Swarm Intelligence

, Volume 7, Issue 2–3, pp 145–172 | Cite as

Towards swarm calculus: urn models of collective decisions and universal properties of swarm performance

  • Heiko Hamann


Methods of general applicability are searched for in swarm intelligence with the aim of gaining new insights about natural swarms and to develop design methodologies for artificial swarms. An ideal solution could be a ‘swarm calculus’ that allows to calculate key features of swarms such as expected swarm performance and robustness based on only a few parameters. To work towards this ideal, one needs to find methods and models with high degrees of generality. In this paper, we report two models that might be examples of exceptional generality. First, an abstract model is presented that describes swarm performance depending on swarm density based on the dichotomy between cooperation and interference. Typical swarm experiments are given as examples to show how the model fits to several different results. Second, we give an abstract model of collective decision making that is inspired by urn models. The effects of positive-feedback probability, that is increasing over time in a decision making system, are understood by the help of a parameter that controls the feedback based on the swarm’s current consensus. Several applicable methods, such as the description as Markov process, calculation of splitting probabilities, mean first passage times, and measurements of positive feedback, are discussed and applications to artificial and natural swarms are reported.


Swarm performance Collective decision-making Urn model Positive feedback 


  1. Arkin, R. C., Balch, T., & Nitz, E. (1993). Communication of behavioral state in multi-agent retrieval tasks. In W. Book & J. Luh (Eds.), IEEE conference on robotics and automation (Vol. 3, pp. 588–594). Los Alamitos: IEEE Press. CrossRefGoogle Scholar
  2. Berman, S., Kumar, V., & Nagpal, R. (2011). Design of control policies for spatially inhomogeneous robot swarms with application to commercial pollination. In S. LaValle, H. Arai, O. Brock, H. Ding, C. Laugier, A. M. Okamura, S. S. Reveliotis, G. S. Sukhatme, & Y. Yagi (Eds.), IEEE international conference on robotics and automation (ICRA’11) (pp. 378–385). Los Alamitos: IEEE Press. CrossRefGoogle Scholar
  3. Bjerknes, J. D., & Winfield, A. (2013). On fault-tolerance and scalability of swarm robotic systems. In A. Martinoli, F. Mondada, N. Correll, G. Mermoud, M. Egerstedt, M. A. Hsieh, L. E. Parker, & K. Støy (Eds.), Springer tracts in advanced robotics: Vol. 83. Distributed autonomous robotic systems (DARS 2010) (pp. 431–444). Berlin: Springer. CrossRefGoogle Scholar
  4. Bjerknes, J. D., Winfield, A., & Melhuish, C. (2007). An analysis of emergent taxis in a wireless connected swarm of mobile robots. In Y. Shi & M. Dorigo (Eds.), IEEE swarm intelligence symposium (pp. 45–52). Los Alamitos: IEEE Press. Google Scholar
  5. Breder, C. M. (1954). Equations descriptive of fish schools and other animal aggregations. Ecology, 35(3), 361–370. CrossRefGoogle Scholar
  6. Camazine, S., Deneubourg, J.-L., Franks, N. R., Sneyd, J., Theraulaz, G., & Bonabeau, E. (2001). Self-organizing biological systems. Princeton: Princeton University Press. Google Scholar
  7. Deneubourg, J.-L., Aron, S., Goss, S., & Pasteels, J. M. (1990). The self-organizing exploratory pattern of the Argentine ant. Journal of Insect Behavior, 3(2), 159–168. CrossRefGoogle Scholar
