Soft Computing

, Volume 17, Issue 10, pp 1883–1892 | Cite as

A study of phase transitions for convergence analysis of spin glasses: application to portfolio selection problems

  • Majid Vafaei Jahan
  • Mohammad-R. Akbarzadeh-T
  • Nasser Shahtahamassbi
Methodologies and Application


To date, the spin glass paradigm has been gainfully used in solving a number of optimization problems by devising a mapping between our understanding of spin interactions within a natural spin glass and the given optimization problems. Among the determining factors in a natural spin glass, phase transition is a physical phenomenon that is controlled by temperature. Depending on the spin glass’s phase, spin glasses behave differently and may or may not reach the globally desired optimum. This study aims to determine this critical temperature below which convergence to a global optimum is more likely. Furthermore, we aim to determine the main parameters that characterize this critical temperature. Specifically, the critical temperature is studied as applied to the portfolio selection problem. It is shown that below the critical temperature, the glass consistently reaches the optimal states, whereas, convergence to optimum becomes increasingly unlikely if temperature exceeds this critical temperature. Application to five of the world’s major financial markets reveals that the critical temperature is directly proportional to covariance and the average return of assets and does not depend on the number of assets. In other words, all stock markets, that have the same asset covariance and average return, also have the same critical temperature. This is confirmed by several empirical tests such as correlation, entropy and hamming distance.


Phase transition Spin glass model Portfolio selection Convergence analysis Entropy and hamming distance 


