Methods for Reconstructing Interbank Networks from Limited Information: A Comparison

  • Piero MazzarisiEmail author
  • Fabrizio Lillo
Conference paper
Part of the New Economic Windows book series (NEW)


In this chapter, we review and compare some methods for the reconstruction of an interbank network from limited information. By exploiting the theory of complex networks and some ideas from statistical physics, we mainly focus on three different methods based on the maximum entropy principle, the relative entropy minimization, and the fitness model. We apply our analysis to the credit network of electronic Market for Interbank Deposit (e-MID) in 2011. In comparing the goodness of fit of the proposed methods, we look at the topological network properties and how reliably each method reproduces the real-world network.


Euro Area Systemic Risk Network Reconstruction Interbank Market Relative Entropy Minimization 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



This work is supported by the European Community H2020 Program under the scheme INFRAIA-1- 2014–2015: Research Infrastructures, grant agreement no. 654024 SoBigData: Social Mining & Big Data Ecosystem (


  1. Allen, F. and Gale, D. (2000). Financial contagion. Journal of political economy, 108(1):1–33.CrossRefGoogle Scholar
  2. Almog, A., Squartini, T., and Garlaschelli, D. (2015). A gdp-driven model for the binary and weighted structure of the international trade network. New Journal of Physics, 17(1):013009.ADSCrossRefGoogle Scholar
  3. Bacharach, M. (1965). Estimating nonnegative matrices from marginal data. International Economic Review, 6(3):294–310.CrossRefzbMATHGoogle Scholar
  4. Bargigli, L. (2014). Statistical ensembles for economic networks. Journal of Statistical Physics, 155(4):810–825.MathSciNetCrossRefzbMATHGoogle Scholar
  5. Bargigli, L., di Iasio, G., Infante, L., Lillo, F., and Pierobon, F. (2015). The multiplex structure of interbank networks. Quantitative Finance, 15:673–691.MathSciNetCrossRefGoogle Scholar
  6. Barucca, P. and Lillo, F. (2015). The organization of the interbank network and how ecb unconventional measures affected the e-mid overnight market.
  7. Barucca, P. and Lillo, F. (2016). Disentangling bipartite and core-periphery structure in financial networks. Chaos, Solitons & Fractals, 88:244–253.ADSMathSciNetCrossRefGoogle Scholar
  8. Battiston, S., Puliga, M., Kaushik, R., Tasca, P., and Caldarelli, G. (2012). Debtrank: Too central to fail? financial networks, the fed and systemic risk. Scientific reports, 2.Google Scholar
  9. Billio, M., Getmansky, M., Lo, A. W., and Pelizzon, L. (2012). Econometric measures of connectedness and systemic risk in the finance and insurance sectors. Journal of Financial Economics, 104(3):535–559.CrossRefGoogle Scholar
  10. Blien, U. and Graef, F. (1991). Entropy optimization in empirical economic research-the estimation of tables from incomplete information. JAHRBUCHER FUR NATIONALOKONOMIE UND STATISTIK, 208(4):399–413.Google Scholar
  11. Boguná, M. and Pastor-Satorras, R. (2003). Class of correlated random networks with hidden variables. Physical Review E, 68(3):036112.ADSCrossRefGoogle Scholar
  12. Bonanno, G., Caldarelli, G., Lillo, F., and Mantegna, R. N. (2003). Topology of correlation-based minimal spanning trees in real and model markets. Physical Review E, 68(4):046130.ADSCrossRefGoogle Scholar
  13. Bregman, L. M. (1967). Proof of the convergence of sheleikhovskii’s method for a problem with transportation constraints. USSR Computational Mathematics and Mathematical Physics, 7(1):191–204.CrossRefGoogle Scholar
  14. Caccioli, F., Shrestha, M., Moore, C., and Farmer, J. D. (2014). Stability analysis of financial contagion due to overlapping portfolios. Journal of Banking & Finance, 46:233–245.CrossRefGoogle Scholar
  15. Caldarelli, G., Capocci, A., De Los Rios, P., and Munoz, M. A. (2002). Scale-free networks from varying vertex intrinsic fitness. Physical review letters, 89(25):258702.ADSCrossRefGoogle Scholar
  16. Caldarelli, G., Chessa, A., Pammolli, F., Gabrielli, A., and Puliga, M. (2013). Reconstructing a credit network. Nature Physics, 9(3):125–126.ADSCrossRefGoogle Scholar
  17. Cimini, G., Squartini, T., Gabrielli, A., and Garlaschelli, D. (2015a). Estimating topological properties of weighted networks from limited information. Physical Review E, 92(4):040802.ADSCrossRefGoogle Scholar
  18. Cimini, G., Squartini, T., Garlaschelli, D., and Gabrielli, A. (2015b). Systemic risk analysis on reconstructed economic and financial networks. Scientific reports, 5.Google Scholar
