Abstract
Complex dynamics is characterized by an irregular, non-periodic time dependence of characteristic quantities. Rare fluctuations which lead to unexpectedly large (or small) values are called extreme events. Since such large deviations from the system’s mean behavior have in many applications huge impact, their statistical characterization and their dynamical origin are of relevance. We discuss recent approaches, with special emphasis on dynamics on networks.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Albeverio, S., Jentsch, V., Kantz, H. (eds.): Extreme Events in Nature and Society. Springer, Berlin (2006)
Altmann, E., Kantz, H.: Recurrence time analysis, long-term correlations, and extreme events. Phys. Rev. E 71, 056106 (2005)
Altmann, E., Hallerberg, S., Kantz, H.: Reactions to extreme events: Moving threshold model. Physica A 364, 435–444 (2006)
Bak, P., Tang, C., Wiesenfeld, K.: Self-organized criticality: an explanation of 1/f noise. Phys. Rev. Lett. 59, 381–384 (1987)
Brier, G.: Verification of forecasts expressed in terms of probability. Mon. Weather Rev. 78, 1–3 (1950)
Bunde, A., Eichner, J., Kantelhardt, J., Havlin, S.: Long-term memory: A natural mechanism for the clustering of extreme events and anomalous residual times in climate records. Phys. Rev. Lett. 94, 048701 (2005)
Caruso, F., Kantz, H.: Prediction of extreme events in the OFC model on a small world network (2010, in review). arXiv:1004.4774v1
Christensen, K., Olami, Z.: Variation of the Gutenberg–Richter b values and nontrivial temporal correlations in a spring-block model for earthquakes. J. Geophys. Res. 97, 8729–8735 (1992)
Coles, S.: An Introduction to Statistical Modeling of Extreme Values. Springer, London (2001)
Egan, J.: Signal Detection Theory and ROC Analysis. Academic Press, New York (1975)
Eurich, C., Ernst, U.: Avalanches of activity in a network of integrate-and-fire neurons with stochastic input. In: Proc. of Int. Conf. on Artificial Neural Networks, ICANN 1999, vol. 2, pp. 545–550 (1999)
Fisher, R., Tippett, L.: Limiting forms of the frequency distribution of the largest and smallest member of a sample. Proc. Camb. Philos. Soc. 24, 180–190 (1928)
Fronczak, P., Fronczak, A., Holyst, J.: Self-organized criticality and coevolution of network structure and dynamics. Phys. Rev. E 73, 046117 (2006)
Garber, A., Kantz, H.: Finite size effects on the statistics of extreme events in the BTW model. Eur. Phys. J. B 67, 437–443 (2009)
Garber, A., Hallerberg, S., Kantz, H.: Predicting extreme avalanches in self-organized critical sandpiles. Phys. Rev. E 80, 026124 (2009)
Gneiting, T., Raftery, A.: Strictly proper scoring rules, prediction, and estimation. Tech. Rep. 436, Department of Statistics (2004)
Goh, K., Lee, D., Kahng, B., Kim, D.: Sandpile on scale-free networks. Phys. Rev. Lett. 91, 148701 (2003)
Gumbel, E.: Statistics of Extremes. Columbia University Press, New York (1958)
Hallerberg, S., Kantz, H.: How does the quality of a prediction depend on the magnitude of the events under study? Nonlinear Process. Geophys. 15, 321–331 (2008)
Hallerberg, S., Altmann, E., Holstein, D., Kantz, H.: Precursors of extreme increments. Phys. Rev. E 75, 016706 (2007)
Hughes, D., Paczuski, M., Dendy, R., Helander, P., McClements, K.: Solar flares as cascades of reconnecting magnetic loops. Phys. Rev. Lett. 90, 131101 (2003)
Kagan, Y., Jackson, D.: Long-term earthquake clustering. Geophys. J. Int. 104, 117–133 (1991)
Olami, Z., Feder, H., Christensen, K.: Self-organized criticality in a continuous, nonconservative cellular automaton modeling earthquakes. Phys. Rev. Lett. 68, 1244–1247 (1992)
Santhanam, M., Kantz, H.: Return interval distribution of extreme events and long-term memory. Phys. Rev. E 78, 051113 (2008)
Sutula, T.: Mechanisms of epilepsy progression: current theories and perspectives from neuroplasticity in adulthood and development. Epilepsy Res. 60, 161–171 (2004)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag London Limited
About this chapter
Cite this chapter
Kantz, H. (2010). Dynamics and Statistics of Extreme Events. In: Estrada, E., Fox, M., Higham, D., Oppo, GL. (eds) Network Science. Springer, London. https://doi.org/10.1007/978-1-84996-396-1_10
Download citation
DOI: https://doi.org/10.1007/978-1-84996-396-1_10
Publisher Name: Springer, London
Print ISBN: 978-1-84996-395-4
Online ISBN: 978-1-84996-396-1
eBook Packages: Computer ScienceComputer Science (R0)