Assessing Particle Swarm Optimizers Using Network Science Metrics

  • Marcos A. C. Oliveira-Júnior
  • Carmelo J. A. Bastos-Filho
  • Ronaldo Menezes
Part of the Studies in Computational Intelligence book series (SCI, volume 476)


Particle Swarm Optimizers (PSOs) have been widely used for optimization problems, but the scientific community still does not have sophisticated mechanisms to analyze the behavior of the swarm during the optimization process. We propose in this paper to use some metrics described in network sciences, specifically the R-value, the number of zero eigenvalues of the Laplacian Matrix, and the Spectral Density, in order to assess the behavior of the particles during the search and diagnose stagnation processes. Assessor methods can be very useful for designing novel PSOs or when one needs to evaluate the performance of a PSO variation applied to a specific problem. In order to apply these metrics, we observed that it is not possible to analyze the dynamics of the swarm by using the communication topology because it does not change. Therefore, we propose in this paper the definition of the influence graph of the swarm. We used this novel concept to assess the dynamics of the swarm. We tested our proposed methodology in three different PSOs in a well-known multimodal benchmark function. We observed that one can retrieve interesting information from the swarm by using this methodology.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Albert, R., Barabasi, A.L.: Statistical mechanics of complex networks. Reviews of Modern Physics 74, 47 (2002), doi:doi:10.1103/RevModPhys.74.47Google Scholar
  2. 2.
    Bastos-Filho, C.J.A., Lima-Neto, F.B., Lins, A.J.C.C., Nascimento, A.I.S., Lima, M.P.: A novel search algorithm based on fish school behavior. In: 2008 IEEE International Conference on Systems, Man and Cybernetics, pp. 2646–2651 (2008)Google Scholar
  3. 3.
    Bratton, D., Kennedy, J.: Defining a standard for particle swarm optimization. In: Swarm Intelligence Symposium, SIS 2007, pp. 120–127. IEEE (2007), doi:10.1109/SIS.2007.368035Google Scholar
  4. 4.
    Clerc, M., Kennedy, J.: The particle swarm - explosion, stability, and convergence in a multidimensional complex space. IEEE Transactions on Evolutionary Computation 6(1), 58–73 (2002), doi:10.1109/4235.985692CrossRefGoogle Scholar
  5. 5.
    Dorigo, M., Caro, G.: Ant colony optimization: A new meta-heuristic. In: Proceedings of the Congress on Evolutionary Computation, pp. 1470–1477. IEEE Press (1999)Google Scholar
  6. 6.
    Engelbrecht, A.P.: Computational Intelligence: An Introduction. Wiley Publishing (2007)Google Scholar
  7. 7.
    Farkas, I.J., Derényi, I., Barabási, A.L., Vicsek, T.: Spectra of ”real-world” graphs: beyond the semicircle law. Phys Rev E Stat Nonlin Soft Matter Phys 64(2 Pt 2) (2001),
  8. 8.
    Karaboga, D.: An idea based on honey bee swarm for numerical optimization. Tech. rep., Erciyes University, Engineering Faculty, Computer Engineering Department (2005)Google Scholar
  9. 9.
    Kennedy, J., Eberhart, R.: Particle swarm optimization, vol. 4, pp. 1942–1948 (1995),, doi:10.1109/ICNN.1995.488968
  10. 10.
    Kennedy, J., Mendes, R.: Population structure and particle swarm performance. In: Proceedings of the 2002 Congress on Evolutionary Computation, CEC 2002, vol. 2, pp. 1671–1676 (2002), doi:10.1109/CEC.2002.1004493Google Scholar
  11. 11.
    Lewis, T.G.: Network Science: Theory and Applications. Wiley Publishing (2009)Google Scholar
  12. 12.
    Oliveira-Júnior, M.A.C., Bastos-Filho, C.J.A., Menezes, R.: Using Network Science to Define a Dynamic Communication Topology for Particle Swarm Optimizers. In: Menezes, R., Evsukoff, A., González, M.C. (eds.) Complex Networks. SCI, vol. 424, pp. 39–47. Springer, Heidelberg (2013), doi:10.1007/978-3-642-30287-9_5CrossRefGoogle Scholar
  13. 13.
    Suganthan, P.: Particle swarm optimiser with neighbourhood operator. In: Proceedings of the 1999 Congress on Evolutionary Computation, CEC 1999, vol. 3, xxxvii+2348 (1999), doi:10.1109/CEC.1999.785514Google Scholar
  14. 14.
    Tang, K., Li, X., Suganthan, P.N., Yang, Z., Weise, T.: Benchmark Functions for the CEC’2010 Special Session and Competition on Large-Scale Global Optimization. Tech. rep., University of Science and Technology of China (USTC), School of Computer Science and Technology, Nature Inspired Computation and Applications Laboratory (NICAL): ChinaGoogle Scholar
  15. 15.
    Watts, D.J., Strogatz, S.H.: Collective dynamics of ’small-world’ networks. Nature 393(6684), 440–442 (1998),, doi:10.1038/30918CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Marcos A. C. Oliveira-Júnior
    • 1
  • Carmelo J. A. Bastos-Filho
    • 1
  • Ronaldo Menezes
    • 2
  1. 1.University of PernambucoRecifeBrazil
  2. 2.Florida Institute of TechnologyMelbourneUSA

Personalised recommendations