Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

The physics of financial networks

Subjects

Abstract

As the total value of the global financial market outgrew the value of the real economy, financial institutions created a global web of interactions that embodies systemic risks. Understanding these networks requires new theoretical approaches and new tools for quantitative analysis. Statistical physics contributed significantly to this challenge by developing new metrics and models for the study of financial network structure, dynamics, and stability and instability. In this Review, we introduce network representations originating from different financial relationships, including direct interactions such as loans, similarities such as co-ownership and higher-order relations such as contracts involving several parties (for example, credit default swaps) or multilayer connections (possibly extending to the real economy). We then review models of financial contagion capturing the diffusion and impact of shocks across each of these systems. We also discuss different notions of ‘equilibrium’ in economics and statistical physics, and how they lead to maximum entropy ensembles of graphs, providing tools for financial network inference and the identification of early-warning signals of system-wide instabilities.

Key points

  • Modelling the financial system as a network is crucial to capture the complex interactions between financial institutions.

  • Such a network is naturally a time-dependent multiplex, because relationships between financial institutions are of many different kinds and keep changing.

  • Models of financial contagion enable understanding of how shocks propagate from one financial institution to another.

  • Missing information on financial networks can be ‘reconstructed’ using maximum entropy approaches borrowed from statistical mechanics.

  • Techniques based on financial networks have been adopted by practitioners and policy institutions.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.

from$8.99

All prices are NET prices.

Fig. 1: Correlation-based networks from multivariate time series.
Fig. 2: Interbank networks and their dynamics.
Fig. 3: Construction of statistical ensembles of financial networks and application to network reconstruction and pattern detection.

References

  1. 1.

    Lazer, D. et al. Computational social science. Science 323, 721–723 (2009).

    Article  Google Scholar 

  2. 2.

    Pentland, A. Social Physics: How Good Ideas Spread-The Lessons from a New Science (Penguin, 2014).

  3. 3.

    Buchanan, M. The Social Atom: Why the Rich Get Richer, Cheaters Get Caught, and Your Neighbor Usually Looks Like You (Bloomsbury, 2008).

  4. 4.

    Ball, P. Why Society is a Complex Matter: Meeting Twenty-first Century Challenges with a New Kind of Science (Springer, 2012).

  5. 5.

    Caldarelli, G., Wolf, S. & Moreno, Y. Physics of humans, physics for society. Nat. Phys. 14, 870–870 (2018).

    Article  Google Scholar 

  6. 6.

    Auyang, S. Y. Foundations of Complex-system Theories: In Economics, Evolutionary Biology, and Statistical Physics (Cambridge Univ. Press, 1998).

  7. 7.

    Mantegna, R. N. & Stanley, H. E. An Introduction to Econophysics: Correlations and Complexity in Finance (Cambridge Univ. Press, 1999).

  8. 8.

    Barabási, A.-L. The network takeover. Nat. Phys. 8, 14–16 (2012).

    Article  Google Scholar 

  9. 9.

    Castellano, C., Fortunato, S. & Loreto, V. Statistical physics of social dynamics. Rev. Mod. Phys. 81, 591–646 (2009).

    Article  ADS  Google Scholar 

  10. 10.

    Pastor-Satorras, R., Castellano, C., Van Mieghem, P. & Vespignani, A. Epidemic processes in complex networks. Rev. Mod. Phys. 87, 925–979 (2015).

    MathSciNet  Article  ADS  Google Scholar 

  11. 11.

    Perc, M. et al. Statistical physics of human cooperation. Phys. Rep. 687, 1–51 (2017).

    MathSciNet  MATH  Article  ADS  Google Scholar 

  12. 12.

    Cimini, G. et al. The statistical physics of real-world networks. Nat. Rev. Phys. 1, 58–71 (2019).

    Article  Google Scholar 

  13. 13.

    May, R. M. & Arinaminpathy, N. Systemic risk: the dynamics of model banking systems. J. R. Soc. Interface 7, 823–838 (2010).

    Article  Google Scholar 

  14. 14.

    Beale, N. et al. Individual versus systemic risk and the regulator’s dilemma. Proc. Natl Acad. Sci. USA 108, 12647–12652 (2011).

    Article  ADS  Google Scholar 

  15. 15.

    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  ADS  Google Scholar 

  16. 16.

    Battiston, S. et al. Complexity theory and financial regulation. Science 351, 818–819 (2016).

    Article  ADS  Google Scholar 

  17. 17.

    Cimini, G., Squartini, T., Garlaschelli, D. & Gabrielli, A. Systemic risk analysis on reconstructed economic and financial networks. Sci. Rep. 5, 15758 (2015). In this paper the entropy-based approach is tailored to the reconstruction of financial networks.

    Article  ADS  Google Scholar 

  18. 18.

    Battiston, S., Caldarelli, G., May, R. M., Roukny, T. & Stiglitz, J. E. The price of complexity in financial networks. Proc. Natl Acad. Sci. USA 113, 10031–10036 (2016).

    MathSciNet  MATH  Article  ADS  Google Scholar 

  19. 19.

    Iori, G., Jafarey, S. & Padilla, F. G. Systemic risk on the interbank market. J. Econ. Behav. Organ. 61, 525–542 (2006).

    Article  Google Scholar 

  20. 20.

    Henry, J. & Kok, C. (eds) A macro stress testing framework for assessing systemic risks in the banking sector. Occasional Paper Series, no. 152. European Central Bank https://www.ecb.europa.eu/pub/pdf/scpops/ecbocp152.pdf (2013).

  21. 21.

    Abad, J. et al. Shedding light on dark markets: first insights from the new EU-wide OTC derivatives dataset. Occasional Paper Series, no. 11. European Systemic Risk Board https://www.esrb.europa.eu/pub/pdf/occasional/20160922_occasional_paper_11.en.pdf (2016).

  22. 22.

    Churm, R. & Nahai-Williamson, P. in Stress Testing: Approaches, Methods and Applications 2nd edn (eds Siddique, A., Hasan, I. & Lynch, D.) (Risk Books, 2019).

  23. 23.

    Allen, F. & Babus, A. in The Network Challenge: Strategy, Profit, and Risk in an Interlinked World (eds Kleindorfer, P. R. & Wind, Y.) 367–382 (Wharton School Publishing, 2009).

  24. 24.

    Elliott, M., Golub, B. & Jackson, M. O. Financial networks and contagion. Am. Econ. Rev. 104, 3115–3153 (2014).

    Article  Google Scholar 

  25. 25.

    Acemoglu, D., Ozdaglar, A. & Tahbaz-Salehi, A. Systemic risk and stability in financial networks. Am. Econ. Rev. 105, 564–608 (2015).

    Article  Google Scholar 

  26. 26.

    Stiglitz, J. E. Risk and global economic architecture: Why full financial integration may be undesirable. Am. Econ. Rev. 100, 388–392 (2010).

    Article  Google Scholar 

  27. 27.

    Haldane, A. G. Rethinking the financial network. Speech given at the Financial Student Association, Amsterdam. Bank of England https://www.bankofengland.co.uk/speech/2009/rethinking-the-financial-network (2009).

  28. 28.

    Yellen, J. L. Interconnectedness and systemic risk – lessons from the financial crisis and policy implications. Speech by Ms Janet L Yellen, Vice Chair of the Board of Governors of the Federal Reserve System, at the American Economic Association/American Finance Association Joint Luncheon, San Diego, California, 4 January 2013. Bank for International Settlements https://www.bis.org/review/r130107a.pdf (2013).

  29. 29.

    Draghi, M. Building on the achievements of post-crisis reforms. Speech by Mario Draghi, President of the ECB and Chair of the European Systemic Risk Board, at the second annual conference of the ESRB, Frankfurt am Main, 21 September 2017. European Central Bank https://www.ecb.europa.eu/press/key/date/2017/html/ecb.sp170921.en.html (2017).

  30. 30.

