page_title: Dragon king theory
section_title: Mechanisms
hierarchical_section_title: Dragon king theory / Mechanisms
caption_attribution_description: Attractor and probability density function of trajectory distances in a chaotic dynamical system with a riddled basin as discussed in Cavalcante, Hugo LD de S., et al. "Predictability and suppression of extreme events in a chaotic system." Physical review letters 111.19 (2013): 198701.
context_page_description: Dragon king is a double metaphor for an event that is both extremely large in size or impact and born of unique origins relative to its peers. DK events are generated by or correspond to mechanisms such as positive feedback, tipping points, bifurcations, and phase transitions, that tend to occur in nonlinear and complex systems, and serve to amplify DK events to extreme levels. By understanding and monitoring these dynamics, some predictability of such events may be obtained.
  The dragon king theory was developed by Didier Sornette, who hypothesizes that many crises are in fact DKs rather than black swans—i.e., they may be predictable to some degree. Given the importance of crises to the long-term organization of a variety of systems, the DK theory urges that special attention be given to the study and monitoring of extremes, and that a dynamic view be taken. From a scientific viewpoint, such extremes are interesting because they may reveal underlying, often hidden, organizing principles.
context_section_description: Physically speaking, dragon kings may be associated with the regime changes, bifurcations, and tipping points of complex out-of-equilibrium systems. For instance, the catastrophe (fold bifurcation) of the global ecology illustrated in the figure could be considered to be a dragon king: Many observers would be surprised by such a dramatic change of state. However, it is well known that in dynamic systems, there are many precursors as the system approaches the catastrophe.
  Positive feedback is also a mechanism that can spawn dragon kings. For instance, in a stampede the number of cattle running increases the level of panic which causes more cattle to run, and so on. In human dynamics such herding and mob behavior has also been observed in crowds, stock markets, and so on (see herd behavior). 
  Dragon kings are also caused by attractor bubbling in coupled oscillator systems. Attractor bubbling is a generic behavior appearing in networks of coupled oscillators where the system typically orbits in an invariant manifold with a chaotic attractor (where the peak trajectories are low), but is intermittently pushed (by noise) into a region where orbits are locally repelled from the invariant manifold (where the peak trajectories are large). These excursions form the dragon kings, as illustrated in the figure. It is claimed that such models can describe many real phenomena such as earthquakes, brain activity, etc. A block and spring mechanical model, considered as a model of geological faults and their earthquake dynamics, produced a similar distribution.
  It could also be the case that dragon kings are created as a result of system control or intervention. That is, trying to suppress the release of stress or death in dynamic complex systems may lead to an accumulation of stress or a maturation towards instability. For instance, brush/forest fires are a natural occurrence in many areas. Such fires are inconvenient and thus we may wish that they are diligently extinguished. This leads to long periods without inconvenient fires, however, in the absence of fires, dead wood accumulates. Once this accumulation reaches a critical point, and a fire starts, the fire becomes so large that it cannot be controlled—a singular event that could be considered to be a dragon king. Other policies, such as doing nothing (allowing for small fires to occur naturally), or performing strategic controlled burning, would avoid enormous fires by allowing for frequent small ones. Another example is monetary policy. Quantitative easing programs and low interest rate policies are common, with the intention of avoiding recessions, promoting growth, etc. However, such programs build instability by increasing income inequality, keeping weak firms alive, and inflating asset bubbles. Ultimately such policies, aimed at smoothing out economic fluctuations, will enable an enormous correction—a dragon king.
caption: Left: Illustration of the system trajectory in the vicinity of a bubbling event. Right: empirical pdf (histogram) of peak heights in trajectories in double logarithmic scale[15]
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