New perspectives in the equilibrium statistical mechanics approach to social and economic sciences

  • Elena Agliari
  • Adriano Barra
  • Raffaella Burioni
  • Pierluigi Contucci
Part of the Modeling and Simulation in Science, Engineering and Technology book series (MSSET)


In this chapter we review some recent development in the mathematical modeling of quantitative sociology by means of statistical mechanics. After a short pedagogical introduction to static and dynamic properties of many body systems, we develop a theory for particle (agents) interactions on random graph.

Our approach is based on describing a social network as a graph whose nodes represent agents and links between two of them stand for a reciprocal interaction. Each agent has to choose among a dichotomic option (i.e., agree or disagree) with respect to a given matter and he is driven by external influences (as media) and peer to peer interactions. These mimic the imitative behavior of the collectivity and may possibly be zero if the two nodes are disconnected.

For this scenario we work out both the dynamics and, given the validity of the detailed balance, the corresponding equilibria (statics). Once the two-body theory is completely explored, we analyze, on the same random graph, a diffusive strategic dynamicswith pairwise interactions, where detailed balance constraint is relaxed. The dynamic encodes some relevant processes which are expected to play a crucial role in the approach to equilibrium in social systems, i.e., diffusion of information and strategic choices. We observe numerically that such a dynamics reaches a well defined steady state that fulfills a shiftproperty: the critical interaction strength for the canonical phase transition is higher with respect to the expected equilibrium one previously obtained with detailed balanced dynamical evolution.

Finally, we show how the stationary states of this kind of dynamics can be described by statistical mechanics equilibria of a diluted p-spin model, for a suitable noninteger real p>2. Several implications from a sociological perspective are discussed together with some general outlooks.


Monte Carlo Statistical Mechanic Critical Exponent Random Graph Thermodynamic Limit 
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.