    Glasserman, P. & Young, H. P. Contagion in financial networks. J. Econ. Lit. 54, 779–831 (2016).

    Article  Google Scholar 

  31. 31.

    International Money Fund. World economic outlook report: October 2018. International Monetary Fund https://www.imf.org/en/Publications/WEO/Issues/2018/09/24/world-economic-outlook-october-2018 (2018).

  32. 32.

    Bardoscia, M., Battiston, S., Caccioli, F. & Caldarelli, G. Pathways towards instability in financial networks. Nat. Commun. 8, 14416 (2017). This paper connects the stability of financial networks to the presence of specific topological structures.

    Article  ADS  Google Scholar 

  33. 33.

    Roukny, T., Battiston, S. & Stiglitz, J. E. Interconnectedness as a source of uncertainty in systemic risk. J. Financial Stab. 35, 93–106 (2018).

    Article  Google Scholar 

  34. 34.

    Colander, D. et al. The financial crisis and the systemic failure of the economics profession. Crit. Rev. 21, 249–267 (2009).

    Article  Google Scholar 

  35. 35.

    Weber, M. Economy and Society: An Outline of Interpretive Sociology (Univ. California Press, 1978).

  36. 36.

    Hill, J. G. & Thomas, R. S. (eds) Research Handbook on Shareholder Power (Edward Elgar Publishing, 2015).

  37. 37.

    Kogut, B. & Walker, G. The small world of Germany and the durability of national networks. Am. Sociol. Rev. 66, 317–335 (2001).

    Article  Google Scholar 

  38. 38.

    Corrado, R. & Zollo, M. Small worlds evolving: governance reforms, privatizations, and ownership networks in Italy. Ind. Corp. Change 15, 319–352 (2006).

    Article  Google Scholar 

  39. 39.

    Fichtner, J., Heemskerk, E. M. & Garcia-Bernardo, J. Hidden power of the Big Three? Passive index funds, re-concentration of corporate ownership, and new financial risk. Bus. Politics 19, 298–326 (2017).

    Article  Google Scholar 

  40. 40.

    Garlaschelli, S., Castri, M. & Servedio, G. The scale free nature of market investment networks. Physica A 350, 491–499 (2005).

    MathSciNet  Article  ADS  Google Scholar 

  41. 41.

    Glattfelder, J. B. & Battiston, S. Backbone of complex networks of corporations: the flow of control. Phys. Rev. E 80, 036104 (2009).

    Article  ADS  Google Scholar 

  42. 42.

    Vitali, S., Glattfelder, J. B. & Battiston, S. The network of global corporate control. PLoS ONE 6, e25995 (2011).

    Article  ADS  Google Scholar 

  43. 43.

    Vitali, S. & Battiston, S. The community structure of the global corporate network. PLoS ONE 9, e104655 (2014).

    Article  ADS  Google Scholar 

  44. 44.

    Vitali, S. & Battiston, S. Geography versus topology in the European ownership network. New J. Phys. 13, 63021 (2011).

    Article  Google Scholar 

  45. 45.

    Glattfelder, J. B. & Battiston, S. The architecture of power: Patterns of disruption and stability in the global ownership network. Social Science Research Network https://ssrn.com/abstract=3314648 (2019).

  46. 46.

    Boss, M., Elsinger, H., Summer, M. & Thurner, S. Network topology of the interbank market. Quant. Finance 4, 677–684 (2004). This is one of the first papers to empirically characterize a real-world (the Austrian) interbank network.

    MATH  Article  Google Scholar 

  47. 47.

    Soramäki, K., Bech, M. L., Arnold, J., Glass, R. J. & Beyeler, W. E. The topology of interbank payment flows. Physica A 379, 317–333 (2007).

    Article  ADS  Google Scholar 

  48. 48.

    Müller, J. Interbank credit lines as a channel of contagion. J. Financial Serv. Res. 29, 37–60 (2006).

    Article  ADS  Google Scholar 

  49. 49.

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

    Article  ADS  Google Scholar 

  50. 50.

    Iori, G., De Masi, G., Precup, O. V., Gabbi, G. & Caldarelli, G. A network analysis of the Italian overnight money market. J. Econ. Dyn. Control 32, 259–278 (2008). This paper provides an empirical characterization of the network of the Italian segment of the European overnight money.

    MATH  Article  Google Scholar 

  51. 51.

    Iazzetta, C. & Manna, M. The topology of the interbank market: developments in Italy since 1990. Social Science Research Network https://ssrn.com/abstract=1478472 (2009).

  52. 52.

    Finger, K., Fricke, D. & Lux, T. Network analysis of the e-MID overnight money market: the informational value of different aggregation levels for intrinsic dynamic processes. Comput. Manag. Sci. 10, 187–211 (2013).

    MathSciNet  MATH  Article  Google Scholar 

  53. 53.

    Demiralp, S., Preslopsky, B. & Whitesell, W. Overnight interbank loan markets. J. Econ. Bus. 58, 67–83 (2006).

    Article  Google Scholar 

  54. 54.

    Bech, M. L. & Atalay, E. The topology of the federal funds market. Physica A 389, 5223–5246 (2010).

    Article  ADS  Google Scholar 

  55. 55.

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

    Google Scholar 

  56. 56.

    Cajueiro, D. O. & Tabak, B. M. The role of banks in the Brazilian interbank market: does bank type matter? Physica A 387, 6825–6836 (2008).

    Article  ADS  Google Scholar 

  57. 57.

    e Santos, E. B. & Cont, R. The Brazilian interbank network structure and systemic risk. Working Paper n. 219. Banco Central do Brasil https://www.bcb.gov.br/pec/wps/ingl/wps219.pdf (2010).

  58. 58.

    Imakubo, K. & Soejima, Y. et al. The transaction network in Japan’s interbank money markets. Monet. Econ. Stud. 28, 107–150 (2010).

    Google Scholar 

  59. 59.

    Squartini, T., van Lelyveld, I. & Garlaschelli, D. Early-warning signals of topological collapse in interbank networks. Sci. Rep. 3, 3357 (2013).

    Article  Google Scholar 

  60. 60.

    León, C. & Berndsen, R. J. Rethinking financial stability: challenges arising from financial networks’ modular scale-free architecture. J. Financial Stab. 15, 241–256 (2014).

    Article  Google Scholar 

  61. 61.

    Craig, B. & Von Peter, G. Interbank tiering and money center banks. J. Financ. Intermed. 23, 322–347 (2014).

    Article  Google Scholar 

  62. 62.

    Martinez-Jaramillo, S., Alexandrova-Kabadjova, B., Bravo-Benitez, B. & Solórzano-Margain, J. P. An empirical study of the Mexican banking system’s network and its implications for systemic risk. J. Econ. Dyn. Control 40, 242–265 (2014).

    Article  Google Scholar 

  63. 63.

    Van Lelyveld, I. et al. Finding the core: Network structure in interbank markets. J. Bank. Finance 49, 27–40 (2014).

    Article  Google Scholar 

  64. 64.

    Gabrieli, S. & Georg, C.-P. A network view on interbank market freezes. Social Science Research Network https://ssrn.com/abstract=2797027 (2014).

  65. 65.

    Fricke, D. & Lux, T. Core–periphery structure in the overnight money market: Evidence from the e-MID trading platform. Comput. Econ. 45, 359–395 (2015).

    Article  Google Scholar 

  66. 66.

    Silva, T. C., Guerra, S. M., Tabak, B. M. & de Castro Miranda, R. C. Financial networks, bank efficiency and risk-taking. J. Financial Stab. 25, 247–257 (2016).

    Article  Google Scholar 

  67. 67.

    Kojaku, S., Cimini, G., Caldarelli, G. & Masuda, N. Structural changes in the interbank market across the financial crisis from multiple core–periphery analysis. J. Netw. Theory Finance 4, 33–51 (2018).

