices, and was one of the minds behind the Lausanne School of economics. He was also responsible for popularising the use of the term elite in social analysis and contributed to elite theory. He has been described as "one of the last Renaissance scholars. Trained in physics and mathematics, he became a polymath whose genius radiated into nearly all other major fields of knowledge."

He introduced the concept of Pareto efficiency and helped develop the field of microeconomics. He was also the first to claim that income follows a Pareto distribution, which is a power law probability distribution. The Pareto principle was named after him, and it was built on his observations that 80% of the wealth in Italy belonged to about 20% of the population. He also contributed to the fields of mathematics and sociology.

Rank–size distribution is the distribution of size by rank, in decreasing order of size. For example, if a data set consists of items of sizes 5, 100, 5, and 8, the rank-size distribution is 100, 8, 5, 5 (ranks 1 through 4). This is also known as the rank–frequency distribution, when the source data are from a frequency distribution. These are particularly of interest when the data vary significantly in scales, such as city size or word frequency. These distributions frequently follow a power law distribution, or less well-known ones such as a stretched exponential function or parabolic fractal distribution, at least approximately for certain ranges of ranks; see below.

A rank-size distribution is not a probability distribution or cumulative distribution function. Rather, it is a discrete form of a quantile function (inverse cumulative distribution) in reverse order, giving the size of the element at a given rank.

In statistics, economics, and econophysics, the king effect is the phenomenon in which the top one or two members of a ranked set show up as clear outliers. These top one or two members are unexpectedly large because they do not conform to the statistical distribution or rank-distribution which the remainder of the set obeys.

Distributions typically followed include the power-law distribution, that is a basis for the stretched exponential function, and parabolic fractal distribution.The King effect has been observed in the distribution of:

In statistics and business, a long tail of some distributions of numbers is the portion of the distribution having many occurrences far from the "head" or central part of the distribution. The distribution could involve popularities, random numbers of occurrences of events with various probabilities, etc. The term is often used loosely, with no definition or an arbitrary definition, but precise definitions are possible.

In statistics, the term long-tailed distribution has a narrow technical meaning, and is a subtype of heavy-tailed distribution. Intuitively, a distribution is (right) long-tailed if, for any fixed amount, when a quantity exceeds a high level, it almost certainly exceeds it by at least that amount: large quantities are probably even larger. Note that there is no sense of the "long tail" of a distribution, but only the property of a distribution being long-tailed.