Next week, I am off to Paris for a workshop, so I am writing my talk. Above is a plot of French cities. The x-axis is the log of the rank of the city, where the ranking of the city is by size, i.e., the first point (shown in pink) is for France’s largest city, Paris, at an a x value of log(1)=0, while the second point is France’s second largest city, Marseille, at a value log(2)=0.30, the third is Lyon at log(3)=0.48, etc. The y axis is the population of the city, raised to the power c = 0.18. This value is a fitting parameter, the value of 0.18 is the one that makes the data closest to a straight line — as you can see for this value of c the data falls on a pretty decent straight line.
Except of course that Paris is an outlier, far above the straight line that does a good job of fitting the size vs rank order of all France’s other cities. Outliers, i.e., objects (here Paris) that are apart from all the others, are often important. Paris is France’s one and only capital city, and people interested in regulating a stock market don’t need to worry about small day-to-day fluctuations but do need to be worry about the rare but devastating price drops that occur in stock market crashes.
Because of this, models have been developed for these extreme outliers. The above plot is from work of Laherrère and Sornette (European Physical Journal B 2 p525 (1998)). Didier Sornette refers to these extreme events as dragon-kings, and is trying to see when they can be predicted, while Nassim Nicholas Taleb calls them black swans, and is less optimistic about predicting them. It may turn out that some extreme events are more easily predicted than others and so some extreme events may be better understood as dragon-kings and others as black swans*.
Which brings me to the anti-HIV drug Ritonavir. In 1998 Ritonavir had been in production for two years and many thousands if not millions of capsules had been manufactured without a problem. But then crystals of a new type (called a polymorph) appeared and started appearing in many capsules, rendering them useless. Ultimately, the appearance of this new form of crystalline Ritonavir caused the drug to have to be completely reformulated at considerable expense.
This annoyed a lot of people, as you would expect, so a lot of work has gone into trying to understand what happened, but we still have only a very limited idea of what is going on. Work of Bauer and coworkers found that the new type of Ritonavir crystal seemingly only starts to form (i.e., nucleates) with great difficulty, unless there is a seed of it already present.
It is possible that this new form may have originally only nucleated once, and then tiny seeds of this crystal spread through the manufactured Ritonavir. If so, this original nucleation event would have been an outlier analogous to Paris in the sense that it was an anomaly: batch after batch of Ritonavir was made and this new form did not appear, and then suddenly after hundreds of batches have been prepared, the new form pops up from nowhere, a black swan, or maybe a dragon-king.
* Maybe outliers intermediate between the two should be called black dragons, or swan-kings.