Swarm Intelligence

, Volume 4, Issue 2, pp 145–171 | Cite as

Trail formation in ants. A generalized Polya urn process

  • Sameena Shah
  • Ravi Kothari
  • Jayadeva
  • Suresh Chandra


Faced with a choice of paths, an ant chooses a path with a higher concentration of pheromone. Subsequently, it drops pheromone on the path chosen. The reinforcement of the pheromone-following behavior favors the selection of an initially discovered path as the preferred path. This may cause a long path to emerge as the preferred path, were it discovered earlier than a shorter path. However, the shortness of the shorter path offsets some of the pheromone accumulated on the initially discovered longer path. In this paper, we model the trail formation behavior as a generalized Polya urn process. For k equal length paths, we give the distribution of pheromone at any time and highlight its sole dependence on the initial pheromone concentrations on paths. Additionally, we propose a method to incorporate the lengths of paths in the urn process and derive how the pheromone distribution alters on its inclusion. Analytically, we show that it is possible, under certain conditions, to reverse the initial bias that may be present in favor of paths that were discovered prior to the discovery of more efficient (shorter) paths. This addresses the Plasticity–Stability dilemma for ants, by laying out the conditions under which the system will remain stable or become plastic and change the path. Finally, we validate our analysis and results using simulations.


Ant trail formation Polya process Positive reinforcement Delayed urn model Plasticity–Stability dilemma 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Abraham, R., Dhersin, J. S., & Ycart, B. (2007). Strong convergence for urn models with reducible replacement policy. Journal of Applied Probability, 44(3), 652–660. zbMATHCrossRefMathSciNetGoogle Scholar
  2. Bai, Z. D., Hu, F., & Rosenberger, W. F. (2002). Asymptotic properties of adaptive designs for clinical trials with delayed response. Annals of Statistics, 30(1), 122–139. zbMATHCrossRefMathSciNetGoogle Scholar
  3. Beckers, R., Deneubourg, J.-L., & Goss, S. (1992). Trails and U-turns in the selection of a path by the ant Lasius niger. Journal of Theoretical Biology, 159, 397–415. CrossRefGoogle Scholar
  4. Carpenter, G., & Grossberg, S. (1987). A massively parallel architecture for a self-organizing neural pattern recognition machine. Computer Vision, Graphics and Image Processing, 37, 54–115. CrossRefGoogle Scholar
  5. Crimaldi, I., & Leisen, F. (2008). Asymptotic results for a generalized Polya urn with “multi-updating” and applications to clinical trials. Communication in Statistics. Theory and Methods, 37(17), 2777–2794. zbMATHCrossRefMathSciNetGoogle Scholar
  6. Deneubourg, J.-L., Aron, S., Goss, S., & Pasteels, J.-M. (1990). The self-organizing exploratory pattern of the Argentine ant. Journal of Insect Behaviour, 3, 159–168. CrossRefGoogle Scholar
  7. Di Caro, G., & Dorigo, M. (1998). Ant colonies for adaptive routing in packet-switched communications networks. In A. E. Eiben, M. Schoenauer, & T. Back (Eds.), LNCS : Vol. 1498. Proceedings of PPSN V—fifth international conference on parallel problem solving from nature (pp. 673–682). Berlin: Springer. CrossRefGoogle Scholar
  8. Dirienzo, A. G. (2000). Using urn models for the design of clinical trials. Sankhyā: The Indian Journal of Statistics Series B, 62(1), 43–69. zbMATHMathSciNetGoogle Scholar
  9. Doerr, B., Neumann, F., Sudholt, D., & Witt, C. (2007). On the runtime analysis of the 1-ANT ACO algorithm. In D. Thierens et al. (Eds.), GECCO’07: Proceedings of the 9th annual conference on genetic and evolutionary computation (pp. 33–40). New York: ACM. CrossRefGoogle Scholar
  10. Dorigo, M., & Blum, C. (2005). Ant colony optimization theory: a survey. Theoretical Computer Science, 344, 243–278. zbMATHCrossRefMathSciNetGoogle Scholar
  11. Dorigo, M., & Gambardella, L. M. (1997). Ant colony system: a cooperative learning approach to the traveling salesman problem. IEEE Transactions on Evolutionary Computation, 1(1), 53–66. CrossRefGoogle Scholar
  12. Dorigo, M., & Stützle, T. (2004). Ant colony optimization. Cambridge: MIT Press. zbMATHGoogle Scholar
  13. Eggenberger, F., & Pólya, G. (1923). Über die Statistik verketteter Vorgänge. ZAMM—Journal of Applied Mathematics and Mechanics, 3(4), 279–289. Google Scholar
  14. Friedman, B. (1965). Bernard Friedman’s urn. Annals of Mathematical Statistics, 36, 956–970. CrossRefMathSciNetGoogle Scholar
  15. Flajolet, P., Dumas, P., & Puyhaubert, V. (2006). Some exactly solvable models of urn process theory. In P. Chassaing (Ed.), Discrete mathematics & theoretical computer science: Vol. AG. Fourth colloquium on mathematics and computer science algorithms, trees, combinatorics and probabilities (pp. 59–118). DMTCS Proceedings, Nancy, France. Google Scholar
