A Pseudo-Boolean Approach to the Market Graph Analysis by Means of the p-Median Model

  • Boris GoldengorinEmail author
  • Anton Kocheturov
  • Panos M. Pardalos
Part of the Springer Optimization and Its Applications book series (SOIA, volume 92)


In the course of recent 10 years algorithms and technologies for network structure analysis have been applied to financial markets among other approaches. The first step of such an analysis is to describe the considered financial market via the correlation matrix of stocks prices over a certain period of time. The second step is to build a graph in which vertices represent stocks and edge weights represent correlation coefficients between the corresponding stocks. In this paper we suggest a new method of analyzing stock markets based on dividing a market into several substructures (called stars) in which all stocks are strongly correlated with a leading (central, median) stock. Our method is based on the p-median model a feasible solution to which is represented by a collection of stars. Our reduction of the adjusted p-Median Problem to the Mixed Boolean pseudo-Boolean Linear Programming Problem is able to find an exact optimal solution for markets with at most 1,000 stocks by means of general purpose solvers like CPLEX. We have designed and implemented a high-quality greedy-type heuristic for large-sized (many thousands of stocks) markets. We observed an important “median nesting” property of returned solutions: the p leading stocks, or medians, of the stars are repeated in the solution for p + 1 stars. Moreover, many leading stocks (medians), for example, in the USA stock market are the well-known market indices and funds such as the Dow Jones, S&P which form the largest stars (clusters).


Stock markets analysis Russia Sweden USA p-Median problem Pseudo-Boolean approach Cluster analysis by stars Leading stocks 


  1. 1.
    Mantegna, R.N.: Hierarchical structure in financial markets. Eur. Phys. J. B 11, 193–197 (1999)CrossRefGoogle Scholar
  2. 2.
    Kullmann, L., Kertesz, J., Mantegna, R.N.: Identification of clusters of companies in stock indices via Potts super-paramagnetic transactions. Physica A 287, 412–419 (2000)CrossRefGoogle Scholar
  3. 3.
    Onnela, J.-P., Chakraborti, A., Kaski, K., Kertesz, J.: Dynamic asset trees and portfolio analysis. Eur. Phys. J. B 30, 285–288 (2002)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Onnela, J.-P., Chakraborti, A., Kaski, K., Kertesz, J., Kanto, A.: Asset trees and asset graphs in financial markets. Phys. Scr. T106, 48–54 (2003)CrossRefzbMATHGoogle Scholar
  5. 5.
    Cukur, S., Eryigit, M., Eryigit, R.: Cross correlations in an emerging market financial data. Physica A 376, 555–564 (2007)CrossRefGoogle Scholar
  6. 6.
    Onnela, J.-P., Chakraborti, A., Kaski, K., Kertesz, J.: Dynamic asset trees and Black Monday. Physica A 324, 247–252 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Kenett, D.Y., Shapira, Y., Madi, A., Bransburg-Zabary, S., Gur-Gershgoren, G., Ben-Jacob, E.: Dynamics of stock market correlations. AUCO Czech Econ. Rev. 4, 330–340 (2010)Google Scholar
  8. 8.
    Kenett, D.Y., Tumminello, M., Madi, A., Gur-Gershgoren, G., Mantegna, R.N.: Dominating clasp of the financial sector revealed by partial correlation analysis of the stock market. PLoS ONE 12(5), 1–14 (2010)Google Scholar
  9. 9.
    Boginski, V., Butenko, S., Pardalos, P.M.: Statistical analysis of financial networks. Comput. Stat. Data Anal. 48, 431–443 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Boginski, V., Butenko, S., Pardalos, P.M.: Mining market data: a network approach. Comput. Oper. Res. 33, 3171–3184 (2006)CrossRefzbMATHGoogle Scholar
  11. 11.
    Jung, W.-S., Chae, S., Yang, J.-S., Moon, H.-T.: Characteristics of the Korean stock market correlations. Physica A 361, 263–271 (2006)CrossRefGoogle Scholar
  12. 12.
    Huang, W.-Q., Zhuang, X.-T., Yao, S.: A network analysis of the Chinese stock market. Physica A 388, 2956–2964 (2009)CrossRefGoogle Scholar
  13. 13.
    Tabak, B.M., Serra, T.R., Cajueiro, D.O.: Topological properties of stock market networks: the case of Brazil. Physica A 389, 3240–3249 (2010)CrossRefGoogle Scholar
  14. 14.
    Jallo, D., Budai, D., Boginski, V., Goldengorin, B., Pardalos, P.M.: Network-based representation of stock market dynamics: an application to American and Swedish stock markets. Springer Proc. Math. Stat. 32, 91–108 (2013)Google Scholar
  15. 15.
    Reese, J.: Solution methods for the p-Median problem: an annotated bibliography. Networks 48(3), 125–142 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Mladenovic, N., Brimberg, J., Hansen, P., Moreno-Perez, J.A.: The p-median problem: a survey of metaheuristic approaches. Eur. J. Oper. Res. 179, 927–939 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Hammer, P.L.: Plant location - a pseudo-Boolean approach. Isr. J. Technol. 6, 330–332 (1968)zbMATHGoogle Scholar
  18. 18.
    Beresnev, V.L.: On a problem of mathematical standardization theory. Upravliajemyje Sistemy 11, 43–54 (1973) [in Russian]Google Scholar
  19. 19.
    AlBdaiwi, B.F., Ghosh, D., Goldengorin, B.: Data aggregation for p-median problems. J. Comb. Optim. 3(21), 348–363 (2011)CrossRefMathSciNetGoogle Scholar
  20. 20.
    AlBdaiwi, B.F., Goldengorin, B., Sierksma, G.: Equivalent instances of the simple plant location problem. Comput. Math. Appl. 57(5), 812–820 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Goldengorin, B., Krushinsky, D.: Complexity evaluation of benchmark instances for the p-median problem. Math. Comput. Model. 53, 1719–1736 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Goldengorin, B., Krushinsky, D., Slomp, J.: Flexible PMP approach for large size cell formation. Oper. Res. 60(5), 1157–1166 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Goldengorin, B., Krushinsky, D., Pardalos, P.M.: Cell Formation in Industrial Engineering. Theory, Algorithms and Experiments. Springer, Berlin, 218 pp. (2013). ISBN:978-1-4614-8001-3Google Scholar
  24. 24.
    Bekker, H., Braad, E.P., Goldengorin, B.: Using bipartite and multidimensional matching to select the roots of a system of polynomial equations. In: Computational Science and Its Applications—ICCSA. Lecture Notes in Computer Science, vol. 3483, pp. 397–406. Springer, Berlin (2005)Google Scholar
  25. 25.
    Goldengorin, B., Krushinsky, D.: A computational study of the pseudo-Boolean approach to the p-median problem applied to cell formation. In: Pahl, J., Reiners, T., Voß, S. (eds.) Network Optimization: Proceedings of 5th International Conference (INOC 2011), Hamburg, 13-16 June 2011. Lecture Notes in Computer Science, vol. 6701, pp. 503–516. Springer, Berlin (2011)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Boris Goldengorin
    • 1
    • 2
    Email author
  • Anton Kocheturov
    • 3
  • Panos M. Pardalos
    • 2
  1. 1.Operations Department, Faculty of Economics and BusinessUniversity of GroningenGroningenThe Netherlands
  2. 2.Center of Applied OptimizationUniversity of FloridaGainesvilleUSA
  3. 3.Laboratory of Algorithms and Technologies for Network AnalysisNational Research University Higher School of EconomicsNizhny NovgorodRussian Federation

Personalised recommendations