Dragon-Kings, Black Swans and the Prediction of Crises

TL;DR

This paper introduces the concept of dragon-kings as significant outliers in event size distributions, linked to phase transitions, and demonstrates their presence across diverse systems with implications for crisis prediction.

Contribution

It develops the theory of dragon-kings, connects them to phase transitions, and provides a generic framework and examples for their identification and prediction.

Findings

Dragon-kings coexist with power laws in various systems.

Presence of dragon-kings indicates underlying phase transitions.

Log-periodic power law method can predict crises and material failures.

Abstract

We develop the concept of ``dragon-kings'' corresponding to meaningful outliers, which are found to coexist with power laws in the distributions of event sizes under a broad range of conditions in a large variety of systems. These dragon-kings reveal the existence of mechanisms of self-organization that are not apparent otherwise from the distribution of their smaller siblings. We present a generic phase diagram to explain the generation of dragon-kings and document their presence in six different examples (distribution of city sizes, distribution of acoustic emissions associated with material failure, distribution of velocity increments in hydrodynamic turbulence, distribution of financial drawdowns, distribution of the energies of epileptic seizures in humans and in model animals, distribution of the earthquake energies). We emphasize the importance of understanding dragon-kings as being often associated with a neighborhood of what can be called equivalently a phase transition, a bifurcation, a catastrophe (in the sense of Rene Thom), or a tipping point. The presence of a phase transition is crucial to learn how to diagnose in advance the symptoms associated with a coming dragon-king. Several examples of predictions using the derived log-periodic power law method are discussed, including material failure predictions and the forecasts of the end of financial bubbles.