**Originally shared by Rob Jongschaap**

**Mathematical Model Reveals the Patterns of How Innovations Arise**

'... The rate at which innovations appear and disappear has been carefully measured. It follows a set of well-characterized patterns that scientists observe in many different circumstances. And yet, nobody has been able to explain how this pattern arises or why it governs innovation.Today, all that changes thanks to the work of Vittorio Loreto at Sapienza University of Rome in Italy and a few pals, who have created the first mathematical model that accurately reproduces the patterns that innovations follow. The work opens the way to a new approach to the study of innovation, of what is possible and how this follows from what already exists.

The notion that innovation arises from the interplay between the actual and the possible was first formalized by the complexity theorist Stuart Kauffmann. In 2002, Kauffmann introduced the idea of the “adjacent possible” as a way of thinking about biological evolution.

The adjacent possible is all those things—ideas, words, songs, molecules, genomes, technologies and so on—that are one step away from what actually exists. It connects the actual realization of a particular phenomenon and the space of unexplored possibilities.

[...]

Nevertheless, even with all this complexity, innovation seems to follow predictable and easily measured patterns that have become known as “laws” because of their ubiquity. One of these is Heaps’ law, which states that the number of new things increases at a rate that is sublinear. In other words, it is governed by a power law of the form V(n) = knβ where β is between 0 and 1.

Words are often thought of as a kind of innovation, and language is constantly evolving as new words appear and old words die out.

This evolution follows Heaps’ law. Given a corpus of words of size n, the number of distinct words V(n) is proportional to n raised to the β power. In collections of real words, β turns out to be between 0.4 and 0.6.

Another well-known statistical pattern in innovation is Zipf’s law, which describes how the frequency of an innovation is related to its popularity. For example, in a corpus of words, the most frequent word occurs about twice as often as the second most frequent word, three times as frequently as the third most frequent word, and so on. In English, the most frequent word is “the” which accounts for about 7 percent of all words, followed by “of” which accounts for about 3.5 percent of all words, followed by “and,” and so on.

This frequency distribution is Zipf’s law and it crops up in a wide range of circumstances, such as the way edits appear on Wikipedia, how we listen to new songs online, and so on.

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https://www.technologyreview.com/s/603366/mathematical-model-reveals-the-patterns-of-how-innovations-arise/

https://www.technologyreview.com/s/603366/mathematical-model-reveals-the-patterns-of-how-innovations-arise/

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