    Article  Google Scholar 

  68. 68.

    In’t Veld, D., van der Leij, M. & Hommes, C. The formation of a core-periphery structure in heterogeneous financial networks. J. Econ. Dyn. Control 119, 103972 (2020).

    MathSciNet  Article  Google Scholar 

  69. 69.

    Barucca, P. & Lillo, F. Disentangling bipartite and core-periphery structure in financial networks. Chaos Solitons Fract. 88, 244–253 (2016).

    MathSciNet  MATH  Article  ADS  Google Scholar 

  70. 70.

    Cont, R. & Kokholm, T. Central clearing of OTC derivatives: bilateral vs multilateral netting. Stat. Risk Model. 31, 3–22 (2014).

    MathSciNet  MATH  Google Scholar 

  71. 71.

    Cont, R. & Minca, A. Credit default swaps and systemic risk. Ann. Oper. Res. 247, 523–547 (2016).

    MathSciNet  MATH  Article  Google Scholar 

  72. 72.

    Duffie, D., Scheicher, M. & Vuillemey, G. Central clearing and collateral demand. J. Financial Econ. 116, 237–256 (2015).

    Article  Google Scholar 

  73. 73.

    Heath, A., Kelly, G., Manning, M., Markose, S. & Shaghaghi, A. R. CCPs and network stability in OTC derivatives markets. J. Financial Stab. 27, 217–233 (2016).

    Article  Google Scholar 

  74. 74.

    Markose, S., Giansante, S. & Shaghaghi, A. R. A systemic risk assessment of otc derivatives reforms and skin-in-the-game for CCPs. Bank of France https://publications.banque-france.fr/sites/default/files/medias/documents/fsr21_web.pdf#page=111 (2017).

  75. 75.

    Duffie, D. & Zhu, H. Does a central clearing counterparty reduce counterparty risk? Rev. Asset Pricing Stud. 1, 74–95 (2011).

    Article  Google Scholar 

  76. 76.

    Poce, G. et al. What do central counterparties default funds really cover? A network-based stress test answer. J. Netw. Theory Finance 4, 43–57 (2018).

    Article  Google Scholar 

  77. 77.

    Huang, X., Vodenska, I., Wang, F., Havlin, S. & Stanley, H. E. Identifying influential directors in the United States corporate governance network. Phys. Rev. E 84, 046101 (2011).

    Article  ADS  Google Scholar 

  78. 78.

    Huang, X., Vodenska, I., Havlin, S. & Stanley, H. E. Cascading failures in bi-partite graphs: model for systemic risk propagation. Sci. Rep. 3, 1219 (2013).

    Article  ADS  Google Scholar 

  79. 79.

    Mantegna, R. N. Hierarchical structure in financial markets. Eur. Phys. J. B 11, 193–197 (1999). This paper builds a hierarchical tree of stocks starting from the correlations of their returns.

    Article  ADS  Google Scholar 

  80. 80.

    Tumminello, M., Aste, T., Di Matteo, T. & Mantegna, R. N. A tool for filtering information in complex systems. Proc. Natl Acad. Sci. USA 102, 10421–10426 (2005).

    Article  ADS  Google Scholar 

  81. 81.

    Kremer, M., Becker, A. P., Vodenska, I., Stanley, H. E. & Schäfer, R. Economic and political effects on currency clustering dynamics. Quant. Finance 19, 705–716 (2019).

    MathSciNet  Article  Google Scholar 

  82. 82.

    Billio, M., Getmansky, M., Lo, A. & L., P. Econometric measures of connectedness and systemic risk in the finance and insurance sectors. J. Financ. Econ. 104, 535–559 (2012).

    Article  Google Scholar 

  83. 83.

    MacMahon, M. & Garlaschelli, D. Community detection for correlation matrices. Phys. Rev. X 5, 021006 (2015).

    Google Scholar 

  84. 84.

    Almog, A., Besamusca, F., MacMahon, M. & Garlaschelli, D. Mesoscopic community structure of financial markets revealed by price and sign fluctuations. PLoS ONE 10, e0133679 (2015).

    Article  Google Scholar 

  85. 85.

    Anagnostou, I., Squartini, T., Kandhai, D. & Garlaschelli, D. Uncovering the mesoscale structure of the credit default swap market to improve portfolio risk modelling. Quant. Finance 10.1080/14697688.2021.1890807 (2021).

  86. 86.

    Verma, A., Buonocore, R. J. & Di Matteo, T. A cluster driven log-volatility factor model: a deepening on the source of the volatility clustering. Quant. Finance 19, 981–996 (2019).

    MathSciNet  MATH  Article  Google Scholar 

  87. 87.

    Bonanno, G., Caldarelli, G., Lillo, F. & Mantegna, R. N. Topology of correlation-based minimal spanning trees in real and model markets. Phys. Rev. E 68, 046130 (2003).

    Article  ADS  Google Scholar 

  88. 88.

    Bartesaghi, P., Benzi, M., Clemente, G. P., Grassi, R. & Estrada, E. Risk-dependent centrality in economic and financial networks. SIAM J. Financ. Math. 11, 526–565 (2020).

    MathSciNet  MATH  Article  Google Scholar 

  89. 89.

    Laloux, L., Cizeau, P., Bouchaud, J.-P. & Potters, M. Noise dressing of financial correlation matrices. Phys. Rev. Lett. 83, 1467 (1999).

    Article  ADS  Google Scholar 

  90. 90.

    Plerou, V. et al. Random matrix approach to cross correlations in financial data. Phys. Rev. E 65, 066126 (2002).

    Article  ADS  Google Scholar 

  91. 91.

    Lillo, F. & Mantegna, R. Spectral density of the correlation matrix of factor models: A random matrix theory approach. Phys. Rev. E 72, 016219 (2005).

    MathSciNet  Article  ADS  Google Scholar 

  92. 92.

    Barucca, P., Kieburg, M. & Ossipov, A. Eigenvalue and eigenvector statistics in time series analysis. Europhys. Lett. 129, 60003 (2020).

    Article  ADS  Google Scholar 

  93. 93.

    Mehta, M. L. Random Matrices (Elsevier, 2004).

  94. 94.

    Livan, G., Novaes, M. & Vivo, P. Introduction to Random Matrices: Theory and Practice Vol. 26 (Springer, 2018).

  95. 95.

    Ahelegbey, D., Billio, M. & R., C. Bayesian graphical models for structural vector autoregressive processes. J. Appl. Econ. 31, 357–386 (2016).

    MathSciNet  Article  Google Scholar 

  96. 96.

    Ahelegbey, D., Billio, M. & R., C. Sparse graphical vector autoregression: a Bayesian approach. Ann. Econ. Stat. 123/124, 333–361 (2016).

    Article  Google Scholar 

  97. 97.

    Bianconi, G. Multilayer Networks: Structure and Function (Oxford Univ. Press, 2018).

  98. 98.

    Battiston, S., Caldarelli, G. & Garas, A. Multiplex and Multilevel Networks (Oxford Univ. Press, 2018).

  99. 99.

    Langfield, S., Liu, Z. & Ota, T. Mapping the UK interbank system. J. Bank. Finance 45, 288–303 (2014).

    Article  Google Scholar 

  100. 100.

    Molina-Borboa, J. L., Martínez-Jaramillo, S., López-Gallo, F. & van der Leij, M. A multiplex network analysis of the Mexican banking system: link persistence, overlap and waiting times. J. Netw. Theory Finance 1, 99–138 (2015).

    Article  Google Scholar 

  101. 101.

    Bargigli, L., di Iasio, G., Infante, L. & Pierobon, F. The multiplex structure of interbank networks. Quant. Finance 15, 673–691 (2015).

    MathSciNet  MATH  Article  Google Scholar 

  102. 102.

    Vodenska, I. et al. Community analysis of global financial markets. Risks 4, 13 (2016).

    Article  Google Scholar 

  103. 103.

