Trail formation in ants. A generalized Polya urn process
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.
KeywordsAnt trail formation Polya process Positive reinforcement Delayed urn model Plasticity–Stability dilemma
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- 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
- Eggenberger, F., & Pólya, G. (1923). Über die Statistik verketteter Vorgänge. ZAMM—Journal of Applied Mathematics and Mechanics, 3(4), 279–289. 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. 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
- Johnson, N., & Kotz, S. (1977). Urn models and their applications. New York: Wiley. Google Scholar
- Leith, C. (2005). Ant algorithms and generalized finite urns. Ph.D. thesis, Queen’s University, Kingston, Ontario, Canada. Google Scholar
- 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
- 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. CrossRefGoogle 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. CrossRefGoogle 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