Quantifying the unexpected: A scientific approach to Black Swans

Many natural and socioeconomic systems are characterized by power-law distributions that make the occurrence of extreme events not negligible. Such events are sometimes referred to as Black Swans, but a quantitative definition of a Black Swan is still lacking. Here, by leveraging on the properties of Zipf-Mandelbrot law, we investigate the relations between such extreme events and the dynamics of the upper cutoff of the inherent distribution. This approach permits a quantification of extreme events and allows to classify them as White, Gray, or Black Swans. Our criterion is in accordance with some previous findings but also allows us to spot new examples of Black Swans, such as Lionel Messi and the Turkish Airlines Flight 981 disaster. The systematic and quantitative methodology we developed allows a scientific and immediate categorization of rare events, also providing insight into the generative mechanism behind Black Swans.