    Vodenska, I., Aoyama, H., Fujiwara, Y., Iyetomi, H. & Arai, Y. Interdependencies and causalities in coupled financial networks. PLoS ONE 11, e0150994 (2016).

    Article  Google Scholar 

  104. 104.

    Curme, C., Stanley, H. E. & Vodenska, I. Coupled network approach to predictability of financial market returns and news sentiments. Int. J. Theor. Appl. Finance 18, 1550043 (2015).

    MathSciNet  MATH  Article  Google Scholar 

  105. 105.

    Berndsen, R. J., León, C. & Renneboog, L. Financial stability in networks of financial institutions and market infrastructures. J. Financ. Stab. 35, 120–135 (2016).

    Article  Google Scholar 

  106. 106.

    Bardoscia, M., Bianconi, G. & Ferrara, G. Multiplex network analysis of the UK over-the-counter derivatives market. Int. J. Finance Econ. 24, 1520–1544 (2019).

    Article  Google Scholar 

  107. 107.

    de Jeude, Jv. L., Aste, T. & Caldarelli, G. The multilayer structure of corporate networks. New J. Phys. 21, 025002 (2019).

    Article  ADS  Google Scholar 

  108. 108.

    Heise, S. & Kühn, R. Derivatives and credit contagion in interconnected networks. Eur. Phys. J. B 85, 115 (2012).

    Article  ADS  Google Scholar 

  109. 109.

    Brunnermeier, M. K. et al. Assessing contagion risks in the CDS market. Occasional Paper Series, no. 4. European Systemic Risk Board https://www.esrb.europa.eu/pub/pdf/occasional/20130917_occasional_paper_4.pdf (2013).

  110. 110.

    Roukny, T., George, C.-P. & Battiston, S. A network analysis of the evolution of the German interbank market. Social Science Research Network https://ssrn.com/abstract=2796998 (2014).

  111. 111.

    D’Errico, M., Battiston, S., Peltonen, T. & Scheicher, M. How does risk flow in the credit default swap market? J. Financ. Stab. 35, 53–74 (2018).

    Article  Google Scholar 

  112. 112.

    Schuldenzucker, S., Seuken, S. & Battiston, S. Default ambiguity: Credit default swaps create new systemic risks in financial networks. Manage. Sci. 66, 1783–2290 (2019).

    Google Scholar 

  113. 113.

    Papp, P. A. & Wattenhofer, R. Default ambiguity: finding the best solution to the clearing problem. Preprint at arXiv https://arxiv.org/abs/2002.07741 (2020).

  114. 114.

    Papp, P. A. & Wattenhofer, R. Sequential defaulting in financial networks. Preprint at arXiv https://arxiv.org/abs/2011.10485 (2020).

  115. 115.

    Eisenberg, L. & Noe, T. H. Systemic risk in financial systems. Manage. Sci. 47, 236–249 (2001). This paper introduces the foundational framework for clearing of interbank obligations and derives key results on the existence of solutions.

    MATH  Article  Google Scholar 

  116. 116.

    Rogers, L. C. & Veraart, L. A. Failure and rescue in an interbank network. Manage. Sci. 59, 882–898 (2013).

    Article  Google Scholar 

  117. 117.

    Banerjee, T. & Feinstein, Z. Impact of contingent payments on systemic risk in financial networks. Math. Financ. Econ. 13, 617–636 (2019).

    MathSciNet  MATH  Article  Google Scholar 

  118. 118.

    Bardoscia, M., Ferrara, G., Vause, N. & Yoganayagam, M. Full payment algorithm. Social Science Research Network https://ssrn.com/abstract=3344580 (2019).

  119. 119.

    Banerjee, T., Bernstein, A. & Feinstein, Z. Dynamic clearing and contagion in financial networks. Preprint at arXiv https://arxiv.org/abs/1801.02091 (2018).

  120. 120.

    Paddrik, M., Rajan, S. & Young, H. P. Contagion in derivatives markets. Manage. Sci. 66, 3295–3798 (2020).

    Google Scholar 

  121. 121.

    Bardoscia, M., Ferrara, G., Vause, N. & Yoganayagam, M. Simulating liquidity stress in the derivatives market. Social Science Research Network https://ssrn.com/abstract=3508655 (2019).

  122. 122.

    Cont, R., Moussa, A. & Santos, E. B. in Handbook on Systemic Risk (eds Fouque, J.-P. & Langsam, J. A.) chap. 13, 327–336 (Cambridge Univ. Press, 2013).

  123. 123.

    Furfine, C. H. Interbank exposures: quantifying the risk of contagion. J. Money Credit Bank. 35, 111–129 (2003).

    Article  Google Scholar 

  124. 124.

    Gleeson, J. P., Hurd, T., Melnik, S. & Hackett, A. in Advances in Network Analysis and its Applications (ed. Kranakis, E.) 27–56 (Springer, 2012).

  125. 125.

    Amini, H., Cont, R. & Minca, A. Resilience to contagion in financial networks. Math. Finance 26, 329–356 (2016).

    MathSciNet  MATH  Article  Google Scholar 

  126. 126.

    Bardoscia, M., Battiston, S., Caccioli, F. & Caldarelli, G. DebtRank: a microscopic foundation for shock propagation. PLoS ONE 10, e0130406 (2015).

    Article  Google Scholar 

  127. 127.

    Bardoscia, M., Caccioli, F., Perotti, J. I., Vivaldo, G. & Caldarelli, G. Distress propagation in complex networks: the case of non-linear DebtRank. PLoS ONE 11, e0163825 (2016).

    Article  Google Scholar 

  128. 128.

    Fink, K., Krüger, U., Meller, B. & Wong, L.-H. The credit quality channel: modeling contagion in the interbank market. J. Financ. Stab. 25, 83–97 (2016).

    Article  Google Scholar 

  129. 129.

    Elsinger, H., Lehar, A. & Summer, M. Risk assessment for banking systems. Manag. Sci. 52, 1301–1314 (2006).

    MATH  Article  Google Scholar 

  130. 130.

    Fischer, T. No-arbitrage pricing under systemic risk: accounting for cross-ownership. Math. Finance 24, 97–124 (2014).

    MathSciNet  MATH  Article  Google Scholar 

  131. 131.

    Barucca, P. Network valuation in financial systems. Math. Finance 30, 1181–1204 (2020). This paper shows that different previously unrelated models of financial contagion can all be mapped to valuation models on a network.

    MathSciNet  MATH  Article  Google Scholar 

  132. 132.

    Bardoscia, M., Barucca, P., Codd, A. B. & Hill, J. Forward-looking solvency contagion. J. Econ. Dyn. Control 108, 103755 (2019).

    MathSciNet  MATH  Article  Google Scholar 

  133. 133.

    Basel Committee on Banking Supervision. Basel III: a global regulatory framework for more resilient banks and banking systems. Bank for International Settlements https://www.bis.org/publ/bcbs189.pdf (2011).

  134. 134.

    Wells, S. Financial interlinkages in the United Kingdom’s interbank market and the risk of contagion. Bank of England http://www.bankofengland.co.uk/archive/Documents/historicpubs/workingpapers/2004/wp230.pdf (2004).

  135. 135.

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

    Article  Google Scholar 

  136. 136.

    Hüser, A.-C., Hałaj, G., Kok, C., Perales, C. & van der Kraaij, A. The systemic implications of bail-in: a multi-layered network approach. J. Financ. Stab. 38, 81–97 (2018).

    Article  Google Scholar 

  137. 137.

    Feinstein, Z. et al. Sensitivity of the Eisenberg–Noe clearing vector to individual interbank liabilities. SIAM J. Financ. Math. 9, 1286–1325 (2018).

    MathSciNet  MATH  Article  Google Scholar 

  138. 138.

    Gai, P., Haldane, A. & Kapadia, S. Complexity, concentration and contagion. J. Monet. Econ. 58, 453–470 (2011).

