Abstract
Systemic risk, i.e., the risk that a local shock propagates throughout a given system due to contagion effects, is of great importance in many fields of our lives. In this summary article, we show how asymptotic methods for random graphs can be used to understand and quantify systemic risk in networks. We define a notion of resilient networks and present criteria that allow us to classify networks as resilient or non-resilient. We further examine the question how networks can be strengthened to ensure resilience. In particular, for financial systems we address the question of sufficient capital requirements. We present the results in random graph models of increasing complexity and relate them to classical results about the phase transition in the Erdös-Rényi model. We illustrate the results by a small simulation study.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Alon, N. & Spencer, J. H. (2016), The Probabilistic Method, 4th edn, Wiley Publishing.
Amini, H. & Fountoulakis, N. (2014), ‘Bootstrap percolation in power-law random graphs’, Journal of Statistical Physics155(1), 72–92. http://dx.doi.org/10.1007/s10955-014-0946-6
Amini, H., Filipović, D. & Minca, A. (2015), ‘Systemic risk and central clearing counterparty design’, Swiss Finance Institute Research Paper pp. 13–34.
Amini, H., Fountoulakis, N. & Panagiotou, K. (2014), ‘Bootstrap percolation in inhomogeneous random graphs’, arXiv:1402.2815
Amini, H., Cont, R. & Minca, A. (2016), ‘Resilience to contagion in financial networks’, Mathematical Finance26(2), 329–365.
Armenti, Y., Crépey, S., Drapeau, S. & Papapantoleon, A. (2018), ‘Multivariate shortfall risk allocation and systemic risk’, SIAM Journal on Financial Mathematics9(1), 90–126.
Artzner, P., Delbaen, F., Eber, J. & Heath, D. (1999), ‘Coherent measures of risk’, Mathematical Finance9(3), 203–228.
Biagini, F., Fouque, J.-P., Fritelli, M. & Meyer-Brandis, T. (2019+), ‘A unified approach to systemic risk measures via acceptance sets’, Mathematical Finance.
Bollobás, B. (2001), Random Graphs, Cambridge Studies in Advanced Mathematics, 2nd edn, Cambridge University Press, Cambridge.
Chalupa, J., Leath, P. L. & Reich, G. R. (1979), ‘Bootstrap percolation on a Bethe lattice’, Journal of Physics C: Solid State Physics12(1), L31–L35.
Chen, C., Iyengar, G. & Moallemi, C. (2013), ‘An axiomatic approach to systemic risk’, Management Science59(6), 1373–1388.
Chong, C. & Klüppelberg, C. (2018), ‘Contagion in financial systems: A Bayesian network approach’, SIAM Journal on Financial Mathematics9(1), 28–53.
Chung, F. & Lu, L. (2002), ‘Connected components in random graphs with given expected degree sequences’, Annals of Combinatorics6(2), 125–145.
Chung, F. & Lu, L. (2003), ‘The average distances in random graphs with given expected degrees’, Internet Mathematics1, 91–114.
Detering, N., Meyer-Brandis, T. & Panagiotou, K. (2015), ‘Bootstrap percolation in directed and inhomogeneous random graphs’, Electronic Journal of Combinatorics, 26(2), 2019, arXiv:1511.07993.
Detering, N., Meyer-Brandis, T., Panagiotou, K. & Ritter, D. (2016), ‘Managing default contagion in inhomogeneous financial networks’, SIAM Journal on Financial Mathematics, 10(2), 2019, arXiv:1610.09542.
Eisenberg, L. & Noe, T. H. (2001), ‘Systemic risk in financial systems’, Management Science47(2), 236–249.
Erdős, P. & Rényi, A. (1960), On the evolution of random graphs, in ‘Publication of the Mathematical Institute of the Hungarian Academy of Sciences’, pp. 17–61.
Feinstein, Z., Rudloff, B. & Weber, S. (2017), ‘Measures of systemic risk’, SIAM Journal on Financial Mathematics8(1), 672–708.
Fouque, J.-P. & Langsam, J. A., eds (2013), Handbook on systemic risk, Cambridge University Press, Cambridge.
Gai, P. & Kapadia, S. (2010), ‘Contagion in Financial Networks’, Proceedings of the Royal Society A466, 2401–2423.
Gandy, A. & Veraart, L. A. M. (2017), ‘A Bayesian methodology for systemic risk assessment in financial networks’, Management Science63(12), 4428–4446.
Gilbert, E. N. (1959), ‘Random Graphs’, Ann. Math. Statist.30(4), 1141–1144. https://doi.org/10.1214/aoms/1177706098
Hoffmann, H., Meyer-Brandis, T. & Svindland, G. (2016), ‘Risk-consistent conditional systemic risk measures’, Stochastic Processes and their Applications126(7), 2014–2037.
Hoffmann, H., Meyer-Brandis, T. & Svindland, G. (2018), ‘Strongly consistent multivariate conditional risk measures’, Mathematics and Financial Economics12(3), 413–444.
Hofstad, R. v. d. (2016), Random Graphs and Complex Networks, Vol. 1 of Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press.
Hurd, T. R. (2016), Contagion! Systemic Risk in Financial Networks, Springer.
Janson, S., Łuczak, T. & Rucinski, A. (2011), Random Graphs, Wiley.
Janson, S., Łuczak, T., Turova, T. & Vallier, T. (2012), ‘Bootstrap percolation on the random graph \(G_{n,p}\)’, Ann. Appl. Probab.22(5), 1989–2047. http://dx.doi.org/10.1214/11-AAP822
Kromer, E., Overbeck, L. & Zilch, K. (2016), ‘Systemic risk measures on general measurable spaces’, Mathematical Methods of Operations Research84(2), 323–357.
Kusnetsov, M. & Veraart, L. A. M. (2016), ‘Interbank Clearing in Financial Networks with Multiple Maturities’, Preprint.
Weber, S. & Weske, S. (2017), ‘The joint impact of bankruptcy costs, cross-holdings and fire sales on systemic risk in financial networks’, Probability, Uncertainty and Quantitative Risk2(9), 1–38.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Detering, N., Meyer-Brandis, T., Panagiotou, K., Ritter, D. (2019). Systemic Risk in Networks. In: Biagini, F., Kauermann, G., Meyer-Brandis, T. (eds) Network Science. Springer, Cham. https://doi.org/10.1007/978-3-030-26814-5_5
Download citation
DOI: https://doi.org/10.1007/978-3-030-26814-5_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-26813-8
Online ISBN: 978-3-030-26814-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)