rankfrequency

Rankfrequency refers to the relationship between the rank of an item when items are ordered by decreasing frequency and the actual frequency of that item. In a rank-frequency distribution, each rank r is assigned a frequency f(r), with f(1) being the most frequent item, f(2) the second most frequent, and so on. This pattern often reveals that a small number of items dominate the total count while many items occur rarely.

A central model is Zipf's law, which states that the frequency of the item at rank r is proportional to 1 divided by r raised to a power s: f(r) = C r^(-s), where C is a normalizing constant and s is typically near 1 in natural language data. On a log-log plot, a Zipfian distribution appears as a straight line with slope approximately -s. When expressed as probabilities, p(r) = f(r) / sum_i f(i), the rank-frequency relation describes the likelihood of observing items at different ranks.

Applications of rankfrequency include word-frequency analysis in languages, city-size distributions, web traffic, and various biological and social systems that exhibit heavy-tailed behavior. The concept helps explain why a few items dominate observations and informs models of growth and preferential attachment.

Limitations and extensions: real data often deviate from perfect power-law behavior, especially at the extreme ranks. Factors such as sampling bias, finite vocabulary, and context effects can affect the observed distribution. Extensions include Zipf-Mandelbrot and other generalized power-law models to better fit empirical data. See also Zipf's law and Pareto distributions.