    Article  Google Scholar 

  139. 139.

    Brandi, G., Clemente, R. D. & Cimini, G. Epidemics of liquidity shortages in interbank markets. Physica A 507, 255–267 (2018).

    Article  ADS  Google Scholar 

  140. 140.

    Cimini, G. & Serri, M. Entangling credit and funding shocks in interbank markets. PLoS ONE 11, e0161642 (2016).

    Article  Google Scholar 

  141. 141.

    Allen, F. & Gale, D. Financial contagion. J. Polit. Econ. 108, 1–33 (2000). This paper builds a model of financial contagion that shows that fully connected networks are more robust than sparser networks.

    Article  Google Scholar 

  142. 142.

    Freixas, X., Parigi, B. M. & Rochet, J.-C. Systemic risk, interbank relations, and liquidity provision by the central bank. J. Money Credit Bank. 32, 611–638 (2000).

    Article  Google Scholar 

  143. 143.

    Battiston, S., Gatti, D. D., Gallegati, M., Greenwald, B. & Stiglitz, J. E. Default cascades: when does risk diversification increase stability? J. Financ. Stab. 8, 138–149 (2012).

    Article  Google Scholar 

  144. 144.

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

    MathSciNet  MATH  Article  Google Scholar 

  145. 145.

    Acemoglu, D., Ozdaglar, A. & Tahbaz-Salehi, A. Systemic risk and stability in financial networks. Am. Econ. Rev. 105, 564–608 (2015).

    Article  Google Scholar 

  146. 146.

    Gai, P. & Kapadia, S. Contagion in financial networks. Proc. R. Soc. A 466, 2401–2423 (2010). This is one of the first papers to show that financial networks may be robust yet fragile, meaning that contagion events are rare but extremely severe.

    MathSciNet  MATH  Article  ADS  Google Scholar 

  147. 147.

    Nier, E., Yang, J., Yorulmazer, T. & Alentorn, A. Network models and financial stability. J. Econ. Dyn. Control 31, 2033–2060 (2007).

    MATH  Article  Google Scholar 

  148. 148.

    Kobayashi, T. Network versus portfolio structure in financial systems. Eur. Phys. J. B 86, 434 (2013).

    MathSciNet  Article  ADS  Google Scholar 

  149. 149.

    Lenzu, S. & Tedeschi, G. Systemic risk on different interbank network topologies. Physica A 391, 4331–4341 (2012).

    Article  ADS  Google Scholar 

  150. 150.

    Roukny, T., Bersini, H., Pirotte, H., Caldarelli, G. & Battiston, S. Default cascades in complex networks: topology and systemic risk. Sci. Rep. 3, 2759 (2013).

    Article  ADS  Google Scholar 

  151. 151.

    Markose, S., Giansante, S. & Shaghaghi, A. R. ‘Too interconnected to fail’ financial network of US CDS market: Topological fragility and systemic risk. J. Econ. Behav. Organ. 83, 627–646 (2012).

    Article  Google Scholar 

  152. 152.

    Glasserman, P. & Young, H. P. How likely is contagion in financial networks? J. Bank. Finance 50, 383–399 (2015).

    Article  Google Scholar 

  153. 153.

    Ramadiah, A. et al. Network sensitivity of systemic risk. J. Netw. Theory Finance 5, 53–72 (2020).

    Google Scholar 

  154. 154.

    Batiz-Zuk, E., López-Gallo, F., Martínez-Jaramillo, S. & Solórzano-Margain, J. P. Calibrating limits for large interbank exposures from a system-wide perspective. J. Financ. Stab. 27, 198–216 (2016).

    Article  Google Scholar 

  155. 155.

    Capponi, A., Dooley, J. M., Oet, M. V. & Ong, S. J. Capital and resolution policies: the US interbank market. J. Financ. Stab. 30, 229–239 (2016).

    Article  Google Scholar 

  156. 156.

    Alter, A., Craig, B. & Raupach, P. Centrality-based capital allocations and bailout. Int. J. Cent. Bank. 11, 329–377 (2015).

    Google Scholar 

  157. 157.

    Minca, A. & Sulem, A. Optimal control of interbank contagion under complete information. Stat. Risk Model. 31, 23–48 (2014).

    MathSciNet  MATH  Google Scholar 

  158. 158.

    Capponi, A. & Chen, P.-C. Systemic risk mitigation in financial networks. J. Econ. Dyn. Control 58, 152–166 (2015).

    MathSciNet  MATH  Article  Google Scholar 

  159. 159.

    Jackson, M. O. & Pernoud, A. Credit freezes, equilibrium multiplicity, and optimal bailouts in financial networks. Social Science Research Network https://ssrn.com/abstract=3735251 (2021).

  160. 160.

    Majdandzic, A. et al. Multiple tipping points and optimal repairing in interacting networks. Nat. Commun. 7, 10850 (2016).

    Article  ADS  Google Scholar 

  161. 161.

    Delpini, D. et al. Evolution of controllability in interbank networks. Sci. Rep. 3, 1626 (2013).

    Article  Google Scholar 

  162. 162.

    Galbiati, M., Delpini, D. & Battiston, S. The power to control. Nat. Phys. 9, 126–128 (2013).

    Article  Google Scholar 

  163. 163.

    Poledna, S. & Thurner, S. Elimination of systemic risk in financial networks by means of a systemic risk transaction tax. Quant. Finance 16, 1599–1613 (2016).

    MathSciNet  MATH  Article  Google Scholar 

  164. 164.

    Thurner, S. & Poledna, S. DebtRank-transparency: Controlling systemic risk in financial networks. Sci. Rep. 3, 1888 (2013).

    Article  ADS  Google Scholar 

  165. 165.

    Diem, C., Pichler, A. & Thurner, S. What is the minimal systemic risk in financial exposure networks?. J. Econ. Dyn. Control 116, 103900 (2020).

    MathSciNet  MATH  Article  Google Scholar 

  166. 166.

    Caccioli, F., Shrestha, M., Moore, C. & Farmer, J. D. Stability analysis of financial contagion due to overlapping portfolios. J. Bank. Finance 46, 233–245 (2014). This paper introduces a network model of contagion due to fire sales and overlapping portfolios, and it shows how contagion through this channel can be modelled as a multitype Galton–Watson process.

    Article  Google Scholar 

  167. 167.

    Greenwood, R., Landier, A. & Thesmar, D. Vulnerable banks. J. Financ. Econ. 115, 471–485 (2015).

    Article  Google Scholar 

  168. 168.

    Corsi, F., Marmi, S. & Lillo, F. When micro prudence increases macro risk: the destabilizing effects of financial innovation, leverage, and diversification. Oper. Res. 64, 1073–1088 (2016).

    MathSciNet  MATH  Article  Google Scholar 

  169. 169.

    Duarte, F. & Eisenbach, T. M. Fire-sale spillovers and systemic risk. Social Science Research Network https://ssrn.com/abstract=2340669 (2018).

  170. 170.

    Cont, R. & Schaanning, E. Fire sales, indirect contagion and systemic stress testing. Social Science Research Network https://ssrn.com/abstract=2541114 (2017).

  171. 171.

    Bouchaud, J.-P., Farmer, J. D. & Lillo, F. in Handbook of Financial Markets: Dynamics and Evolution (eds Hens, T. & Schenk-Hoppé, K. R.) 57–160 (Elsevier, 2009).

  172. 172.

    Watts, D. J. A simple model of global cascades on random networks. Proc. Natl Acad. Sci. USA 99, 5766–5771 (2002).

    MathSciNet  MATH  Article  ADS  Google Scholar 

  173. 173.

    Ramadiah, A., Fricke, D. & Caccioli, F. Backtesting macroprudential stress tests. Social Science Research Network https://ssrn.com/abstract=3674323 (2020).

  174. 174.

