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Understanding the occurrence of rare events

The rich can get richer more often than traditional statistics would predict. This is an example of a statistical distribution with a “heavy tail.” These patterns are common in the real world, including everything from “hundred-year floods” to stock market bubbles and crashes. Now scientists propose a basic principle underlying these systems, which could help researchers devise better models for these events, report findings detailed in the Proceedings of the National Academy of Sciences.

Heavy tails, also known as fat tails or power-law tails, suggest that events one might think of as rare outliers may actually be common than one might suspect.

“Some recent books — ‘The Black Swan’ by Taleb, for example — note that the reason that economists mis-predict stock market bubbles and crashes is because we don’t have math models that properly treat risk,” says statistical physicist Ken Dill, director at the Laufer Center for Physical and Quantitative Biology at Stony Brook University in New York. “The ‘Black Swan’ idea is about rare events. Rare events are captured in ‘fat-tailed’ distributions, not Gaussians” — what are commonly known as bell-curve distributions.

Heavy tails are not only seen in patterns of fluctuations in financial markets, but also in how people generally prefer larger cities to smaller ones, among many other things. While Gaussians are often found in physics, heavy-tailed distributions “often occur in social physics where people or games are involved,” Dill says.

Dill and his colleagues explored whether an underlying principle existed for heavy tails. They discovered this principle may be that of maximum entropy, which is used in statistical physics to select the best scenario that matches data from all possible scenarios meeting specific constraints.

The framework the researchers devised involves a particle that pays some form of cost, such as energy, to join a given cluster of particles. This scenario could in principle model how people populate cities, or links are added to websites, or papers accumulate citations.

This framework assumes the cost of joining is subject to an economy of scale — the price for a particle to join a cluster of particles goes down the larger the cluster is. In these situations, maximizing the entropy, the measure of how disorderly the system is, predicted heavy-tailed distributions that were excellent matches with 13 real-world data sets, ranging from friendship links in social networks, to protein–protein interactions, to the severity of terrorist attacks.

This model could help better model scenarios where outliers are more common than expected.

“Now it remains to apply the general framework to the many specific problems people are interested in: stock market bubbles, why the populations of the world’s cities have power-law distributions, why income is distributed according to power laws — why do the rich get richer? — the distributions of links in the Internet or in terrorist organizations, and so on,” Dill says.

Categories: Applied Mathematics
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