Zipf's Law — an empirical observation, not a theoretical law — states that the frequency of a word is roughly inversely proportional to its rank in the frequency table. For instance, one statistics-gathering exercise found that "the" is the most frequently-occurring word, and all by itself accounts for nearly 7% of all word occurrences (69971 out of slightly over 1 million). The second-place word "of" accounts for slightly over 3.5% of words (36411 occurrences) — half as frequent — followed by "and" (28852) — roughly a third of the frequency. Zipf's Law is a PowerLaw.

Zipf's Law also appears to apply in other situations, such as the value of each member accessible to you via a network. If this holds in a given case, then the value of the network is <tt>1/1 + 1/2 + … + 1/(n-1)</tt>, or <tt>O(log n)</tt>, leading to the conclusion that the value of an entire network grows as <tt>O(n log n)</tt>. See [http://www.spectrum.ieee.org/jul06/4109 Metcalfe's Law is Wrong] (by Briscoe, Odlyzko and Tilly) for an extended version of this argument.

== References ==

[1] Lada A. Adamic. [http://ginger.hpl.hp.com/shl/papers/ranking/ranking.html  Zipf, Power-laws, and Pareto - a ranking tutorial].

== Links ==

* http://planetmath.org/encyclopedia/ZipfsLaw.html
* http://en.wikipedia.org/wiki/Zipf%27s_law

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Do other PowerLaw""s also give <tt>O(n log n)</tt> as the value of a network? I can't remember my series summation facts. -- ChrisPurcell