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The Construction and Properties of Assortative Configuration Graphs

  • T. R. HurdEmail author
Chapter
Part of the Fields Institute Communications book series (FIC, volume 79)

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.

Keywords

Skeleton Systemic risk Banking network Configuration graph Assortativity Random graph simulation Large graph asymptotics Laplace method Locally tree-like Percolation theory 

References

  1. 1.
    M. Bech and E. Atalay. The topology of the federal funds market. Physica A: Statistical Mechanics and its Applications, 389(22):5223–5246, 2010.CrossRefGoogle Scholar
  2. 2.
    B. Bollobás. A probabilistic proof of an asymptotic formula for the number of labelled regular graphs. European Journal of Combinatorics, 1:311, 1980.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    B. Bollobás. Random Graphs. Cambridge studies in advanced mathematics. Cambridge University Press, 2 edition, 2001.Google Scholar
  4. 4.
    Ningyuan Chen and Mariana Olvera-Cravioto. Directed random graphs with given degree distributions. Stochastic Systems, 3(1):147–186, 2013.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Philippe Deprez and Mario V. Wüthrich. Construction of directed assortative configuration graphs. arXiv:1510.00575, October 2015.Google Scholar
  6. 6.
    Arthur Erdélyi. Asymptotic expansions. Dover, New York, 1956.zbMATHGoogle Scholar
  7. 7.
    P. Erdös and A. Rényi. On random graphs. I. Publ. Math. Debrecen, 6:290–297, 1959.zbMATHGoogle Scholar
  8. 8.
    T. R. Hurd. Saddlepoint approximation. In Rama Cont, editor, Encyclopedia of Quantitative Finance. John Wiley & Sons, Ltd, 2010.Google Scholar
  9. 9.
    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.CrossRefzbMATHGoogle Scholar
  10. 10.
    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.
  11. 11.
    Svante Janson. The probability that a random multigraph is simple. Combinatorics, Probability and Computing, 18:205–225, 3 2009.Google Scholar
  12. 12.
    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.CrossRefGoogle Scholar
  13. 13.
    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.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    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.Google Scholar
  15. 15.
    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.

Copyright information

© Springer Science+Business Media LLC 2017

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsMcMaster UniversityHamiltonCanada

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