Scientists seek mathematical insights for taming and explaining ‘dragon kings’

The Mother of All Dragon Kings sounds like a character from Game of Thrones.

But in fact, it’s a mix of wartime rhetoric and a technical scientific term. A “dragon king,” in the lingo of scientists who study complex systems, is an outlier. It’s an event, or effect, or activity, that’s literally off the scale — so big, so calamitous, that it doesn’t fit in the range of expected magnitudes. Huge earthquakes, sudden economic depressions, companies worth $600 billion are the dragon kings of the natural and socioeconomic worlds. In older times, prime dragon king examples included outsized political entities, like the Roman Empire, or epidemics like the Black Death.

“Dragon” is part of the term because of its extra­ordinary abilities, not on the same page with the rest of the world’s animals. “King” reflects a typical monarch’s extra­ordinary income, off the scale of incomes for the kingdom’s subjects. Kings and dragons play by different rules, and so make a shambles of ordinary attempts to explain, understand and predict the magnitude of “dragon king” phenomena.

But perhaps dragon kings aren’t truly unpredictable outliers. Maybe they emerge from complicated interactions in a social or natural system that just aren’t initially apparent.

All scientists agree that many phenomena and events are not randomly distributed in size. Many distributions — such as the populations of cities and the magnitudes of earthquakes — can instead be mathematically described by a power law. That’s simply an equation with an exponent (a “power”); such equations can be used to accurately plot a graph of a distribution of magnitudes. Power laws explain that any given country has many small towns, a fair number of midsized cities and a few pretty big urban centers. Same with earthquakes: There are a lot of small ones, too weak to notice; several that rattle a dish or two; a few that make headlines; and a very few that take out coastal cities with tsunamis.

These power laws are not mysteries. They arise naturally, from the strains that build up to trigger quakes or the patterns of interaction by which people congregate in communities. Sometimes, though, a quake or a city, or an empire or company, gets so big that it appears to defy the power-law pattern, earning dragon king status.

But maybe that’s because the system hasn’t been analyzed properly. In a new paper, Russian researchers argue that, at least for social systems, dragon kings can be mathematically tamed.

“We demonstrate that in social systems dragon-king is not a random outlier unexplainable by power-law statistics, but a natural effect,” Artemy Malkov and colleagues write in the November 1 issue of Physica A.

Take cities, for instance. A country’s largest city, often its capital, tends to look like a dragon king, lying outside the power-law scale for the country’s other cities. But capitals are special — they’re part of a network of the capital cities around the world; in that network, their populations are usually not out of line at all.

Another source of apparent dragon kings is interaction between the two biggest players in the game. When the second-biggest firm in an industry goes out of business or merges with the leader, a dragon king company then stands out from the rest. Such dragon kings are usually short-lived, as other firms grow closer in size to the leader.

In other situations, though, long-lasting dragon kings may emerge when factors conspire to allow an entity to overwhelm the whole system that it’s part of. Malkov and colleagues identify two such situations, one involving growing stress and the other marked by enhanced communication.

Avalanches, for instance, occur when growing stress (more snow, more gravity) overcomes the snow’s cohesion. Ordinarily, avalanches of all sizes occur, many small and some big, as power laws ordain. But if not enough smaller ones happen, the snowscape becomes “supercritical,” existing above the usual avalanche threshold. Then some trigger can send the whole mountainside into the valley, an avalanche off the usual scale.

In social systems, similar dragon king effects can occur when communication suddenly becomes global. In typical social systems, clusters form with strong ties between individual members. Communication links between clusters are weaker (more costly, or technologically difficult) than links within clusters. But when a powerful innovation (say, the Internet) allows communications to percolate more efficiently, small clusters can link to form supersized clusters, such as dragon king networks like Facebook.

Actually, Malkov and colleagues point out, growing stress and higher connectivity are not really distinct processes — they can work together. So combining the math describing both may be the key to predicting dragon king events. And that’s a much saner scenario than the one in which outlier events are utterly beyond understanding, as Didier Sornette, a pioneer in dragon king research, recently emphasized.

“In a world where catastrophes (in human-controlled activities, for instance) are pure surprises,” Sornette and colleague Guy Ouillon write online at, “no one can be responsible.”

SN Prime | September 17, 2012 | Vol. 2, No. 35

Tom Siegfried is a contributing correspondent. He was editor in chief of Science News from 2007 to 2012 and managing editor from 2014 to 2017.

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