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Bootstrapping Topological Properties and Systemic Risk of Complex Networks Using the Fitness Model


In this paper we present a novel method to reconstruct global topological properties of a complex network starting from limited information. We assume to know for all the nodes a non-topological quantity that we interpret as fitness. In contrast, we assume to know the degree, i.e. the number of connections, only for a subset of the nodes in the network. We then use a fitness model, calibrated on the subset of nodes for which degrees are known, in order to generate ensembles of networks. Here, we focus on topological properties that are relevant for processes of contagion and distress propagation in networks, i.e. network density and k-core structure, and we study how well these properties can be estimated as a function of the size of the subset of nodes utilized for the calibration. Finally, we also study how well the resilience to distress propagation in the network can be estimated using our method. We perform a first test on ensembles of synthetic networks generated with the Exponential Random Graph model, which allows to apply common tools from statistical mechanics. We then perform a second test on empirical networks taken from economic and financial contexts. In both cases, we find that a subset as small as 10 % of nodes can be enough to estimate the properties of the network along with its resilience with an error of 5 %.

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  1. Battiston, S., Gatti, D., Gallegati, M., Greenwald, B., Stiglitz, J.: Liaisons dangereuses: increasing connectivity, risk sharing, and systemic risk. J. Econ. Dyn. Control 36(8), 1121–1141 (2012).

    Article  Google Scholar 

  2. Battiston, S., Puliga, M., Kaushik, R., Tasca, P., Caldarelli, G.: DebtRank: too central to fail? Financial networks, the fed and systemic risk. Sci. Rep. 2, 541 (2012)

    Article  Google Scholar 

  3. Caldarelli, G., Capocci, A., De Los Rios, P., Muñoz, M.: Scale-free networks from varying vertex intrinsic fitness. Phys. Rev. Lett. 89(25), 258702 (2002)

    Article  ADS  Google Scholar 

  4. Clauset, A., Moore, C., Newman, M.: Hierarchical structure and the prediction of missing links in networks. Nature 453(7191), 98–101 (2008)

    Article  ADS  Google Scholar 

  5. Degryse, H., Nguyen, G.: Interbank exposures: an empirical examination of contagion risk in the Belgian banking system. Int. J. Cent. Bank. 3(2), 123–171 (2007)

    Google Scholar 

  6. De Masi, G., Iori, G., Caldarelli, G.: Fitness model for the Italian interbank money market. Phys. Rev. E 74(6), 066112 (2006)

    Article  ADS  Google Scholar 

  7. Dorogovtsev, S.: Lectures on complex networks. Phys. J. 9(11), 51 (2010)

    Google Scholar 

  8. Fagiolo, G.: The international-trade network: gravity equations and topological properties. J. Econ. Interact. Coord. 5(1), 1–25 (2010)

    Article  Google Scholar 

  9. Garlaschelli, D., Loffredo, M.: Fitness-dependent topological properties of the world trade web. Phys. Rev. Lett. 93(18), 188,701 (2004)

    Article  Google Scholar 

  10. Garlaschelli, D., Battiston, S., Castri, M., Servedio, V., Caldarelli, G.: The scale-free topology of market investments. Physica A 350(2), 491–499 (2005)

    Article  MathSciNet  ADS  Google Scholar 

  11. Garlaschelli, D., Capocci, A., Caldarelli, G.: Self-organized network evolution coupled to extremal dynamics. Nat. Phys. 3(5), 813–817 (2007)

    Article  Google Scholar 

  12. Kitsak, M., Gallos, L., Havlin, S., Liljeros, F., Muchnik, L., Stanley, H., Makse, H.: Identification of influential spreaders in complex networks. Nat. Phys. 6(11), 888–893 (2010)

    Article  Google Scholar 

  13. Mastromatteo, I., Zarinelli, E., Marsili, M.: Reconstruction of financial networks for robust estimation of systemic risk. J. Stat. Mech. Theory Exp. 2012(03), P03011 (2012)

    Article  Google Scholar 

  14. Mistrulli, P.: Assessing financial contagion in the interbank market: maximum entropy versus observed interbank lending patterns. J. Bank. Finance 35(5), 1114–1127 (2011)

    Article  Google Scholar 

  15. Park, J., Newman, M.: Statistical mechanics of networks. Phys. Rev. E 70(6), 066117 (2004)

    Article  MathSciNet  ADS  Google Scholar 

  16. van Lelyveld, I., Liedorp, F.: Interbank contagion in the dutch banking sector. Int. J. Cent. Bank. 2, 99–134 (2006)

    Google Scholar 

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We thank support from the European project FET-Open FOC (255987) and the Italian PNR project CRISIS-Lab.

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Correspondence to Guido Caldarelli.

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Musmeci, N., Battiston, S., Caldarelli, G. et al. Bootstrapping Topological Properties and Systemic Risk of Complex Networks Using the Fitness Model. J Stat Phys 151, 720–734 (2013).

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  • Complex networks
  • Financial systems