King effect

The king effect is the name given by Jean Laherrère and Didier Sornette^[2] to the phenomenon in natural distributions where the top one or two members of a ranked set are clear outliers. These top one or two members are unexpectedly large and do not conform to the statistical distribution or rank-distribution which the remainder of the set follows.^[3]

Distributions typically followed include the power-law distribution,^[4] that is a basis for the stretched exponential function,^[1][5] and parabolic fractal distribution. Laherrere and Sornette noted the King effect in the distributions of:

• French city sizes (where the point representing Paris is the "king", failing to conform to the stretched exponential^[1]), and similarly for other countries with a primate city, such as the United Kingdom (London), and the extreme case of Bangkok (see list of cities in Thailand).
• Country populations (where only the points representing China and India fail to fit a stretched exponential^[1]).

Note, however, that the king effect is not limited to outliers with a positive evaluation attached to their rank: for rankings on an undesirable attribute, there may exist a pauper effect, with a similar detachment of extremely ranked data points from the reasonably distributed portion of the data set.^{[citation needed]}