Journal of Statistical Physics

, Volume 165, Issue 2, pp 351–370 | Cite as

A Statistical Test of Walrasian Equilibrium by Means of Complex Networks Theory

  • Leonardo Bargigli
  • Stefano Viaggiu
  • Andrea Lionetto


We represent an exchange economy in terms of statistical ensembles for complex networks by introducing the concept of market configuration. This is defined as a sequence of nonnegative discrete random variables \(\{w_{ij}\}\) describing the flow of a given commodity from agent i to agent j. This sequence can be arranged in a nonnegative matrix W which we can regard as the representation of a weighted and directed network or digraph G. Our main result consists in showing that general equilibrium theory imposes highly restrictive conditions upon market configurations, which are in most cases not fulfilled by real markets. An explicit example with reference to the e-MID interbank credit market is provided.


Exchange economy General equilibrium Complex networks Canonical ensembles Graph temperature Thermodynamics 


  1. 1.
    Samuelson, P.: Economics, 8th edn. McGraw-Hill, New York (1970)MATHGoogle Scholar
  2. 2.
    Smith, E., Foley, D.K.: Classical thermodynamics and economic general equilibrium theory. J. Econ. Dyn. Control 32, 7–65 (2008)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Reguera, D., Reiss, H., Rawlings, P.K.: Entropic basis of the Pareto law. Physica A 343, 643–652 (2004)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Yakovenko, V.M.: Statistical mechanics approach to the probability distribution of money. In: Ganssmann, H. (ed.) New Approaches to Monetary Theory: Interdisciplinary Perspectives, pp. 104–123 (2011)Google Scholar
  5. 5.
    Banerjee, A., Yakovenko, V.M.: Universal patterns of inequality. New J. Phys. 12, 075032 (2010)ADSCrossRefGoogle Scholar
  6. 6.
    Yakovenko, V.M., Rosser, J.B.: Colloquium: statistical mechanics of money, wealth, and income. Rev. Mod. Phys. 81, 1703–1725 (2009)ADSCrossRefGoogle Scholar
  7. 7.
    Saslow, W.M.: An economic analogy to thermodynamics. Am. J. Phys. 67(12), 1239–1247 (1999)ADSCrossRefGoogle Scholar
  8. 8.
    McCauley, J.L.: Thermodynamic analogies in economics and finance: instability of markets. Physica A 329, 199–212 (2003)ADSCrossRefMATHGoogle Scholar
  9. 9.
    McCauley, J.L.: Response to ?Worrying Trends in Econophysics? Phys. A 371, 601–609 (2006)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Dragulescu, A.A., Yakovenko, V.M.: Statistical mechanics of money. Eur. Phys. J. B 17, 723–729 (2000)ADSCrossRefGoogle Scholar
  11. 11.
    Dragulescu, A.A., Yakovenko, V.M.: Exponential and power-law probability distributions of wealth and income in the United Kingdom and the United States. Physica A 299, 213–221 (2001)ADSCrossRefMATHGoogle Scholar
  12. 12.
    Dragulescu, A.A., Yakovenko, V.M.: Evidence for the exponential distribution of income in the USA. Eur. Phys. J. B 20, 585–589 (2001)ADSCrossRefMATHGoogle Scholar
  13. 13.
    Angle, J.: The surplus theory of social stratification and the size distribution of Personal Wealth. Soc. Forces 65, 293–326 (1986)CrossRefGoogle Scholar
  14. 14.
    Bennati, E.: Un metodo di simulazione statistica per l’analisi della distribuzione del reddito. Rivista Internazionale di Scienze Economiche e Commerciali 35, 735–756 (1988)Google Scholar
  15. 15.
    Garibaldi, U., Scalas, E.: Finitary Probabilistic Methods in Econophysics. Cambridge University Press, Cambridge, UK (2010)CrossRefMATHGoogle Scholar
  16. 16.
    Viaggiu, S., Lionetto, A., Bargigli, L., Longo, M.: Statistical ensembles for money and debt. Physica A 391(21), 4839–4849 (2012)ADSCrossRefGoogle Scholar
  17. 17.
    Upper, C.: Simulation results to assess the danger of contagion in interbank markets. J. Financ. Stab. 7, 111–125 (2011)CrossRefGoogle Scholar
  18. 18.
    Chinazzi, M., Fagiolo., G.: Systemic risk, contagion, and financial networks: A survey. LEM Papers Series 2013/08, Laboratory of Economics and Management (LEM), Sant’Anna School of Advanced Studies, Pisa, Italy (2013)Google Scholar