  8. Dussutour, A., Fourcassié, V., Helbing, D., & Deneubourg, J.-L. (2004). Optimal traffic organization in ants under crowded conditions. Nature, 428, 70–73. CrossRefGoogle Scholar
  9. Dussutour, A., Beekman, M., Nicolis, S. C., & Meyer, B. (2009). Noise improves collective decision-making by ants in dynamic environments. Proceedings of the Royal Society of London, Series B, 276, 4353–4361. CrossRefGoogle Scholar
  10. Edelstein-Keshet, L. (2006). Mathematical models of swarming and social aggregation. Robotica, 24(3), 315–324. CrossRefGoogle Scholar
  11. Ehrenfest, P., & Ehrenfest, T. (1907). Über zwei bekannte Einwände gegen das Boltzmannsche H-Theorem. Physikalische Zeitschrift, 8, 311–314. zbMATHGoogle Scholar
  12. Eigen, M., & Winkler, R. (1993). Laws of the game: how the principles of nature govern chance. Princeton: Princeton University Press. Google Scholar
  13. Galam, S. (2004). Contrarian deterministic effect on opinion dynamics: the “hung elections scenario”. Physica A, 333(1), 453–460. MathSciNetCrossRefGoogle Scholar
  14. Gardiner, C. W. (1985). Handbook of stochastic methods for physics, chemistry and the natural sciences. Berlin: Springer. Google Scholar
  15. Gautrais, J., Theraulaz, G., Deneubourg, J.-L., & Anderson, C. (2002). Emergent polyethism as a consequence of increased colony size in insect societies. Journal of Theoretical Biology, 215(3), 363–373. CrossRefGoogle Scholar
  16. Goldberg, D., & Matarić, M. J. (1997). Interference as a tool for designing and evaluating multi-robot controllers. In B. J. Kuipers & B. Webber (Eds.), Proc. of the fourteenth national conference on artificial intelligence (AAAI’97) (pp. 637–642). Cambridge: MIT Press. Google Scholar
  17. Graham, R., Knuth, D., & Patashnik, O. (1998). Concrete mathematics: a foundation for computer science. Reading: Addison–Wesley. Google Scholar
  18. Grinstead, C. M., & Snell, J. L. (1997). Introduction to probability. Providence: American Mathematical Society. zbMATHGoogle Scholar
  19. Hamann, H. (2006). Modeling and investigation of robot swarms. Master’s thesis, University of Stuttgart, Germany. Google Scholar
  20. Hamann, H. (2010). Space-time continuous models of swarm robotics systems: supporting global-to-local programming. Berlin: Springer. CrossRefGoogle Scholar
  21. Hamann, H. (2012). Towards swarm calculus: universal properties of swarm performance and collective decisions. In M. Dorigo, M. Birattari, C. Blum, A. L. Christensen, A. P. Engelbrecht, R. Groß, & T. Stützle (Eds.), Lecture notes in computer science: Vol. 7461. Swarm intelligence: 8th international conference, ANTS 2012 (pp. 168–179). Berlin: Springer. Google Scholar
  22. Hamann, H., & Wörn, H. (2007). Embodied computation. Parallel Processing Letters, 17(3), 287–298. MathSciNetCrossRefGoogle Scholar
  23. Hamann, H., & Wörn, H. (2008). Aggregating robots compute: an adaptive heuristic for the Euclidean Steiner tree problem. In M. Asada, J. C. Hallam, J.-A. Meyer, & J. Tani (Eds.), Lecture notes in artificial intelligence: Vol. 5040. The tenth international conference on simulation of adaptive behavior (SAB’08) (pp. 447–456). Berlin: Springer. Google Scholar
  24. Hamann, H., Meyer, B., Schmickl, T., & Crailsheim, K. (2010). A model of symmetry breaking in collective decision-making. In S. Doncieux, B. Girard, A. Guillot, J. Hallam, J.-A. Meyer, & J.-B. Mouret (Eds.), Lecture notes in artificial intelligence: Vol. 6226. From animals to animats 11 (pp. 639–648). Berlin: Springer. CrossRefGoogle Scholar
