Universality of Zipf’s law

After a long enough break, we are back to feature a 2010 paper by CSLab fellows. We are talking about former member Bernat Corominas-Murtra and group leader Ricard Solé, and their long-lasting effort to clear up Zipf’s law. As you may know, Zipf’s law is one of the most famous statistical regularities that we can find everywhere: from the size of cities within a country to the frequency of words in a text. It’s enchantments fascinate many, as it appears linked to many odd phenomena that still demand an explanation; and its presence everywhere led many others to think that it is meaningless.

Universality of Zipf’s law is a very elegant analytic tour de force that explains how Zipf’s law emerges as the only limiting solution between complete randomness and trivial complexity. This emergence happens in boundless situations, where a system is allowed to occupy random states of an ever-growing ensemble. In this situation, only the Zipf’s law retains some meaningful complexity! Any other solution — and here we speak of statistical regularities of some observable — will either be completely random or trivial.

This intends to shed some light about those very diverse phenomena that display Zipf’s law and whose nature is likely open-ended. In a following paper our colleagues, together with Jordi Fortuny, applied a refined version of the reasoning in Universality of… to an actual system: the evolution of communication, in which taking into account the path dependence of evolution is crucial.

There is something dark and fascinating about Zipf’s law that seems to invoke unbounded systems, path dependence (somehow implying non ergodicity), criticality, and many others of our favorite pets is complex systems. What’s going to be the next breakthrough here?

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