  19. 19.
    Mastromatteo, I., Zarinelli, E., Marsili, M.: Reconstruction of financial network for robust estimation of systemic risk. J. Stat. Mech. 2012, P03011 (2012)CrossRefGoogle Scholar
  20. 20.
    Fagiolo, G., Squartini, T., Garlaschelli, D.: Null models of economic networks: the case of the world trade web. J. Econ. Interact. Coord. 8(1), 75–107 (2013)CrossRefGoogle Scholar
  21. 21.
    Park, J., Newman, M.E.J.: The origin of degree correlations in the Internet and other networks. Phys. Rev. E 68, 026112 (2003)ADSCrossRefGoogle Scholar
  22. 22.
    Garlaschelli, D., Loffredo, M.I.: Maximum likelihood: extracting unbiased information from complex networks. Phys. Rev. E 73, 015101(R) (2006)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    Arrow, K.J., Debreu, G.: Existence of an equilibrium for a competitive economy. Econometrica 22, 265–290 (1954)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Serrano, M.A., Boguñá, M.: Weighted Configuration Model, AIP Conference Proceedings, vol. 776 (1), pp. 101–107 (2005)Google Scholar
  25. 25.
    Bianconi, G.: Statistical mechanics of multiplex networks: entropy and overlap. Phys. Rev. E 87, 062806 (2013)ADSCrossRefGoogle Scholar
  26. 26.
    Foley, D.K.: A statistical equilibrium theory of markets. J. Econ. Theory 62(2), 321–345 (1994)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Huang, K.: Statistical Mechanics, 2nd edn. Wiley, New York (1987)MATHGoogle Scholar
  28. 28.
    Garlaschelli D., Ahnert S.E., Fink T.M.A., Caldarelli G.: Temperature in complex network, arXiv:cond-mat/0606805
  29. 29.
    Dmitri, K., Fragkiskos, P., Vahdat, A., Boguná, M.: Curvature and temperature of complex networks. Phys. Rev. E 80, 035101(R) (2009)Google Scholar
  30. 30.
    Iori, G., Mantegna, R.N., Marotta, L., Micciché, S., Porter, S., Tumminello, M.: Networked relationships in the e-MID interbank market: a trading model with memory. J. Econ. Dyn. Control 50, 98–116 (2015)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Hatzopoulosa, V., Iori, G., Mantegna, R.N., Miccichè, S., Tumminello, M.: Quantifying preferential trading in the e-MID interbank market. Quant. Financ. 15(4), 693–710 (2015)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Clauset, A., Shalizi, C.R., Newman, M.E.J.: Power-law distributions in empirical data. SIAM Rev. 51, 661–703 (2009)ADSMathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Samuelson, P.: Structure of a minimum equilibrium system. In: Stiglitz, J.E. (ed.) The Collected Scientific Papers of Paul A, pp. 651–686. Samuelson, MIT Press, Cambridge, MA (1966)Google Scholar
  34. 34.
    Bargigli, L.: Statistical ensembles for economic networks. J. Stat. Phys. 155(4), 810–825 (2014)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Musmeci, N., Battiston, S., Caldarelli, G., Puliga, M., Gabrielli, A.: Bootstrapping topology and systemic risk of complex network using the fitness model. J. Stat. Phys. 151(3–4), 720–734 (2013)Google Scholar
  36. 36.
    Squartini, T., Van Lelyveld, I., Garlaschelli, D.: Early-warning signals of topological collapse in interbank networks. Sci. Rep. 3, 3357 (2013)ADSGoogle Scholar
  37. 37.
    Bargigli, L., Gallegati, M.: Finding communities in credit networks. Economics 7, 2013–2017 (2013)Google Scholar
  38. 38.
    Karrer, B., Newman, M.E.J.: Stochastic blockmodels and community structure in networks. Phys. Rev. E 83, 016107 (2011)ADSMathSciNetCrossRefGoogle Scholar
  39. 39.
    Bargigli, L., Gallegati, M.: Random digraphs with given expected degree sequences: a model for economic networks. J. Econ. Behav. Organ. 78(3), 396–411 (2011)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Dipartimento di Scienze per l’Economia e l’ImpresaUniversità di FirenzeFlorenceItaly
  2. 2.Dipartimento di MatematicaUniversitá ‘Tor Vergata’RomeItaly
  3. 3.Dipartimento di FisicaUniversitá ‘Tor Vergata’RomeItaly

Personalised recommendations