Skip to main content

The discovery of population interaction with a power law distribution in brain storm optimization


Brain storm optimization (BSO) is a novel evolutionary algorithm which originates from the human brainstorming process. The successful applications of BSO on various problems demonstrate its validity and efficiency. To theoretically analyze the performance of algorithm from the viewpoint of population evolution, the population interaction network (PIN) is used to construct the relationship among individuals in BSO. Four experiments in different dimensions, parameters, combinatorial parameter settings and related algorithms are implemented, respectively. The experimental results indicate the frequency of average degree of BSO meets a power law distribution in the functions with low dimension, which shows the best performance of algorithm among three kinds of dimensions. The parameters of BSO are investigated to find the influence of the population interaction with the power law distribution on the performance of algorithm, and respective parameter can change the relationship among individuals. In addition, the mutual effect among parameters is analyzed to find the best combinatorial result to significantly enhance the performance of BSO. The contrast among BSO, DE and PSO demonstrates a power law distribution is more effective for boosting the population interaction to enhance the performance of algorithm.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5


  1. Barabási AL (2009) Scale-free networks: a decade and beyond. Science 325(5939):412–413

    MathSciNet  Article  MATH  Google Scholar 

  2. Bell R, Dean P (1966) Properties of vitreous silica: analysis of random network models. Nature 212(5068):1354–1356

    Article  Google Scholar 

  3. Boccaletti S, Latora V, Moreno Y, Chavez M, Hwang DU (2006) Complex networks: structure and dynamics. Phys Rep 424(4):175–308

    MathSciNet  Article  MATH  Google Scholar 

  4. Cao Z, Hei X, Wang L, Shi Y, Rong X (2015) An improved brain storm optimization with differential evolution strategy for applications of ANNs. Math Probl Eng 2015:1–8

    Google Scholar 

  5. Cao Z, Shi Y, Rong X, Liu B, Du Z, Yang B (2015) Random grouping brain storm optimization algorithm with a new dynamically changing step size. In: Tan Y, Shi Y, Buarque F, Gelbukh A, Das S, Engelbrecht A (eds) International conference in swarm intelligence. Springer, Cham, pp 357–364

    Google Scholar 

  6. Cheng S, Qin Q, Chen J, Shi Y (2016) Brain storm optimization algorithm: a review. Artif Intell Rev 46(4):445–458

    Article  Google Scholar 

  7. Cheng S, Shi Y, Qin Q, Gao S (2013) Solution clustering analysis in brain storm optimization algorithm. In: 2013 IEEE symposium on swarm intelligence (SIS). IEEE, pp 111–118

  8. Clauset A, Shalizi CR, Newman ME (2009) Power-law distributions in empirical data. SIAM Rev 51(4):661–703

    MathSciNet  Article  MATH  Google Scholar 

  9. Derrac J, García S, Molina D, Herrera F (2011) A practical tutorial on the use of nonparametric statistical tests as a methodology for comparing evolutionary and swarm intelligence algorithms. Swarm Evolut Comput 1(1):3–18

    Article  Google Scholar 

  10. Dorronsoro B, Bouvry P (2011) Improving classical and decentralized differential evolution with new mutation operator and population topologies. IEEE Trans Evolut Comput 15(1):67–98

    Article  Google Scholar 

  11. Gao S, Wang Y, Wang J, Cheng JJ (2017) Understanding differential evolution: a poisson law derived from population interaction network. J Comput Sci 21:140–149

    Article  Google Scholar 

  12. Ishibuchi H, Sakane Y, Tsukamoto N, Nojima Y (2011) Implementation of cellular genetic algorithms with two neighborhood structures for single-objective and multi-objective optimization. Soft Comput 15(9):1749–1767

    Article  Google Scholar 

  13. Janson S, Middendorf M (2005) A hierarchical particle swarm optimizer and its adaptive variant. IEEE Trans Syst Man Cybern B (Cybern) 35(6):1272–1282

    Article  Google Scholar 

  14. Jia Z, Duan H, Shi Y (2016) Hybrid brain storm optimisation and simulated annealing algorithm for continuous optimisation problems. Int J Bio-Inspired Comput 8(2):109–121

    Article  Google Scholar 

  15. Jugulum R, Taguchi S et al (2004) Computer-based robust engineering: essentials for DFSS. ASQ Quality Press, Milwaukee

    Google Scholar 

  16. Mendes R, Kennedy J, Neves J (2004) The fully informed particle swarm: simpler, maybe better. IEEE Trans Evolut Comput 8(3):204–210

    Article  Google Scholar 

  17. Newman ME (2004) Analysis of weighted networks. Phys Rev E 70(5):056,131

    Article  Google Scholar 

  18. Qi J, Rong Z (2013) The emergence of scaling laws search dynamics in a particle swarm optimization. Phys A Stat Mech Appl 392(6):1522–1531

    Article  Google Scholar 

  19. Shi Y (2011) Brain storm optimization algorithm. In: Tan Y, Shi Y, Wang G (eds) International conference in swarm intelligence. Springer, Berlin, pp 303–309

    Google Scholar 

  20. Suganthan PN, Hansen N, Liang JJ, Deb K, Chen YP, Auger A, Tiwari S (2005) Problem definitions and evaluation criteria for the cec 2005 special session on real-parameter optimization. KanGAL Rep 2005005:2005

    Google Scholar 

  21. Watts DJ, Strogatz SH (1998) Collective dynamics of ‘small-world’ networks. Nature 393(6684):440

    Article  MATH  Google Scholar 

  22. Whitacre JM, Sarker RA, Pham QT (2008) The self-organization of interaction networks for nature-inspired optimization. IEEE Trans Evolut Comput 12(2):220–230

    Article  Google Scholar 

  23. Yang Y, Shi Y, Xia S (2015) Advanced discussion mechanism-based brain storm optimization algorithm. Soft Comput 19(10):2997–3007

    Article  Google Scholar 

  24. Yang Z, Shi Y (2015) Brain storm optimization with chaotic operation. In: 2015 seventh international conference on advanced computational intelligence (ICACI). IEEE, pp 111–115

  25. Yao X, Liu Y, Lin G (1999) Evolutionary programming made faster. IEEE Trans Evolut Comput 3(2):82–102

    Article  Google Scholar 

  26. Zhan Z, Chen W, Lin Y, Gong Y, Li Y, Zhang J (2013) Parameter investigation in brain storm optimization. In: 2013 IEEE symposium on swarm intelligence (SIS). IEEE, pp 103–110

  27. Zhang C, Yi Z (2011) Scale-free fully informed particle swarm optimization algorithm. Inf Sci 181(20):4550–4568

    MathSciNet  Article  MATH  Google Scholar 

  28. Zhou D, Shi Y, Cheng S (2012) Brain storm optimization algorithm with modified step-size and individual generation. In: Proceedings of the international conference on swarm intelligence (ICSI), 2012. pp 243–252

  29. Zhu H, Shi Y (2015) Brain storm optimization algorithms with k-medians clustering algorithms. In: 2015 seventh international conference on advanced computational intelligence (ICACI). IEEE, pp 107–110

Download references


This research was partially supported by the JSPS KAKENHI Grant Number JP17K12751.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Shangce Gao.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Wang, Y., Gao, S., Yu, Y. et al. The discovery of population interaction with a power law distribution in brain storm optimization. Memetic Comp. 11, 65–87 (2019).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • Population interaction
  • Power law distribution
  • Brain storm optimization
  • Average degree