  25. Hamann, H., Schmickl, T., Wörn, H., & Crailsheim, K. (2012). Analysis of emergent symmetry breaking in collective decision making. Neural Computing & Applications, 21(2), 207–218. CrossRefGoogle Scholar
  26. Ingham, A. G., Levinger, G., Graves, J., & Peckham, V. (1974). The Ringelmann effect: studies of group size and group performance. Journal of Experimental Social Psychology, 10(4), 371–384. CrossRefGoogle Scholar
  27. Jeanne, R. L., & Nordheim, E. V. (1996). Productivity in a social wasp: per capita output increases with swarm size. Behavioral Ecology, 7(1), 43–48. CrossRefGoogle Scholar
  28. Jeanson, R., Fewell, J. H., Gorelick, R., & Bertram, S. M. (2007). Emergence of increased division of labor as a function of group size. Behavioral Ecology and Sociobiology, 62, 289–298. CrossRefGoogle Scholar
  29. Karsai, I., & Wenzel, J. W. (1998). Productivity, individual-level and colony-level flexibility, and organization of work as consequences of colony size. Proceedings of the National Academy of Sciences of the United States of America, 95, 8665–8669. CrossRefGoogle Scholar
  30. Kennedy, J., & Eberhart, R. C. (2001). Swarm intelligence. San Mateo: Morgan Kaufmann. Google Scholar
  31. Klein, M. J. (1956). Generalization of the Ehrenfest urn model. Physical Review, 103(1), 17–20. zbMATHCrossRefGoogle Scholar
  32. Krafft, O., & Schaefer, M. (1993). Mean passage times for triangular transition matrices and a two parameter Ehrenfest urn model. Journal of Applied Probability, 30(4), 964–970. MathSciNetzbMATHCrossRefGoogle Scholar
  33. Lerman, K., & Galstyan, A. (2002). Mathematical model of foraging in a group of robots: effect of interference. Autonomous Robots, 13, 127–141. zbMATHCrossRefGoogle Scholar
  34. Lerman, K., Martinoli, A., & Galstyan, A. (2005). A review of probabilistic macroscopic models for swarm robotic systems. In E. Şahin & W. M. Spears (Eds.), Lecture notes in computer science: Vol. 3342. Swarm robotics—SAB 2004 international workshop (pp. 143–152). Berlin: Springer. Google Scholar
  35. Levenberg, K. (1944). A method for the solution of certain non-linear problems in least squares. Quarterly of Applied Mathematics, 2, 164–168. MathSciNetzbMATHGoogle Scholar
  36. Lighthill, M. J., & Whitham, G. B. (1955). On kinematic waves. II. A theory of traffic flow on long crowded roads. Proceedings of the Royal Society of London, Series A, 229(1178), 317–345. MathSciNetzbMATHCrossRefGoogle Scholar
  37. Mahmassani, H. S., Dong, J., Kim, J., Chen, R. B., & Park, B. (2009). Incorporating weather impacts in traffic estimation and prediction systems. Technical Report FHWA-JPO-09-065, U.S. Department of Transportation. Google Scholar
  38. Mallon, E. B., Pratt, S. C., & Franks, N. R. (2001). Individual and collective decision-making during nest site selection by the ant leptothorax albipennis. Behavioral Ecology and Sociobiology, 50, 352–359. CrossRefGoogle Scholar
  39. Marquardt, D. (1963). An algorithm for least-squares estimation of nonlinear parameters. SIAM Journal on Applied Mathematics, 11(2), 431–441. MathSciNetzbMATHCrossRefGoogle Scholar
  40. Milutinovic, D., & Lima, P. (2007). Cells and robots: modeling and control of large-size agent populations. Berlin: Springer. Google Scholar
  41. Miramontes, O. (1995). Order-disorder transitions in the behavior of ant societies. Complexity, 1(1), 56–60. CrossRefGoogle Scholar