    Sakamoto, Y. & Vodenska, I. Systemic risk propagation in bank-asset network: new perspective on Japanese banking crisis in the 1990s. J. Complex Netw. 5, 315–333 (2016).

    Google Scholar 

  175. 175.

    Sakamoto, Y. & Vodenska, I. Impact of bankruptcy through asset portfolios. Eur. Phys. J. Spec. Top. 225, 1311–1316 (2016).

    Article  Google Scholar 

  176. 176.

    Smolyak, A., Levy, O., Vodenska, I., Buldyrev, S. & Havlin, S. Mitigation of cascading failures in complex networks. Sci. Rep. 10, 16124 (2020).

    Article  ADS  Google Scholar 

  177. 177.

    Adrian, T. & Shin, H. S. Liquidity and leverage. J. Financ. Intermed. 19, 418–437 (2010).

    Article  Google Scholar 

  178. 178.

    Shin, H. S. Risk and Liquidity (Oxford Univ. Press, 2010).

  179. 179.

    Cont, R. & Schaanning, E. Monitoring indirect contagion. J. Bank. Finance 104, 85–102 (2019).

    Article  Google Scholar 

  180. 180.

    Guo, W., Minca, A. & Wang, L. The topology of overlapping portfolio networks. Stat. Risk Model. 33, 139–155 (2016).

    MathSciNet  MATH  Google Scholar 

  181. 181.

    Braverman, A. & Minca, A. Networks of common asset holdings: aggregation and measures of vulnerability. J. Netw. Theory Finance 4, 53–78 (2018).

    Article  Google Scholar 

  182. 182.

    Fricke, C. & Fricke, D. Vulnerable asset management? The case of mutual funds. Social Science Research Network https://ssrn.com/abstract=2866301 (2017).

  183. 183.

    Baranova, Y., Coen, J., Noss, J. & Lowe, P. Simulating stress across the financial system: the resilience of corporate bond markets and the role of investment funds. Social Science Research Network https://ssrn.com/abstract=3134656 (2017).

  184. 184.

    Fricke, D. & Wilke, H. Connected funds. Social Science Research Network https://ssrn.com/abstract=3684270 (2020).

  185. 185.

    Delpini, D., Battiston, S., Caldarelli, G. & Riccaboni, M. Systemic risk from investment similarities. PLoS ONE 14, e0217141 (2019).

    Article  Google Scholar 

  186. 186.

    Farmer, J. D., Kleinnijenhuis, A. M., Nahai-Williamson, P. & Wetzer, T. Foundations of system-wide financial stress testing with heterogeneous institutions. Social Science Research Network https://ssrn.com/abstract=3601846 (2020).

  187. 187.

    Caccioli, F., Ferrara, G. & Ramadiah, A. Modelling fire sale contagion across banks and non-banks. Social Science Research Network https://ssrn.com/abstract=3647204 (2020).

  188. 188.

    Kusnetsov, M. & Veraart, L. A. M. Interbank clearing in financial networks with multiple maturities. SIAM J. Financ. Math. 10, 37–67 (2019).

    MathSciNet  MATH  Article  Google Scholar 

  189. 189.

    Feinstein, Z. Obligations with physical delivery in a multilayered financial network. SIAM J. Financ. Math. 10, 877–906 (2019).

    MathSciNet  MATH  Article  Google Scholar 

  190. 190.

    Burkholz, R., Leduc, M. V., Garas, A. & Schweitzer, F. Systemic risk in multiplex networks with asymmetric coupling and threshold feedback. Physica D 323, 64–72 (2016).

    MathSciNet  MATH  Article  ADS  Google Scholar 

  191. 191.

    Brummitt, C. D. & Kobayashi, T. Cascades in multiplex financial networks with debts of different seniority. Phys. Rev. E 91, 062813 (2015).

    Article  ADS  Google Scholar 

  192. 192.

    Poledna, S., Molina-Borboa, J. L., Martínez-Jaramillo, S., van der Leij, M. & Thurner, S. The multi-layer network nature of systemic risk and its implications for the costs of financial crises. J. Financ. Stab. 20, 70–81 (2015).

    Article  Google Scholar 

  193. 193.

    Montagna, M. & Kok, C. Multi-layered interbank model for assessing systemic risk. Working Paper Series, no. 1944. European Central Bank https://www.ecb.europa.eu/pub/pdf/scpwps/ecbwp1944.en.pdf (2016).

  194. 194.

    Bookstaber, R. & Kenett, D. Y. Looking deeper, seeing more: a multilayer map of the financial system. OFR Brief Series, no. 16-06. Office of Financial Research https://www.financialresearch.gov/briefs/files/OFRbr_2016-06_Multilayer-Map.pdf (2016).

  195. 195.

    Cifuentes, R., Ferrucci, G. & Shin, H. S. Liquidity risk and contagion. J. Eur. Econ. Assoc. 3, 556–566 (2005).

    Article  Google Scholar 

  196. 196.

    Caccioli, F., Farmer, J. D., Foti, N. & Rockmore, D. Overlapping portfolios, contagion, and financial stability. J. Econ. Dyn. Control 51, 50–63 (2015).

    MathSciNet  MATH  Article  Google Scholar 

  197. 197.

    Poledna, S., Martínez-Jaramillo, S., Caccioli, F. & Thurner, S. Quantification of systemic risk from overlapping portfolios in the financial system. J. Financial Stab. 52, 100808 (2021).

    Article  Google Scholar 

  198. 198.

    Squartini, T. & Garlaschelli, D. Maximum-Entropy Networks. Pattern Detection, Network Reconstruction and Graph Combinatorics (Springer, 2017).

  199. 199.

    Squartini, T. & Garlaschelli, D. Stationarity, non-stationarity and early warning signals in economic networks. J. Complex Netw. 3, 1–21 (2015).

    MathSciNet  MATH  Article  Google Scholar 

  200. 200.

    Mastrandrea, R., Squartini, T., Fagiolo, G. & Garlaschelli, D. Enhanced reconstruction of weighted networks from strengths and degrees. New J. Phys. 16, 043022 (2014).

    Article  ADS  Google Scholar 

  201. 201.

    Acemoglu, D., Ozdaglar, A. E. & Tahbaz-Salehi, A. Systemic risk in endogenous financial networks. Social Science Research Network https://ssrn.com/abstract=2553900 (2015).

  202. 202.

    Cabrales, A., Gottardi, P. & Vega-Redondo, F. Risk sharing and contagion in networks. Rev. Financ. Stud. 30, 3086–3127 (2017).

    Article  Google Scholar 

  203. 203.

    Farboodi, M. Intermediation and voluntary exposure to counterparty risk. Social Science Research Network https://ssrn.com/abstract=2535900 (2014).

  204. 204.

    Bargigli, L., Lionetto, A. & Viaggiu, S. A statistical test of Walrasian equilibrium by means of complex networks theory. J. Stat. Phys. 165, 351–370 (2016).

    MathSciNet  MATH  Article  ADS  Google Scholar 

  205. 205.

    Squartini, T., Caldarelli, G., Cimini, G., Gabrielli, A. & Garlaschelli, D. Reconstruction methods for networks: the case of economic and financial systems. Phys. Rep. 208, 1–47 (2018).

    MathSciNet  MATH  Article  ADS  Google Scholar 

  206. 206.

    Upper, C. & Worms, A. Estimating bilateral exposures in the German interbank market: Is there a danger of contagion? Eur. Econ. Rev. 48, 827–849 (2004).

    Article  Google Scholar 

  207. 207.

    Parisi, F., Squartini, T. & Garlaschelli, D. A faster horse on a safer trail: generalized inference for the efficient reconstruction of weighted networks. New J. Phys. 22, 053053 (2020).

    MathSciNet  Article  ADS  Google Scholar 

  208. 208.

    Anand, K., Craig, B. & Von Peter, G. Filling in the blanks: Network structure and interbank contagion. Quant. Finance 15, 625–636 (2015).

