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AN URN MODEL FOR CASCADING FAILURES ON A LATTICE

Published online by Cambridge University Press:  30 July 2012

Pasquale Cirillo  and
Jürg Hüsler
Pasquale Cirillo
Affiliation:
Institute of Mathematical Statistics and Actuarial Sciences, University of Bern, Sidlerstrasse 5, Bern CH3008, Switzerland E-mails: pasquale.cirillo@stat.unibe.ch; juerg.huesler@stat.unibe.ch
Jürg Hüsler
Affiliation:
Institute of Mathematical Statistics and Actuarial Sciences, University of Bern, Sidlerstrasse 5, Bern CH3008, Switzerland E-mails: pasquale.cirillo@stat.unibe.ch; juerg.huesler@stat.unibe.ch
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Abstract

A cascading failure is a failure in a system of interconnected parts, in which the breakdown of one element can lead to the subsequent collapse of the others. The aim of this paper is to introduce a simple combinatorial model for the study of cascading failures. In particular, having in mind particle systems and Markov random fields, we take into consideration a network of interacting urns displaced over a lattice. Every urn is Pólya-like and its reinforcement matrix is not only a function of time (time contagion) but also of the behavior of the neighboring urns (spatial contagion), and of a random component, which can represent either simple fate or the impact of exogenous factors. In this way a non-trivial dependence structure among the urns is built, and it is used to study default avalanches over the lattice. Thanks to its flexibility and its interesting probabilistic properties, the given construction may be used to model different phenomena characterized by cascading failures such as power grids and financial networks.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 26 , Issue 4 , October 2012 , pp. 509 - 534
Copyright
Copyright © Cambridge University Press 2012