  42. Mondada, F., Bonani, M., Guignard, A., Magnenat, S., Studer, C., & Floreano, D. (2005). Superlinear physical performances in a SWARM-BOT. In M. S. Capcarrere (Ed.), Lecture notes in computer science: Vol. 3630. Proc. of the 8th European conference on artificial life (ECAL) (pp. 282–291). Berlin: Springer. Google Scholar
  43. Nembrini, J., Winfield, A. F. T., & Melhuish, C. (2002). Minimalist coherent swarming of wireless networked autonomous mobile robots. In B. Hallam, D. Floreano, J. Hallam, G. Hayes, & J.-A. Meyer (Eds.), Proceedings of the seventh international conference on simulation of adaptive behavior on from animals to animats (pp. 373–382). Cambridge: MIT Press. Google Scholar
  44. Nicolis, S. C., Zabzina, N., Latty, T., & Sumpter, D. J. T. (2011). Collective irrationality and positive feedback. PLoS ONE, 6, e18901. CrossRefGoogle Scholar
  45. Okubo, A. (1986). Dynamical aspects of animal grouping: swarms, schools, flocks, and herds. Advances in Biophysics, 22, 1–94. CrossRefGoogle Scholar
  46. Okubo, A., & Levin, S. A. (2001). Diffusion and ecological problems: modern perspectives. Berlin: Springer. CrossRefGoogle Scholar
  47. Østergaard, E. H., Sukhatme, G. S., & Matarić, M. J. (2001). Emergent bucket brigading: a simple mechanisms for improving performance in multi-robot constrained-space foraging tasks. In E. André, S. Sen, C. Frasson, & J. P. Müller (Eds.), Proceedings of the fifth international conference on autonomous agents (AGENTS’01) (pp. 29–35). New York: ACM. CrossRefGoogle Scholar
  48. Prorok, A., Correll, N., & Martinoli, A. (2011). Multi-level spatial models for swarm-robotic systems. The International Journal of Robotics Research, 30(5), 574–589. CrossRefGoogle Scholar
  49. Saffre, F., Furey, R., Krafft, B., & Deneubourg, J.-L. (1999). Collective decision-making in social spiders: dragline-mediated amplification process acts as a recruitment mechanism. Journal of Theoretical Biology, 198, 507–517. CrossRefGoogle Scholar
  50. Schmickl, T., & Hamann, H. (2011). BEECLUST: a swarm algorithm derived from honeybees. In Y. Xiao (Ed.), Bio-inspired computing and communication networks. Boca Raton: CRC Press. Google Scholar
  51. Schneider-Fontán, M., & Matarić, M. J. (1996). A study of territoriality: the role of critical mass in adaptive task division. In P. Maes, S. W. Wilson, & M. J. Matarić (Eds.), From animals to animats IV (pp. 553–561). Cambridge: MIT Press. Google Scholar
  52. Seeley, T. D., Camazine, S., & Sneyd, J. (1991). Collective decision-making in honey bees: how colonies choose among nectar sources. Behavioral Ecology and Sociobiology, 28(4), 277–290. CrossRefGoogle Scholar
  53. Strogatz, S. H. (2001). Exploring complex networks. Nature, 410(6825), 268–276. CrossRefGoogle Scholar
  54. Vicsek, T., & Zafeiris, A. (2012). Collective motion. Physics Reports, 517(3–4), 71–140. CrossRefGoogle Scholar
  55. Wong, G., & Wong, S. (2002). A multi-class traffic flow model—an extension of LWR model with heterogeneous drivers. Transportation Research. Part A, Policy and Practice, 36(9), 827–841. CrossRefGoogle Scholar
  56. Yates, C. A., Erban, R., Escudero, C., Couzin, I. D., Buhl, J., Kevrekidis, I. G., Maini, P. K., & Sumpter, D. J. T. (2009). Inherent noise can facilitate coherence in collective swarm motion. Proceedings of the National Academy of Sciences of the United States of America, 106(14), 5464–5469. CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of PaderbornPaderbornGermany

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