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Trail formation in ants. A generalized Polya urn process

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Abstract

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.

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References

  • 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.

    Article  MATH  MathSciNet  Google Scholar 

  • 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.

    Article  MATH  MathSciNet  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  MATH  MathSciNet  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Chapter  Google Scholar 

  • 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.

    MATH  MathSciNet  Google Scholar 

  • 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.

    Chapter  Google Scholar 

  • Dorigo, M., & Blum, C. (2005). Ant colony optimization theory: a survey. Theoretical Computer Science, 344, 243–278.

    Article  MATH  MathSciNet  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Dorigo, M., & Stützle, T. (2004). Ant colony optimization. Cambridge: MIT Press.

    MATH  Google Scholar 

  • Eggenberger, F., & Pólya, G. (1923). Über die Statistik verketteter Vorgänge. ZAMM—Journal of Applied Mathematics and Mechanics, 3(4), 279–289.

    Google Scholar 

  • Friedman, B. (1965). Bernard Friedman’s urn. Annals of Mathematical Statistics, 36, 956–970.

    Article  MathSciNet  Google Scholar 

  • 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.

  • Gouet, R. (1993). Martingale functional central limit theorems for a generalized Polya urn. Annals of Probability, 21(3), 1624–1639.

    Article  MATH  MathSciNet  Google Scholar 

  • Gutjahr, W. J. (2000). A graph-based ant system and its convergence. Future Generation Computer Systems, 16, 873–888.

    Article  Google Scholar 

  • 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.

    Article  MATH  MathSciNet  Google Scholar 

  • 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 

  • Hardwick, J., Oehmke, R., & Stout, Q. F. (2006). New adaptive designs for delayed response models. Journal of Sequential Planning and Inference, 136, 1940–1955.

    Article  MATH  MathSciNet  Google Scholar 

  • Hu, F., & Zhang, L.-X. (2004). Asymptotic normality of urn models for clinical trials with delayed response. Bernoulli, 10(3), 447–463.

    Article  MATH  MathSciNet  Google Scholar 

  • Johnson, N., & Kotz, S. (1977). Urn models and their applications. New York: Wiley.

    Google Scholar 

  • Kotz, S., Mahmoud, H. M., & Robert, P. (2000). On generalized Pólya urn models. Statistics and Probability Letters, 49, 163–173.

    Article  MATH  MathSciNet  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Leith, C. (2005). Ant algorithms and generalized finite urns. Ph.D. thesis, Queen’s University, Kingston, Ontario, Canada.

  • 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 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Merkle, D., & Middendorf, M. (2002). Modelling the dynamics of ant colony optimization algorithms. Evolutionary Computation, 10(3), 253–262.

    Article  Google Scholar 

  • Merkle, D., Middendorf, M., & Schmeck, H. (2002). Ant colony optimization for resource-constrained project scheduling. IEEE Transactions on Evolutionary Computation, 6(4), 333–346.

    Article  Google Scholar 

  • 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 

  • 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.

    Chapter  Google Scholar 

  • Neumann, F., Sudholt, D., & Witt, C. (2009). Analysis of different MMAS ACO algorithms on unimodal functions and plateaus. Swarm Intelligence, 3(1), 35–68.

    Article  Google Scholar 

  • 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.

    Chapter  Google Scholar 

  • 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 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Indian Institute of Technology, Delhi, Hauz Khas, New Delhi, 110016, India

    Sameena Shah,  Jayadeva & Suresh Chandra

  2. IBM India Research Lab., 4 Block C, Institutional Area, Vasant Kunj, New Delhi, 110070, India

    Ravi Kothari

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  1. Sameena Shah
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  2. Ravi Kothari
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  3. Jayadeva
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  4. Suresh Chandra
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Correspondence to Sameena Shah.

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Shah, S., Kothari, R., Jayadeva et al. Trail formation in ants. A generalized Polya urn process. Swarm Intell 4, 145–171 (2010). https://doi.org/10.1007/s11721-010-0041-9

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  • DOI: https://doi.org/10.1007/s11721-010-0041-9

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