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References

1.Ahmad, M.I. (1988). Applications of statistics in flood frequency analysis. PhD thesis, University of St. Andrews.Google Scholar
2.Aldous, D.J. (1985). Exchangeability and related topics. Lecture Notes in Mathematics vol. 117, New York: SpringerGoogle Scholar
3.Anderson, T.W. & Darling, D.A. (1952). Asymptotic theory of certain goodness-of-fit criteria based on stochastic processes. Annals of Mathematical Statistics 23: 193–212.CrossRefGoogle Scholar
4.Aoki, M. (2000). Cluster size distributions of economic agents of many types in a market. Journal of Mathematical Analysis and Applications 249: 32–52.CrossRefGoogle Scholar
5.Balakrishnan, N. (1997). Advances in combinatorial methods and applications to probability and statistics. Berlin: Birkhäuser.Google Scholar
6.Bernoulli, J. (1713). Ars Conjectandi. German version in Wahrscheinlichkeitsrechnung: Ars conjectandi. 1., 2., und 4. Teil (1998). Berlin: Harry Deutsch.Google Scholar
7.Berti, P., Pratelli, L. & Rigo, P. (2004). Limit theorems for a class of identically distributed random variables. Annals of Probability 32: 2029–2052.CrossRefGoogle Scholar
8.Bhargava, S.C. & Mukherjee, A. (1994). Evolution and technological growth in a model based on stochastic cellular automata. In Leydesdorff, L., Van den Besselaar, P., (eds.), Evolutionary economics and chaos theory: new directions in technology studies. London: Pinter.Google Scholar
9.Blume, L.E. & Durlauf, S.N. (2003). Equilibrium concepts for social interaction models. International Game Theory Review 5: 193–209.CrossRefGoogle Scholar
10.Bottazzi, G. & Secchi, A. (2003). Why are distributions of firm growth rates tent-shaped? Economics Letters 80: 415–420.CrossRefGoogle Scholar
11.Choulakian, V. & Stephens, M.A. (2001). Goodness-of-fit tests for the generalized Pareto distribution. Technometrics 43: 478–484.CrossRefGoogle Scholar
12.Cifarelli, D.M. & Regazzini, E. (1978). Problemi statistici non parametrici in condizioni di scambiabilità parziale. Impiego di medie associative. In: Istituto di Matematica Finanziaria dell'Università di Torino, Serie III, No. 12. English translation available from http://www.unibocconi.it/wps/allegatiCTP/CR-Scamb-parz[1].20080528.135739.pdfGoogle Scholar
13.Cirillo, P. (2008). New urn approaches to shock and default models. PhD thesis, Milan: Bocconi University.Google Scholar
14.Cirillo, P. (2012). A simple model of spatially dependent urns. Working Paper, University of Bern.Google Scholar
15.Cirillo, P., & Hüsler, J. (2009). An urn-based approach to generalized extreme shock models. Statistics and Probability Letters 79: 969–976.CrossRefGoogle Scholar
16.Cirillo, P., & Hüsler, J. (2009). On the upper tail of Italian firms size distribution. Physica A Statistical Mechanics and Applications 388: 1546–1554.CrossRefGoogle Scholar
17.Cirillo, P., & Hüsler, J. (2011). Shock models for defaults: parametric and nonparametric approaches. In Hunter, D.R., Richards, D.S.P., Rosenberger, J.L. (eds.) Nonparametric statistics and mixture models. Singapore: WSP.Google Scholar
18.Clauset, A., Shalizi, C.R. & Newman, M.E.J. (2009). Power-law distributions in empirical data. SIAM Review 51(4): 661–678.CrossRefGoogle Scholar
19.Costantini, D., Donadio, S., Garibaldi, U. & Viarengo, P. (2005). Herding and clustering: Ewens vs. Simon-Yule models. Physica A Statistical Mechanics and Applications 355: 224–231.CrossRefGoogle Scholar
20.Dai Pra, P, Runggaldier, W.J., Sartori, E. & Tolotti, M. (2009). Large portfolio losses: a dynamic contagion model. Annals of Applied Probabability 19: 347–394.Google Scholar
21.de Finetti, B. (1975). Theory of probability. New York: John Wiley and Sons.Google Scholar
22.Delli Gatti, D., Gallegati, M., Greenwald, B.C., Russo, A. & Stiglitz, J.E. (2009). Business fluctuations and bankruptcy avalanches in an evolving network economy. Journal of Econimic Interaction and Coordination 4: 195–212.CrossRefGoogle Scholar
23.Dobson, I., Carreras, B.A., Lynch, V.E. & Newman, D.E. (2007). Complex systems analysis of series of blackouts: cascading failure, critical points, and self-organization. Chaos 17, 026103.CrossRefGoogle ScholarPubMed
24.Dobson, I., Wierzbicki, K.R., Carreras, B.A., Lynch, V.E. & Newman, D. (2006). An estimator of propagation of cascading failures. Thirty-ninth Hawaii International IEEE Conference on System Sciences Proceedings.CrossRefGoogle Scholar
25.Drees, H., de Haan, L. & Li, D. (2006). Approximations to the tail empirical distribution function with application to testing extreme value conditions. Journal of Statistical Planning and Inference 136: 3498–3538.CrossRefGoogle Scholar
26.Durrett, R. (2010). Some features of the spread of epidemics and information on a random graph. Proc. Natl. Acad. Sei. 107: 4491–4498.CrossRefGoogle ScholarPubMed
27.Eggenberger, F. & Pólya, G. (1923). Über die Statistik verketteter Vorgänge. Zeitschrift für Angerwandte Mathematik and Mechanik 1: 279–289.CrossRefGoogle Scholar
28.Embrechts, P., Klüppelberg, C. & Mikosch, T. (1997). Modelling extremal events for insurance and finance. Berlin: Springer.CrossRefGoogle Scholar
29.Falk, M., Hüsler, J. & Reiss, R-D. (2004). Laws of small numbers: extremes and rare events. Basel: Birkhäuser.CrossRefGoogle Scholar
30.Flajolet, P., Dumas, P. & Puyhaubert, V. (2006). Some exactly solvable models of urn process theory. Fourth Colloquium on Mathematics and Computer Science DMTCS 59.Google Scholar
31.Frey, R. & Backhaus, J. (2008). Pricing and hedging of portfolio credit derivatives with interacting default intensities. International Journal of Theoretical and Applied Finance 11: 611–631.CrossRefGoogle Scholar
32.Giesecke, K. & Weber, S. (2006). Credit contagion and aggregate losses. Journal of Economic Dynamics and Control 30: 741–767.CrossRefGoogle Scholar
33.Giudici, P., Mezzetti, M. & Muliere, P. (2003). Mixtures of products of Dirichlet processes for variable selection in survival analysis. Journal of Statistical Planning and Inference 111: 101–115.CrossRefGoogle Scholar
34.Ghosh, J.K. & Ramamoorthi, R.V. (2002). Bayesian nonparametrics. New York: Springer.Google Scholar
35.Hall, P. & Heyde, C.C. (1980). Martingale limit theory and its applications. San Diego: Academic Press.Google Scholar
36.Johnson, N.L. & Kotz, S. (1977). Urn models and their applications. New York: Wiley.Google Scholar
37.Johnson, R.A. & Wichern, D.A. (2007). Applied multivariate analysis. New Jersey: Prentice Hall.Google Scholar
38.Kindermann, R. & Snell, J.L. (1980). Markov random fields and their applications. San Francisco: AMS Press.CrossRefGoogle Scholar
39.Liggett, T.M., Steif, J.E. & Tóth, B. (2007). Statistical mechanical systems on complete graphs, infinite exchangeability, finite extensions and a discrete finite moment problem. Annals of Probability 35: 867–914.CrossRefGoogle Scholar
40.Liggett, T. (2009). Stochastic interacting systems: Contact, voter and exclusion processes. New York: Springer.Google Scholar
41.Lindley, V. & Singpurwalla, N.D. (2002). On exchangeable, causal and cascading failures. Statistical Science 17: 209–219.CrossRefGoogle Scholar
42.Lorenz, J., Battiston, S. & Schweitzer, F. (2009). Systemic risk in an unifying framework for cascading processes on networks. European Physical Journal B 71: 441–460.CrossRefGoogle Scholar
43.Mahmoud, H.M. (2009). Polya Urn Models. Boca Raton: CRP Press.Google Scholar
44.Marshall, A.W. & Olkin, I. (1993). Bivariate life distributions from Polya's urn model for contagion. Journal of Applied Probability 30: 497–508.CrossRefGoogle Scholar
45.Marsili, M. & Valleriani, A. (1998). Self organization of interacting Polya urn. European Physical Journal B 3: 417–420.CrossRefGoogle Scholar
46.May, C., Secchi, P. & Paganoni, A. (2002). On a two-color generalized Polya urn. Metron LXIII, 115–134.Google Scholar
47.Muliere, P., Paganoni, A. & Secchi, P. (2006). A randomly reinforced urn. Journal of Statistical Planning and Inference 136: 1853–1874.CrossRefGoogle Scholar
48.Muliere, P., Secchi, P. & Walker, S.G. (2000). Urn schemes and reinforced random walks. Stochastic Processes and their Applications 88: 59–78.CrossRefGoogle Scholar
49.Murri, N. & Pinto, N. (2002). Cluster size distribution in self-organized systems. Physica B Condensed Matter 321: 404–407.CrossRefGoogle Scholar
50.Paganoni, A.M. & Secchi, P. (2004). Interacting reinforced-urn systems. Advances in Applied Probability 36: 791–804.CrossRefGoogle Scholar
51.Pemantle, R. (2007). A survey of random processes with reinforcement. Probability Surveys 4: 1–79.CrossRefGoogle Scholar
52.Pérez, P. (1998). Markov random fields and images. CWI Quarterly 11: 413–437.Google Scholar
53.Swift, A. (2008). Stochastic models of cascading failures. Journal of Applied Probability 45: 907–921.CrossRefGoogle Scholar
54.Wang, J., Liu, Y. & Jiao, Y. (2009). A new cascading failure model with delay time in congested complex networks. Journal of System Science and System Engineering 18: 369–381.CrossRefGoogle Scholar
55.Young, A.P. (1998). Spin glasses and random fields. Singapore: WSP.Google Scholar