    MathSciNet  MATH  Article  Google Scholar 

  209. 209.

    Gandy, A. & Veraart, L. A. A Bayesian methodology for systemic risk assessment in financial networks. Manage. Sci. 63, 4428–4446 (2016).

    Article  Google Scholar 

  210. 210.

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

    MathSciNet  Article  ADS  Google Scholar 

  211. 211.

    Squartini, T. & Garlaschelli, D. Analytical maximum-likelihood method to detect patterns in real networks. New J. Phys. 13, 083001 (2011).

    MATH  Article  ADS  Google Scholar 

  212. 212.

    Shannon, C. E. A mathematical theory of communication. Bell Syst. Tech. J. 27, 623–656 (1948).

    MathSciNet  MATH  Article  Google Scholar 

  213. 213.

    Jaynes, E. T. Information theory and statistical mechanics. Phys. Rev. 106, 620–630 (1957). Milestone paper showing that equilibrium statistical mechanics provides an unbiased prescription to make inferences from partial information.

    MathSciNet  MATH  Article  ADS  Google Scholar 

  214. 214.

    Gabrielli, A., Mastrandrea, R., Caldarelli, G. & Cimini, G. Grand canonical ensemble of weighted networks. Phys. Rev. E 99, 030301 (2019).

    Article  ADS  Google Scholar 

  215. 215.

    Cimini, G., Squartini, T., Gabrielli, A. & Garlaschelli, D. Estimating topological properties of weights networks from limited information. Phys. Rev. E 92, 040802 (2015).

    Article  ADS  Google Scholar 

  216. 216.

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

    Article  ADS  Google Scholar 

  217. 217.

    Anand, K. et al. The missing links: a global study on uncovering financial network structures from partial data. J. Financ. Stab. 35, 107–119 (2018). This paper performs an extensive comparison of reconstruction methods on various empirical financial networks.

    Article  Google Scholar 

  218. 218.

    Mazzarisi, P. & Lillo, F. in Econophysics and Sociophysics: Recent Progress and Future Directions (eds Abergel, F. et al.) 201–215 (Springer, 2017).

  219. 219.

    Ramadiah, A., Caccioli, F. & Fricke, D. Reconstructing and stress testing credit networks. J. Econ. Dyn. Control 111, 103817 (2020).

    MathSciNet  MATH  Article  Google Scholar 

  220. 220.

    Lebacher, M., Cook, S., Klein, N. & Kauermann, G. In search of lost edges: a case study on reconstructing financial networks. Preprint at arXiv https://arxiv.org/abs/1909.01274 (2019).

  221. 221.

    Serrano, M. & Boguna, M. Topology of the world trade web. Phys. Rev. E 68, 015101 (2003).

    Article  ADS  Google Scholar 

  222. 222.

    Fagiolo, G., Reyes, J. & Schiavo, S. World-trade web: topological properties, dynamics, and evolution. Phys. Rev. E 79, 036115 (2009).

    MathSciNet  Article  ADS  Google Scholar 

  223. 223.

    Barigozzi, M., Fagiolo, G. & Garlaschelli, D. Multinetwork of international trade: a commodity-specific analysis. Phys. Rev. E 81, 046104 (2010).

    Article  ADS  Google Scholar 

  224. 224.

    Fronczak, A. Structural Hamiltonian of the international trade network. Acta Phys. Pol. B 5, 31–46 (2012).

    Google Scholar 

  225. 225.

    Fronczak, A. & Fronczak, P. Statistical mechanics of the international trade network. Phys. Rev. E 85, 056113 (2012).

    MATH  Article  ADS  Google Scholar 

  226. 226.

    Duenas, M. & Fagiolo, G. Modeling the international-trade network: a gravity approach. J. Econ. Interact. Coord. 8, 155–178 (2013).

    Article  Google Scholar 

  227. 227.

    Fagiolo, G., Squartini, T. & Garlaschelli, D. Null models of economic networks: the case of the world trade web. J. Econ. Interact. Coord. 8, 75–107 (2012).

    Article  Google Scholar 

  228. 228.

    Gualdi, S., Cimini, G., Primicerio, K., Di Clemente, R. & Challet, D. Statistically validated network of portfolio overlaps and systemic risk. Sci. Rep. 6, 39467 (2016).

    Article  ADS  Google Scholar 

  229. 229.

    Saracco, F., Di Clemente, R., Gabrielli, A. & Squartini, T. Randomizing bipartite networks: the case of the World Trade Web. Sci. Rep. 5, 10595 (2015).

    Article  ADS  Google Scholar 

  230. 230.

    Saracco, F. et al. Inferring monopartite projections of bipartite networks: an entropy-based approach. New J. Phys. 19, 053022 (2017).

    Article  ADS  Google Scholar 

  231. 231.

    Newman, M. E., Strogatz, S. H. & Watts, D. J. Random graphs with arbitrary degree distributions and their applications. Phys. Rev. E 64, 026118 (2001).

    Article  ADS  Google Scholar 

  232. 232.

    Bianconi, G. Entropy of network ensembles. Phys. Rev. E 79, 036114 (2009).

    MathSciNet  Article  ADS  Google Scholar 

  233. 233.

    Coolen, A. C. C., De Martino, A. & Annibale, A. Constrained Markovian dynamics of random graphs. J. Stat. Phys. 136, 1035–1067 (2009).

    MathSciNet  MATH  Article  ADS  Google Scholar 

  234. 234.

    Del Genio, C. I., Kim, H., Toroczkai, Z. & Bassler, K. E. Efficient and exact sampling of simple graphs with given arbitrary degree sequence. PLoS ONE 5, e10012 (2010).

    Article  Google Scholar 

  235. 235.

    Artzy-Randrup, Y. & Stone, L. Generating uniformly distributed random networks. Phys. Rev. E 72, 056708 (2005).

    MathSciNet  Article  ADS  Google Scholar 

  236. 236.

    Blitzstein, J. & Diaconis, P. A sequential importance sampling algorithm for generating random graphs with prescribed degrees. Internet Math. 6, 489–522 (2011).

    MathSciNet  MATH  Article  Google Scholar 

  237. 237.

    Tumminello, M., Micciché, S., Lillo, F., Piilo, J. & Mantegna, R. N. Statistically validated networks in bipartite complex systems. PLoS ONE 6, e17994 (2011).

    Article  ADS  Google Scholar 

  238. 238.

    Strona, G., Nappo, D., Boccacci, F., Fattorini, S. & San-Miguel-Ayanz, J. A fast and unbiased procedure to randomize ecological binary matrices with fixed row and column totals. Nat. Commun. 5, 4114 (2014).

    Article  ADS  Google Scholar 

  239. 239.

    Carstens, C. J. Proof of uniform sampling of binary matrices with fixed row sums and column sums for the fast curveball algorithm. Phys. Rev. E 91, 042812 (2015).

    MathSciNet  Article  ADS  Google Scholar 

  240. 240.

    Basel Committee on Banking Supervision. Macroeconomic impact assessment of OTC derivatives regulatory reforms. Bank for International Settlements https://www.bis.org/publ/othp20.pdf (2013).

  241. 241.

    Basel Committee on Banking Supervision. Making supervisory stress tests more macroprudential: Considering liquidity and solvency interactions and systemic risk. Bank for International Settlements https://www.bis.org/bcbs/publ/wp29.pdf (2015).

  242. 242.

    Alves, I. et al. Network analysis of the EU insurance sector. Occasional Paper, no. 7. European Systemic Risk Board https://www.esrb.europa.eu/pub/pdf/occasional/20150713_occasional_paper_7.pdf (2015).

  243. 243.

    Bank of England. Stress testing the UK banking system: 2016 results. Bank of England https://www.bankofengland.co.uk/-/media/boe/files/stress-testing/2016/stress-testing-the-uk-banking-system-2016-results.pdf (2016).

  244. 244.