  16. Gouet, R. (1993). Martingale functional central limit theorems for a generalized Polya urn. Annals of Probability, 21(3), 1624–1639. zbMATHCrossRefMathSciNetGoogle Scholar
  17. Gutjahr, W. J. (2000). A graph-based ant system and its convergence. Future Generation Computer Systems, 16, 873–888. CrossRefGoogle Scholar
  18. Gutjahr, W. J., & Sebastiani, G. (2008). Runtime analysis of ant colony optimization with best-so-far reinforcement. Methodology and Computing in Applied Probability, 10, 409–433. zbMATHCrossRefMathSciNetGoogle Scholar
  19. Hardwick, J., Oehmke, R., & Stout, Q. F. (2001). Optimal adaptive designs for delayed response models: exponential case. In A. Atkinson, P. Hackl, & W. Müller (Eds.), MODA6: model oriented data analysis (pp. 127–134). Heidelberg: Physica Verlag. Google Scholar
  20. Hardwick, J., Oehmke, R., & Stout, Q. F. (2006). New adaptive designs for delayed response models. Journal of Sequential Planning and Inference, 136, 1940–1955. zbMATHCrossRefMathSciNetGoogle Scholar
  21. Hu, F., & Zhang, L.-X. (2004). Asymptotic normality of urn models for clinical trials with delayed response. Bernoulli, 10(3), 447–463. zbMATHCrossRefMathSciNetGoogle Scholar
  22. Johnson, N., & Kotz, S. (1977). Urn models and their applications. New York: Wiley. Google Scholar
  23. Kotz, S., Mahmoud, H. M., & Robert, P. (2000). On generalized Pólya urn models. Statistics and Probability Letters, 49, 163–173. zbMATHCrossRefMathSciNetGoogle Scholar
  24. Lamb, A. E., & Ollason, J. G. (1994). Trail-laying and recruitment to sugary foods by foraging red wood-ants Formica aquilonia Yarrow (Hymenoptera: Formicidae). Behavioural Processes, 31, 111–124. CrossRefGoogle Scholar
  25. Leith, C. (2005). Ant algorithms and generalized finite urns. Ph.D. thesis, Queen’s University, Kingston, Ontario, Canada. Google Scholar
  26. Mahmoud, H. M. (2003). Pólya urn models and connection to random trees: a review. Journal of the Iranian Statistical Society, 2(1), 53–114. Google Scholar
  27. Mailleux, A.-C., Detrain, C., & Deneubourg, J.-L. (2004). Triggering and persistence of trail laying in foragers of the ant Lasius niger. Journal of Insect Physiology, 51, 297–304. CrossRefGoogle Scholar
  28. Maniezzo, V., & Colorni, A. (1999). The ant system applied to the quadratic assignment problem. IEEE Transactions on Data and Knowledge Engineering, 11(5), 769–778. CrossRefGoogle Scholar
  29. Merkle, D., & Middendorf, M. (2002). Modelling the dynamics of ant colony optimization algorithms. Evolutionary Computation, 10(3), 253–262. CrossRefGoogle Scholar
  30. Merkle, D., Middendorf, M., & Schmeck, H. (2002). Ant colony optimization for resource-constrained project scheduling. IEEE Transactions on Evolutionary Computation, 6(4), 333–346. CrossRefGoogle Scholar
  31. Neumann, F., & Witt, C. (2006). Runtime analysis of a simple ant colony optimization algorithm. In T. Asano (Ed.), LNCS : Vol. 4288. Proceedings of the 17th international symposium on algorithms and computation, ISAAC 2006 (pp. 618–627). Berlin: Springer. Google Scholar
  32. Neumann, F., Sudholt, D., & Witt, C. (2008). Rigorous analyses for the combination of ant colony optimization and local search. In M. Dorigo, M. Birattari, C. Blum, M. Clerc, T. Stützle, & A. F. T. Winfield (Eds.), LNCS : Vol. 5217. Sixth international conference on ant colony optimization and swarm intelligence, ANTS 2008 (pp. 132–143). Berlin: Springer. CrossRefGoogle Scholar
  33. Neumann, F., Sudholt, D., & Witt, C. (2009). Analysis of different MMAS ACO algorithms on unimodal functions and plateaus. Swarm Intelligence, 3(1), 35–68. CrossRefGoogle Scholar
  34. Reimann, M., Doerner, K., & Hartl, R. F. (2003). Analyzing a unified ant system for the VRP and some of its variants. In G. Raidl, S. Cagnoni, J. J. R. Cardalda, D. W. Corne, J. Gottlieb, A. Guillot, E. Hart, C. G. Johnson, E. Marchiori, J. A. Meyer, & M. Middendorf (Eds.), LNCS : Vol. 2611. Applications of evolutionary computing: EvoWorkshops (pp. 300–310). Berlin: Springer. CrossRefGoogle Scholar
  35. Shah, S., Kothari, R., Jayadeva, & Chandra, S. (2008). Mathematical modeling and convergence analysis of trail formation. In D. Fox, & C. P. Gomes (Eds.), Proceedings of the twenty-third AAAI conference on artificial intelligence (pp. 170–175). Menlo Park: AAAI Press. Google Scholar
  36. Stützle, T., & Dorigo, M. (2002). A short convergence proof for a class of ant colony optimization algorithms. IEEE Transactions on Evolutionary Computation, 6(4), 358–365. CrossRefGoogle Scholar
  37. Zhang, L.-X., Hu, F., & Cheung, S. H. (2006). Asymptotic theorems of sequential estimation-adjusted urn models. The Annals of Applied Probability, 16(1), 340–369. zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science + Business Media, LLC 2010

Authors and Affiliations

  • Sameena Shah
    • 1
  • Ravi Kothari
    • 2
  • Jayadeva
    • 1
  • Suresh Chandra
    • 1
  1. 1.Indian Institute of Technology, DelhiNew DelhiIndia
  2. 2.IBM India Research Lab.New DelhiIndia

Personalised recommendations