  19. Cont, R., Moussa, A., et al. (2010). Network structure and systemic risk in banking systems. Edson Bastos e, Network Structure and Systemic Risk in Banking Systems (December 1, 2010).Google Scholar
  20. De Masi, G., Iori, G., and Caldarelli, G. (2006). Fitness model for the italian interbank money market. Physical Review E, 74(6):066112.ADSCrossRefGoogle Scholar
  21. Di Gangi, D., Lillo, F., and Pirino, D. (2015). Assessing systemic risk due to fire sales spillover through maximum entropy network reconstruction. Available at SSRN 2639178.Google Scholar
  22. Gai, P. and Kapadia, S. (2010). Contagion in financial networks. In Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, page rspa20090410. The Royal Society.Google Scholar
  23. Garlaschelli, D. and Loffredo, M. I. (2004). Fitness-dependent topological properties of the world trade web. Physical review letters, 93(18):188701.ADSCrossRefGoogle Scholar
  24. Garlaschelli, D. and Loffredo, M. I. (2008). Maximum likelihood: Extracting unbiased information from complex networks. Physical Review E, 78(1):015101.ADSCrossRefGoogle Scholar
  25. Garlaschelli, D. and Loffredo, M. I. (2009). Generalized bose-fermi statistics and structural correlations in weighted networks. Physical review letters, 102(3):038701.ADSCrossRefGoogle Scholar
  26. Geyer, C. J. and Thompson, E. A. (1992). Constrained monte carlo maximum likelihood for dependent data. Journal of the Royal Statistical Society. Series B (Methodological), pages 657–699.Google Scholar
  27. Iori, G., De Masi, G., Precup, O. V., Gabbi, G., and Caldarelli, G. (2008). A network analysis of the italian overnight money market. Journal of Economic Dynamics and Control, 32(1):259–278.CrossRefzbMATHGoogle Scholar
  28. Mastromatteo, I., Zarinelli, E., and Marsili, M. (2012). Reconstruction of financial networks for robust estimation of systemic risk. Journal of Statistical Mechanics: Theory and Experiment, 2012(03):P03011.CrossRefGoogle Scholar
  29. Mistrulli, P. E. (2011). Assessing financial contagion in the interbank market: Maximum entropy versus observed interbank lending patterns. Journal of Banking & Finance, 35(5):1114–1127.CrossRefGoogle Scholar
  30. Musmeci, N., Battiston, S., Caldarelli, G., Puliga, M., and Gabrielli, A. (2013). Bootstrapping topological properties and systemic risk of complex networks using the fitness model. Journal of Statistical Physics, 151(3-4):720–734.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. Newman, M. E. (2002). Assortative mixing in networks. Physical review letters, 89(20):208701.ADSCrossRefGoogle Scholar
  32. Newman, M. E. (2003). Mixing patterns in networks. Physical Review E, 67(2):026126.ADSMathSciNetCrossRefGoogle Scholar
  33. Park, J. and Newman, M. E. (2004). Statistical mechanics of networks. Physical Review E, 70(6):066117.ADSMathSciNetCrossRefGoogle Scholar
  34. Servedio, V. D., Caldarelli, G., and Butta, P. (2004). Vertex intrinsic fitness: How to produce arbitrary scale-free networks. Physical Review E, 70(5):056126.ADSCrossRefGoogle Scholar
  35. Shannon, C. E. (1948). A mathematical theory of communication. Bell System Tech. J, 27:623.MathSciNetCrossRefzbMATHGoogle Scholar
  36. Sheldon, G., Maurer, M., et al. (1998). Interbank lending and systemic risk: an empirical analysis for switzerland. REVUE SUISSE D ECONOMIE POLITIQUE ET DE STATISTIQUE, 134:685–704.Google Scholar
  37. Silvestri, L. and Cont, R. (2015). Essays on systemic risk, financial networks and macro-prudential regulation. PhD Thesis.Google Scholar
  38. Snijders, T. A. (2002). Markov chain monte carlo estimation of exponential random graph models. Journal of Social Structure, 3(2):1–40.Google Scholar
  39. Squartini, T. and Garlaschelli, D. (2011). Analytical maximum-likelihood method to detect patterns in real networks. New Journal of Physics, 13(8):083001.ADSCrossRefGoogle Scholar
  40. Squartini, T., Mastrandrea, R., and Garlaschelli, D. (2015). Unbiased sampling of network ensembles. New Journal of Physics, 17(2):023052.ADSCrossRefGoogle Scholar
  41. Upper, C. and Worms, A. (2004). Estimating bilateral exposures in the german interbank market: Is there a danger of contagion? European Economic Review, 48(4):827–849.CrossRefGoogle Scholar
  42. Van Duijn, M. A., Gile, K. J., and Handcock, M. S. (2009). A framework for the comparison of maximum pseudo-likelihood and maximum likelihood estimation of exponential family random graph models. Social Networks, 31(1):52–62.CrossRefGoogle Scholar
  43. Wells, S. J. (2004). Financial interlinkages in the united kingdom’s interbank market and the risk of contagion.Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Scuola Normale SuperiorePisaItaly

Personalised recommendations