Abstract
In the new field of financial systemic risk, the network of interbank counterparty relationships can be described as a directed random graph. In cascade models of systemic risk, this skeleton acts as the medium through which financial contagion is propagated. It has been observed in real networks that such counterparty relationships exhibit negative assortativity, meaning that a bank’s counterparties are more likely to have unlike characteristics. This paper introduces and studies a general class of random graphs called the assortative configuration model, parameterized by an arbitrary node-type distribution P and edge-type distribution Q. The first main result is a law of large numbers that says the empirical edge-type distributions converge in probability to Q as the number of nodes N goes to infinity. The second main result is a formula for the large N asymptotic probability distribution of general graphical objects called configurations. This formula exhibits a key property called locally tree-like that in simpler models is known to imply strong results of percolation theory on the size of large connected clusters. Thus this paper provides the essential foundations needed to prove rigorous percolation bounds and cascade mappings in assortative networks.
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References
M. Bech and E. Atalay. The topology of the federal funds market. Physica A: Statistical Mechanics and its Applications, 389(22):5223–5246, 2010.
B. Bollobás. A probabilistic proof of an asymptotic formula for the number of labelled regular graphs. European Journal of Combinatorics, 1:311, 1980.
B. Bollobás. Random Graphs. Cambridge studies in advanced mathematics. Cambridge University Press, 2 edition, 2001.
Ningyuan Chen and Mariana Olvera-Cravioto. Directed random graphs with given degree distributions. Stochastic Systems, 3(1):147–186, 2013.
Philippe Deprez and Mario V. Wüthrich. Construction of directed assortative configuration graphs. arXiv:1510.00575, October 2015.
Arthur Erdélyi. Asymptotic expansions. Dover, New York, 1956.
P. Erdös and A. Rényi. On random graphs. I. Publ. Math. Debrecen, 6:290–297, 1959.
T. R. Hurd. Saddlepoint approximation. In Rama Cont, editor, Encyclopedia of Quantitative Finance. John Wiley & Sons, Ltd, 2010.
T. R. Hurd. Contagion! Systemic Risk in Financial Networks. SpringerBriefs in Quantitative Finance. Springer Verlag, Berlin Heidelberg New York, 2016. Available at http://ms.mcmaster.ca/tom/tom.html.
T. R. Hurd, Davide Cellai, Sergey Melnik, and Quentin Shao. Double cascade model of financial crises. International Journal of Theoretical and Applied Finance, (to appear), 2016. http://arxiv.org/abs/1310.6873v3.
Svante Janson. The probability that a random multigraph is simple. Combinatorics, Probability and Computing, 18:205–225, 3 2009.
Robert M. May and Nimalan Arinaminpathy. Systemic risk: the dynamics of model banking systems. Journal of The Royal Society Interface, 7(46):823–838, 2010.
Michael Molloy and Bruce Reed. A critical point for random graphs with a given degree sequence. Random Structures & Algorithms, 6(2–3):161–180, 1995.
Kimmo Soramäki, M. Bech, J. Arnold, R. Glass, and W. Beyeler. The topology of interbank payment flows. Physica A: Statistical Mechanics and its Applications, 379(1):317–333, 2007.
R. van der Hofstad. Random Graphs and Complex Networks. unpublished, available at http://www.win.tue.nl/rhofstad/NotesRGCN.html, 2016. Book, to be published.
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Hurd, T.R. (2017). The Construction and Properties of Assortative Configuration Graphs. In: Melnik, R., Makarov, R., Belair, J. (eds) Recent Progress and Modern Challenges in Applied Mathematics, Modeling and Computational Science. Fields Institute Communications, vol 79. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-6969-2_11
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DOI: https://doi.org/10.1007/978-1-4939-6969-2_11
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