A dragon king event is one that is both extremely large in size or effect (a "king") and born of unique origins (a "dragon") relative to other events from the same system. 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 dragon king events to extreme levels. By understanding and monitoring these dynamics, some predictability of such events may be obtained.D. Sornette, Dragon-Kings, Black Swans and the Prediction of Crises, International Journal of Terraspace Science and Engineering 1(3), 1–17 (2009) (https://arxiv.org/abs/0907.4290) and (http://ssrn.com/abstract=1470006)D. Sornette, Predictability of catastrophic events: material rupture, earthquakes, turbulence, financial crashes and human birth, Proc. Natl. Acad. Sci. USA 99, SUPP1 (2002), 2522–2529.
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. Practically speaking, one should study extreme risks, but not forget that significant uncertainty will almost always be present, and should be rigorously considered in decisions regarding risk management and design.
The dragon king theory is related to concepts such as black swan theory, outliers, complex systems, nonlinear dynamics, power laws, extreme value theory, prediction, extreme risks, and risk management.
Black swans and dragon kings A black swan can be considered a metaphor for an event that is surprising (to the observer), has a major effect, and, after being observed, is rationalized in hindsight. The theory of black swans is epistemological, relating to the limited knowledge and understanding of the observer. The term was introduced and popularized by Nassim Taleb and has been associated with concepts such as heavy tails, non-linear payoffs, model error, and even Knightian uncertainty, whose "unknowable unknown" event terminology was popularized by former United States Secretary of Defense Donald Rumsfeld. Taleb claims that black swan events are not predictable, and in practice, the theory encourages one to "prepare rather than predict", and limit one's exposure to extreme fluctuations.
The black swan concept is important and poses a valid criticism of people, firms, and societies that are irresponsible in the sense that they are overly confident in their ability to anticipate and manage risk. However, claiming that extreme events are—in general—unpredictable may also lead to a lack of accountability in risk management roles. In fact, it is known that in a wide range of physical systems that extreme events are predictable to some degree.Albeverio, Sergio, Volker Jentsch, and Holger Kantz. Extreme events in nature and society. Springer Science & Business Media, 2006. One simply needs to have a sufficiently deep understanding of the structure and dynamics of the focal system, and the ability to monitor it. This is the domain of the dragon kings. Such events have been referred to as "grey swans" by Taleb. A more rigorous distinction between black swans, grey swans, and dragon kings is difficult as black swans are not precisely defined in physical and mathematical terms. However, technical elaboration of concepts in the Black Swan book are elaborated in the Silent Risk document. An analysis of the precise definition of a black swan in a risk management context was written by professor Terje Aven.Aven, Terje. "On the meaning of a black swan in a risk context." Safety science 57 (2013): 44–51.
Beyond power laws
Many phenomena in both the natural and social sciences have power law statistics (Pareto distribution).Mitzenmacher, Michael. "A brief history of generative models for power law and lognormal distributions." Internet mathematics 1.2 (2004): 226–251.Newman, Mark EJ. "Power laws, Pareto distributions and Zipf's law." Contemporary physics 46.5 (2005): 323–351.Sornette, Didier. "Critical Phenomena in Natural Sciences: Chaos, Fractals, Selforganization and Disorder: Concepts and Tools (Springer Series in Synergetics)." (2006). Furthermore, from extreme value theory, it is known that a broad range of distributions (the Frechet class) have tails that are asymptotically power law. The result of this is that, when dealing with crises and extremes, power law tails are the "normal" case. The unique property of power laws is that they are scale-invariant, self-similar and fractal. This property implies that all events—both large and small—are generated by the same mechanism, and thus there will be no distinct precursors by which the largest events may be predicted. A well-known conceptual framework for events of this type is self-organized criticality. Such concepts are compatible with the theory of the black swan. However Taleb has also stated that considering the power law as a model instead of a model with lighter tails (e.g., a Gaussian) "converts black swans into gray ones", in the sense that the power law model gives non-negligible probability to large events.
In a variety of studies it has been found that, despite the fact that a power law models the tail of the empirical distribution well, the largest events are significantly outlying (i.e., much larger than what would be expected under the model).Wheatley, Spencer, and Didier Sornette. "Multiple Outlier Detection in Samples with Exponential & Pareto Tails: Redeeming the Inward Approach & Detecting Dragon Kings." arXiv preprint arXiv:1507.08689 (2015).Pisarenko, V. F., and D. Sornette. "Robust statistical tests of Dragon-Kings beyond power law distributions." The European Physical Journal Special Topics 205.1 (2012): 95–115. Such events are interpreted as dragon kings as they indicate a departure from the generic process underlying the power law. Examples of this include the largest radiation release events occurring in nuclear power plant accidents, the largest city (agglomeration) within the sample of cities in a country, the largest crashes in financial markets, and intraday wholesale electricity prices.
Mechanisms
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). The role of positive feedback loops in dragon king formation is well documented, particularly in oscillatory and cascading networks. Such a system periodically reaches a tipping point, becoming unstable and triggering a self-amplifying cascade of cascades, leading to a dragon king.
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.Shaw, Bruce E., Jean M. Carlson, and James S. Langer. "Patterns of seismic activity preceding large earthquakes." Journal of Geophysical Research: Solid Earth (1978–2012) 97.B1 (1992): 479–488.
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.Sornette, Didier, and Peter Cauwels. "1980–2008: The illusion of the perpetual money machine and what it bodes for the future." Risks 2.2 (2014): 103–131.Sornette, Didier, and Peter Cauwels. "Managing risk in a creepy world." Journal of Risk Management in Financial Institutions 8.1 (2015): 83–108. Ultimately such policies, aimed at smoothing out economic fluctuations, will enable an enormous correction—a dragon king.
As discussed in the previous section, dragon kings often arise in systems where other events follow a power law distribution. Mechanisms generating power laws are well-studied under the framework of self-organized criticality (SOC). Many systems exhibiting dragon kings alongside power laws can be understood by how their dynamics differ from pure SOC. SOC typically involves self-organization (SO) around a continuous absorbing-state phase transition (ASPT), where SO refers to an emergent balance between external driving and internal dissipation, and ASPT marks a transition between active and inactive (absorbing) phases. If this phase transition is first-order (discontinuous), it produces hysteresis, leading to large dragon king events. Even continuous transitions, which typically result in SOC, can enter regimes that produce dragon kings, depending on the nuances of the driving and dissipation. The figure on the right illustrates the taxonomy of dragon kings based on this classification: Set A includes all dragon kings, while sets B and C distinguish between those with self-organization near continuous and discontinuous transitions. Sets D and E further focus on cases with specifically absorbing-state transitions.
Dragon kings as statistical outliers