Similarity-Based Analysis of Population Dynamics in Genetic Programming Performing Symbolic Regression

  • Stephan M. WinklerEmail author
  • Michael Affenzeller
  • Bogdan Burlacu
  • Gabriel Kronberger
  • Michael Kommenda
  • Philipp Fleck
Part of the Genetic and Evolutionary Computation book series (GEVO)


Population diversity plays an important role in the evolutionary dynamics of genetic programming (GP). In this paper we use structural and semantic similarity measures to investigate the evolution of diversity in three GP algorithmic flavors: standard GP, offspring selection GP (OS-GP), and age-layered population structure GP (ALPS-GP). Empirical measurements on two symbolic regression benchmark problems reveal important differences between the dynamics of the tested configurations. In standard GP, after an initial decrease, population diversity remains almost constant until the end of the run. The higher variance of the phenotypic similarity values suggests that small changes on individual genotypes have significant effects on their corresponding phenotypes. By contrast, strict offspring selection within the OS-GP algorithm causes a significantly more pronounced diversity loss at both genotypic and, in particular, phenotypic levels. The pressure for adaptive change increases phenotypic robustness in the face of genotypic perturbations, leading to less genotypic variability on the one hand, and very low phenotypic diversity on the other hand. Finally, the evolution of similarities in ALPS-GP follows a periodic pattern marked by the time interval when the bottom layer is reinitialized with new individuals. This pattern is easily noticed in the lower layers characterized by shorter migration intervals, and becomes less and less noticeable on the upper layers.


Genetic programming Symbolic regression Population dynamics Genetic diversity Phenotypic diversity Offspring selection Age-layered population structure ALPS 



The work described in this paper was done within the COMET Project Heuristic Optimization in Production and Logistics (HOPL), #843532 funded by the Austrian Research Promotion Agency (FFG).


  1. 1.
    Affenzeller, M., Winkler, S., Wagner, S., Beham, A.: Genetic algorithms and genetic programming: modern concepts and practical applications. In: Numerical Insights. CRC Press, Singapore (2009). CrossRefGoogle Scholar
  2. 2.
    Burke, E.K., Gustafson, S., Kendall, G.: Diversity in genetic programming: an analysis of measures and correlation with fitness. IEEE Trans. Evol. Comput. 8(1), 47–62 (2004). CrossRefGoogle Scholar
  3. 3.
    Draper, N.R., Smith, H.: Applied Regression Analysis, 3rd edn. Wiley, Hoboken (1998)zbMATHGoogle Scholar
  4. 4.
    Hornby, G.S.: Alps: the age-layered population structure for reducing the problem of premature convergence. In: Proceedings of the 8th Annual Conference on Genetic and Evolutionary Computation, GECCO ’06, pp. 815–822. ACM, New York (2006).
  5. 5.
    Hornby, G.S.: A steady-state version of the age-layered population structure EA. In: Genetic Programming Theory and Practice VII, pp. 87–102. Springer, Boston (2010)Google Scholar
  6. 6.
    Koza, J.R.: Genetic Programming: On the Programming of Computers by Means of Natural Selection. MIT Press, Cambridge (1992)zbMATHGoogle Scholar
  7. 7.
    Luke, S.: Two fast tree-creation algorithms for genetic programming. IEEE Trans. Evol. Comput. 4(3), 274–283 (2000)CrossRefGoogle Scholar
  8. 8.
    Poli, R.: A simple but theoretically-motivated method to control bloat in genetic programming. In: Ryan, C., Soule, T., Keijzer, M., Tsang, E., Poli, R., Costa, E. (eds.) Genetic Programming, Proceedings of EuroGP’2003, LNCS, vol. 2610, pp. 204–217. Springer, Essex (2003). zbMATHGoogle Scholar
  9. 9.
    Poli, R., Langdon, W.B., McPhee, N.F.: A Field Guide to Genetic Programming. ACM, New York (2008). Published via and freely available at
  10. 10.
    Schaper, S., Louis, A.A.: The arrival of the frequent: how bias in genotype-phenotype maps can steer populations to local optima. PLoS One 9(2), e86,635 (2014). CrossRefGoogle Scholar
  11. 11.
    Valiente, G.: An efficient bottom-up distance between trees. In: Proceedings of the 8th International Symposium of String Processing and Information Retrieval, pp. 212–219. IEEE, Piscataway (2001)Google Scholar
  12. 12.
    Vladislavleva, E.J., Smits, G.F., Den Hertog, D.: Order of nonlinearity as a complexity measure for models generated by symbolic regression via Pareto genetic programming. IEEE Trans. Evol. Comput. 13(2), 333–349 (2009)CrossRefGoogle Scholar
  13. 13.
    Wagner, S., Affenzeller, M.: SexualGA: gender-specific selection for genetic algorithms. In: Callaos, N., Lesso, W., Hansen, E. (eds.) Proceedings of the 9th World Multi-Conference on Systemics, Cybernetics and Informatics (WMSCI) 2005, vol. 4, pp. 76–81. International Institute of Informatics and Systemics, Winter Garden (2005)Google Scholar
  14. 14.
    Wagner, S., Kronberger, G., Beham, A., Kommenda, M., Scheibenpflug, A., Pitzer, E., Vonolfen, S., Kofler, M., Winkler, S.M., Dorfer, V., Affenzeller, M.: Architecture and design of the heuristiclab optimization environment. Adv. Methods Appl. Comput. Intell. Top. Intell. Eng. Inform. 6, 197–261 (2013)Google Scholar
  15. 15.
    White, D.R., McDermott, J., Castelli, M., Manzoni, L., Goldman, B.W., Kronberger, G., Jaskowski, W., O’Reilly, U.M., Luke, S.: Better GP benchmarks: community survey results and proposals. Genet. Program Evolvable Mach. 14(1), 3–29 (2013). CrossRefGoogle Scholar
  16. 16.
    Winkler, S.M.: Structural versus evaluation based solutions similarity in genetic programming based system identification. In: González, J.R., Pelta, D.A., Cruz, C., Terrazas, G., Krasnogor, N. (eds.) Nature Inspired Cooperative Strategies for Optimization, NICSO 2010. Studies in Computational Intelligence, vol. 284, pp. 269–282. Springer, Granada (2010). Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Stephan M. Winkler
    • 1
    • 2
    Email author
  • Michael Affenzeller
    • 1
    • 2
  • Bogdan Burlacu
    • 1
    • 2
  • Gabriel Kronberger
    • 1
  • Michael Kommenda
    • 1
    • 2
  • Philipp Fleck
    • 1
  1. 1.Heuristic and Evolutionary Algorithms LaboratoryUniversity of Applied Sciences Upper AustriaHagenbergAustria
  2. 2.Institute for Formal Models and VerificationJohannes Kepler UniversityLinzAustria

Personalised recommendations