Black swans, extreme risks, and the e-pile model of self-organized criticality

Alexander V. Milovanov*, Jens Juul Rasmussen, Bertrand Groslambert

*Corresponding author for this work

Research output: Contribution to journalJournal articleResearchpeer-review


A new model of self-organized criticality (SOC) is described which differs from ordinary sand-pile SOC models in that it uses electrically charged particles of different kinds to propagate activities and to generate the critical state. The model, arguably called the e-pile model (as an alternative to sand-pile, with “e-” standing for “electric charge”), is motivated by the problem of dielectric relaxation in self-asembling random lattices with disorder under the action of electrostatic forces, but in principle it may be applied to different SOC processes with the random injection scheme and charitable lattice-redistribution rule. We show that the critical state is that of self-evolving random percolation clusters at the edge of percolation and is also different from known “self-organized” versions of the percolation problem based on the directed percolation. A set of critical exponents is obtained based on the random walks, using the Kramers-Kronig relation and the formalism of frequency-dependent complex conductivity. The relaxation of a supercritical system to SOC is shown to obey the Mittag-Leffler pattern and fractional relaxation equation, with a broad distribution of durations of relaxation events. A peculiar feature of the e-pile system is that it is characterized by an intrinsic bias disregarding a vast majority of incipient relaxation events (by killing them “in their egg”), while it also favors other events that may, then, grow into extremely large sizes. We use this observation to explain the phenomenon of the “black swan,” by which one means a family of rare, large events, whose sudden occurrence is very difficult to predict, despite the important impact this type of event may have over the entire system. The event-size distribution of the black swans is found to be a power law, with the drop-off exponent being sensitive to the complexity features of the underlying percolation clusters. On the basis of this observation, we show that the black swans are less probable in more complex environments, and we use this argument to explain why driven, dissipative systems would develop complex structures as a result of dynamical evolution. The e-pile model reveals the upper critical dimension dc=6 for which a crossover to mean-field SOC is found. In six and higher dimensions, the probability density to observe a black swan of a given size behaves as the inverse cube of the size. Below dc the decay is always slower than the inverse-cube and is flatter at lower dimensions. The implications of the mean-field law in country risk assessment, business, and finance are discussed. Finally, we show that the occurrence of the power-law tails is limited to the rate of the driving, and that for too high a rate a different branch of extreme events may result, with the defining features enabling to associate this branch with the phenomenon of dragon kings.

Original languageEnglish
Article number110665
JournalChaos, Solitons and Fractals
Number of pages21
Publication statusPublished - 2021

Bibliographical note

Funding Information:
The authors acknowledge the hospitality and support for the NIAS workshop on ?Mechanisms for Extreme Events Generation? at the Lorentz Center@Snellius at Leiden, The Netherlands, during July 8?12, 2019, as well as constructive and stimulating discussions with (in alphabetical order) Markus Aschwanden, Devraj Basu, Alessio Biondo, Norma Crosby, Alexander Iomin, Boris Kozelov, Kristoffer Rypdal, A. Surjalal Sharma, Lawrence Schulman, and Didier Sornette. The dragon, shown in Figs. 4 and 5, is the official dragon of the Workshop at Leiden. AVM acknowledges the hospitality and support at the International Space Science Institute (ISSI) at Bern, Switzerland, where this study was initiated.

Publisher Copyright:
© 2021 Elsevier Ltd


  • Extreme events
  • Fractional kinetics
  • Power laws
  • Random processes
  • Self-organized criticality


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