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  1. 1.
    Agliari E., Barra A., ` F.: Criticality in diluted ferromagnet. J. Stat. Mech., P10003, (2008)Google Scholar
  2. 2.
    Agliari E., Burioni R., Cassi D., Vezzani A.: Diffusive thermal dynamics for the spin-S Ising ferromagnet. Eur. Phys. J. B, 46, 109–116, (2005)CrossRefGoogle Scholar
  3. 3.
    Agliari E., Burioni R., Cassi D., Vezzani A.: Fractal geometry of Ising magnetic patterns: signatures of criticality and diffusive dynamics. Eur. Phys. J. B 49, 119–125, (2006)CrossRefGoogle Scholar
  4. 4.
    Agliari E., Burioni R., Cassi D., Vezzani A.: Random walks interacting with evolving energy landscapes. Eur. Phys. J. B, 48, 529–536, (2006)CrossRefGoogle Scholar
  5. 5.
    Agliari E., Burioni R., Contucci P.: A Diffusive Strategy in Group Competition. J. Stat. Phys. 139, 478–491, (2010)MATHCrossRefGoogle Scholar
  6. 6.
    Aizenman M., Contucci P.: On the stability of the quenched state in mean field spin glass models. J. Stat. Phys. 92, 765–783 (1998)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Amit D.J.: Modeling brain function. The world of attractor neural network. Cambridge Univerisity Press, Cambridge (1992)Google Scholar
  8. 8.
    Barra A.: Irreducible free energy expansion and overlap locking in mean field spin glasses. J. Stat. Phys. 123, 601–614 (2006)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Barra A.: The mean field Ising model trought interpolating techniques. J. Stat. Phys. 132, 787–809 (2008a)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Barra A.: Notes on ferromagnetic P-spin and REM. Math. Met. Appl. Sci. 94, 354–367, (2008b)Google Scholar
  11. 11.
    Binder K.: Applications of Monte Carlo methods to statistical physics. Rep. Prog. Phys. 60, 487–559, (1997)CrossRefGoogle Scholar
  12. 12.
    Bouchaud J.P.: Economics needs a scientific revolution. Nature, 455, 1181–1187, (2008)CrossRefGoogle Scholar
  13. 13.
    Buonsante P., Burioni R., Cassi D., Vezzani A.: Diffusive thermal dynamics for the Ising ferromagnet. Phys. Rev. E, 66, 036121–036128 (2002)CrossRefGoogle Scholar
  14. 14.
    Chowdhury D., Santen L., Schadschneider A.: Statistical Physics of Vehicular Traffic and some related systems. Phys. Rep. 199, 256–287, (2000)MathSciNetGoogle Scholar
  15. 15.
    Contucci P., Giardinà C.: Spin-Glass Stochastic Stability: a Rigorous Proof. Annales H. Poincaré, 6, Vol.5 (2005)Google Scholar
  16. 16.
    Contucci P., Graffi S.: How can mathematics contribute to social sciences. Qual. Quant. 41, 531–537, (2007)CrossRefGoogle Scholar
  17. 17.
    Coolen A.C.C.: The Mathematical Theory of Minority Games – Statistical Mechanics of Interacting Agents. Oxford Finance, Oxford (2005)MATHGoogle Scholar
  18. 18.
    Durlauf S.N.: How can statistical mechanics contribute to social science? P.N.A.S. 96, 10582–10584, (1999)Google Scholar
  19. 19.
    Evans D., Morris G.: Statistical mechanics of non equilibrium liquids. Cambridge University Press, Cambridge (1990)Google Scholar
  20. 20.
    Guerra F.: The cavity method in the mean field spin glass model. Functional reperesentations of thermodynamic variables. In: Albeverio S. (Ed.), Advances in dynamical systems and quantum physics. Singapore (1995)Google Scholar
  21. 21.
    Guerra F., Toninelli F.L.: The high temperature region of the Viana-Bray diluted spin glass model. J. Stat. Phys. 115, 456–467, (2004)CrossRefMathSciNetGoogle Scholar
  22. 22.
    Huang K.: Lectures on statistical physics and protein folding. World Scientific Publishing, London (2007)Google Scholar
  23. 23.
    Landau L., Lifshitz E.M.: Course of theoretical physics, Vol. 5. Butterworth-Heinemann, Oxford (1980)Google Scholar
  24. 24.
    Liggett T.M.: Stochastic interacting systems: contact, voter, and exclusion processes. Springer-Verlag, New York (1999)MATHGoogle Scholar
  25. 25.
    Mc Fadden, D.: Economic choices. Am. Econ. Rev. 91, 351–378, (2001)Google Scholar
  26. 26.
    Mézard M., Parisi G., Virasoro M.A.: Spin glass theory and beyond. World Scientific, Singapore (1987)MATHGoogle Scholar
  27. 27.
    Newman M.E.J., Barkema G.T.: Monte Carlo methods in Statistical Physics. Oxford University Press, Oxford (2001)Google Scholar
  28. 28.
    Parisi G.: A simple model for the immune network. P.N.A.S. 87, 429–433, (1990)Google Scholar
  29. 29.
    Parisi G.: Stochastic stability. Sollich P., Coolen T. (Eds.), Proceedings of the Conference Disordered and Complex Systems, London (2000)Google Scholar
  30. 30.
    Walras L.: Economique et Mecanique. Bulletin de la Societe Vaudoise de Sciences Naturelles, 45, 313–318, (1909)Google Scholar

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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Elena Agliari
    • 1
  • Adriano Barra
    • 2
  • Raffaella Burioni
    • 3
  • Pierluigi Contucci
    • 4
  1. 1.Dipartimento di FisicaUniversità di ParmaParmaItaly
  2. 2.Dipartimento di Fisica, Sapienza Università di Roma and Dipartimento di MatematicaUniversità di BolognaBolognaItaly
  3. 3.Dipartimento di FisicaUniversità di Parma and INFN, Gruppo Collegato di ParmaParmaItaly
  4. 4.Dipartimento di MatematicaUniversità di BolognaBolognaItaly

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