  1. Ackley DH, Hinton GE, Sejnowski TJ (1985) A learning algorithm for Boltzmann machines. Cogn Sci 9:147–169CrossRefGoogle Scholar
  2. Bar-Yam Y (1997) Dynamics of Complex Systems. Addison Wesley Longman Inc., Amsterdam, pp 146–180Google Scholar
  3. Bar-Yam Y (2005) About Engineering Complex Systems: Multiscale Analysis and Evolutionary Engineering. Springer Verlag, Berlin, pp 16–31Google Scholar
  4. Berthier L, Young AP (2004) Time and length scales in spin glasses. J Phys Condens Matter 16:S729–S734CrossRefGoogle Scholar
  5. Boettcher S (1999) Extremal Optimization of graph partitioning at the percolation threshold. J Phys A Math Gen 32:5201–5211MathSciNetCrossRefzbMATHGoogle Scholar
  6. Boettcher S (2004) Extremal optimization at the phase transition of the 3-coloring problem. Phys Rev E Stat Nonlin Soft Matter Phys 69(6 Pt 2):066703CrossRefGoogle Scholar
  7. Bolthausen E, Bovier A (2007) “Spin Glasses,” Springer-Verlag, BerlinGoogle Scholar
  8. Bulatov AA, Skvortsov ES (2008) “Phase transition for Local Search on planted SAT,” arXiv:0811.2546v1Google Scholar
  9. Ciliberti S, Mezard M (2007) Risk minimization through portfolio replication. Europ Phy J B 57:175–180MathSciNetCrossRefzbMATHGoogle Scholar
  10. Coppersmith D, Gamarnik D, Hajiaghayi M, Sorkin GB (2003) Random max sat, random max cut, and their phase transitions. J Rand Struct Algorithm 24(4):502–545MathSciNetCrossRefGoogle Scholar
  11. Gabor A, Kondor I (1999) Portfolio with nonlinear constraints and spin glasses. Phys A 274:222–228CrossRefGoogle Scholar
  12. Galluccio S, Bouchaud JP, Potters M (1998) Rational decisions, random matrices and spin glasses. J Phy A 259:449–456Google Scholar
  13. Gent IP, Walsh T (1996) The TSP phase transition. Artif Intell 88:349–358MathSciNetCrossRefzbMATHGoogle Scholar
  14. Hartmann AK, Rieger H (2002) “Optimization Algorithms in Physics,” Wiley-VCH Verlag Co, CambridgeGoogle Scholar
  15. Hartmann AK, Weigt M (2005) “Phase Transitions in Combinatorial Optimization Problems, Basics, Algorithms and Statistical Mechanics,” Wiley-VCH Verlag Co, CambridgeGoogle Scholar
  16. Hinton GE, Sejnowski TJ, Rumelhart DE, McClelland JL (1986) Learning and Relearning in Boltzmann Machines. Cambridge MIT Press, Cambridge, pp 282–317Google Scholar
  17. Horiguchi T, Takahashi H, Hayashi K, Yamaguchi C (2004) Ising model for packet routing control. J Phy Lett A 330:192–197MathSciNetCrossRefzbMATHGoogle Scholar
  18. Hubermann BA, Hogg T (1987) Phase transitions in artificial intelligence systems. J Artif Intell 33:155–171CrossRefGoogle Scholar
  19. Ingber L (1993) Simulated annealing: Practice versus theory. Math Comput Model 18(11):29–57MathSciNetCrossRefzbMATHGoogle Scholar
  20. Lotov AV (2005) Approximation and Visualization of Pareto Frontier in the Framework of Classical Approach to Multi-Objective Optimization. In: Dagstuhl Seminar Proceedings 04461, Practical Approaches to Multi-Objective Optimization, pp 235Google Scholar
  21. Markowitz H (1952) Portfolio Selection. J Finan 7:77–91Google Scholar
  22. Mooij JM, Kappen HJ (2004) Spin-glass phase transitions on real-world graphs. arXiv:0408378v2Google Scholar
  23. Nishimori H (2001) Statistical Physics of Spin Glasses and Information Processing: An introduction. Clarendon press Oxford, OxfordGoogle Scholar
  24. Nishimori H (2007) Spin glasses and information. Phys A 384:94–99MathSciNetCrossRefGoogle Scholar
  25. Nordblad Per (2004) Spin glasses: model systems for non-equilibrium dynamics. J Phys Condens Matter 16:S715–S722CrossRefGoogle Scholar
  26. Sarkar P (2000) A brief history of cellular automata. ACM Comput Surv 32(1):80–107CrossRefGoogle Scholar
  27. Sivanandam SN, Deepa SN (2008) “Introduction to Genetic Algorithms,” Springer-Verlag, BerlinGoogle Scholar
  28. Vafaei Jahan M, Akbarzadeh-T MR (2010) From local search to global conclusions: migrating spin glass-based distributed portfolio selection. IEEE Trans Evolut Comput 14(2):591–601Google Scholar
  29. Vafaei Jahan M, Akbarzadeh Totonchi MR (2012a) Composing local and global behavior: higher performance of spin glass based portfolio selection. J Comput Sci 3(4):238–245CrossRefGoogle Scholar
  30. Vafaei Jahan M, Akbarzadeh Totonchi MR (2012b) Extremal optimization vs. learning automata: strategies for spin selection in portfolio selection problems. Appl Soft Comput 12(10):3276–3284CrossRefGoogle Scholar
  31. Waelbroeck H, Zertuche F (1999) Discrete chaos. J Phys A Math Gen 32(1):175–189MathSciNetCrossRefzbMATHGoogle Scholar
  32. Wang F, Landau DP (2001) An efficient, multiple range random walk algorithm to calculate the density of states. Phys Rev Lett 86(10):2050–2053CrossRefGoogle Scholar
  33. Young AP (2007) Phase transitions in spin glasses. J Magn Magn Mater 310:1482–1486CrossRefGoogle Scholar
  34. Zhang W, Korf R (1996) A study of complexity transitions on the asymmetric traveling salesman problem. J Artif Intell 81(2):223–239MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Majid Vafaei Jahan
    • 1
  • Mohammad-R. Akbarzadeh-T
    • 2
  • Nasser Shahtahamassbi
    • 3
  1. 1.Department of Computer EngineeringIslamic Azad University- Mashhad Branch MashhadMashhadIran
  2. 2.Cognitive Computing LabCenter for Applied Research On Soft Computing and Intelligent Systems, Ferdowsi University of MashhadMashhadIran
  3. 3.Nano Research Centre and Condensed Matter PhysicsFerdowsi University of MashhadMashhadIran

Personalised recommendations