    Bank of England. Stress testing the UK banking system: 2017 results. Bank of England https://www.bankofengland.co.uk/-/media/boe/files/stress-testing/2017/stress-testing-the-uk-banking-system-2017-results.pdf (2017).

  245. 245.

    Basel Committee on Banking Supervision. Global systemically important banks: updated assessment methodology and the higher loss absorbency requirement. Bank for International Settlements https://www.bis.org/publ/bcbs255.pdf (2013).

  246. 246.

    Anand, K., Gai, P. & Marsili, M. Rollover risk, network structure and systemic financial crises. J. Econ. Dyn. Control 36, 1088–1100 (2012).

    MathSciNet  MATH  Article  Google Scholar 

  247. 247.

    Mazzarisi, P., Barucca, P., Lillo, F. & Tantari, D. A dynamic network model with persistent links and node-specific latent variables, with an application to the interbank market. Eur. J. Oper. Res. 281, 50–65 (2020).

    MathSciNet  MATH  Article  Google Scholar 

  248. 248.

    Hatzopoulos, V., Iori, G., Mantegna, R. N., Miccichè, S. & Tumminello, M. Quantifying preferential trading in the e-MID interbank market. Quant. Finance 15, 693–710 (2015).

    MathSciNet  MATH  Article  Google Scholar 

  249. 249.

    Silva, T. C., da Silva, M. A. & Tabak, B. M. Systemic risk in financial systems: a feedback approach. J. Econ. Behav. Organ. 144, 97–120 (2017).

    Article  Google Scholar 

  250. 250.

    Silva, T. C., da Silva Alexandre, M. & Tabak, B. M. Bank lending and systemic risk: a financial-real sector network approach with feedback. J. Financial Stab. 38, 98–118 (2018).

    Article  Google Scholar 

  251. 251.

    Cortes, G. S., Silva, T. C. & Van Doornik, B. F. N. Credit shock propagation in firm networks: evidence from government bank credit expansions. Working Papers, no. 507. Banco Central do Brasil https://www.bcb.gov.br/pec/wps/ingl/wps507.pdf (2019).

  252. 252.

    Battiston, S., Mandel, A., Monasterolo, I., Schütze, F. & Visentin, G. A climate stress-test of the financial system. Nat. Clim. Change 7, 283–288 (2017).

    Article  ADS  Google Scholar 

  253. 253.

    Upper, C. Simulation methods to assess the danger of contagion in interbank markets. J. Financ. Stab. 7, 111–125 (2011).

    Article  Google Scholar 

  254. 254.

    Bacharach, M. Estimating nonnegative matrices from marginal data. Int. Econ. Rev. 6, 294–310 (1965).

    MATH  Article  Google Scholar 

  255. 255.

    Baral, P. & Fique, J. P. Estimation of bilateral exposures – A copula approach. CIRANO http://www.cirano.qc.ca/conferences/public/pdf/networks2012/02-BARAL-FIQUE-Estimation_of_Bilateral_Exposures-A_Copula_Approach.pdf (2012).

  256. 256.

    Di Gangi, D., Lillo, F. & Pirino, D. Assessing systemic risk due to fire sales spillover through maximum entropy network reconstruction. J. Econ. Dyn. Control 94, 117–141 (2018).

    MathSciNet  MATH  Article  Google Scholar 

  257. 257.

    Drehmann, M. & Tarashev, N. Measuring the systemic importance of interconnected banks. J. Financ. Intermed. 22, 586–607 (2013).

    Article  Google Scholar 

  258. 258.

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

    Article  Google Scholar 

  259. 259.

    Moussa, A. Contagion and systemic risk in financial networks. Ph.D. thesis, Columbia Univ. (2011).

  260. 260.

    Montagna, M. & Lux, T. Contagion risk in the interbank market: A probabilistic approach to cope with incomplete structural information. Quant. Finance 17, 101–120 (2017).

    MathSciNet  MATH  Article  Google Scholar 

  261. 261.

    Hałaj, G. & Kok, C. Assessing interbank contagion using simulated networks. Comput. Manag. Sci. 10, 157–186 (2013).

    MathSciNet  MATH  Article  Google Scholar 

  262. 262.

    Fronczak, A. in Encyclopedia of Social Network Analysis and Mining (eds Alhajj, R. & Rokne, J.) (Springer, 2014).

Download references

Acknowledgements

G.C. acknowledges support from the EU project HumanE-AI-Net, no. 952026. D.G. acknowledges support from the Dutch Econophysics Foundation (Stichting Econophysics, Leiden, the Netherlands) and the Netherlands Organization for Scientific Research (NWO/OCW). F.S. and T.S. acknowledge support from the EU project SoBigData++, no. 871042.

Author information

Affiliations

Authors

Contributions

All authors contributed equally to this manuscript.

Corresponding author

Correspondence to Guido Caldarelli.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Disclaimer

Any views expressed are solely those of the author(s) and so cannot be taken to represent those of the Bank of England or to state Bank of England policy.

Peer review information

Nature Reviews Physics thanks Ladislav Krištoufek, Helmut Elsinger and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Glossary

Bipartite networks

Networks in which nodes are of two different types, say A and B (for instance, companies and directors), and links exist only between nodes of different type (for instance, a director being connected to a company if they sit on the board of that company).

Multiplex networks

Collections of networks (also called layers) with the same set of nodes, but with different links. In this context, multiplex networks are used to represent different kinds of linkages between financial institutions.

Assets

Items on the balance sheet of an institution that have a positive economic value because they generate present or future income.

Small-world

A network structure characterized by a large clustering coefficient and a small average shortest path length.

Community structure

A network characterization in which nodes can be grouped into sets such that each set of nodes is densely connected internally.

Path

On a network, a sequence of consecutive edges connecting a sequence of distinct nodes. The shortest path between two nodes is the path of minimal length connecting them.

Disassortativity

The tendency of nodes to be linked to other nodes with dissimilar degrees. Conversely, assortativity is the tendency for nodes to be linked to other nodes with similar degrees.

One-mode projections

A one-mode projection of a bipartite network contains only nodes of one type (say A) and any two such nodes are connected to each other with an intensity proportional to the number of their common neighbours of the other type in the original bipartite network (for instance, two directors are connected by a link indicating the number of common boards on which they sit).

Filtered matrix

Matrix, for instance representing correlations, that has been statistically validated (or otherwise processed to eliminate the effects of noise) so that, ideally, only statistically significant information is retained.

Liquidity

Refers to the case in which the liquid assets (such as cash) of one institution are larger than its short-term liabilities (such as loans to be paid back overnight).

Liabilities

Items on the balance sheet of an institution that have a negative economic value because they represent debt to be repaid, potentially at different times (or maturities) in the future.

Equity

In accounting, equity is defined by the balance sheet identity as the difference between assets and liabilities. Therefore, it represents the net worth of the institution.

Solvency

Solvency refers to the case in which the assets of one institution are larger than its liabilities and, therefore, its equity is positive.

Value-at-risk

(VaR). Risk measure defined as a (typically) large quantile of the probability distribution of losses. For example, when the quantile is 0.99 and the distribution of losses is over a time horizon of 1 year, it is interpreted as the loss that occurs once every 100 years.

Expected shortfall

(ES). Risk measure defined as the mean loss exceeding a (typically) large quantile of the probability distribution of losses. It is always larger than the value-at-risk at the same quantile.

Constraints

Quantities representing the structural properties either to be enforced in the network reconstruction process or to be discounted in the network validation process.

Shannon entropy

Functional quantifying the amount of uncertainty associated with a probability distribution (see Box 2). Its maximum is attained for a uniform distribution.

Density

The fraction of possible connections that are actually realized in a network.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Bardoscia, M., Barucca, P., Battiston, S. et al. The physics of financial networks. Nat Rev Phys (2021). https://doi.org/10.1038/s42254-021-00322-5

Download citation

Search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing