Swarm Intelligence

, Volume 11, Issue 2, pp 155–179 | Cite as

The impact of agent density on scalability in collective systems: noise-induced versus majority-based bistability

  • Yara Khaluf
  • Carlo Pinciroli
  • Gabriele Valentini
  • Heiko Hamann


In this paper, we show that non-uniform distributions in swarms of agents have an impact on the scalability of collective decision-making. In particular, we highlight the relevance of noise-induced bistability in very sparse swarm systems and the failure of these systems to scale. Our work is based on three decision models. In the first model, each agent can change its decision after being recruited by a nearby agent. The second model captures the dynamics of dense swarms controlled by the majority rule (i.e., agents switch their opinion to comply with that of the majority of their neighbors). The third model combines the first two, with the aim of studying the role of non-uniform swarm density in the performance of collective decision-making. Based on the three models, we formulate a set of requirements for convergence and scalability in collective decision-making.


Bistable system Swarm density Noise Collective decision-making Non-uniform spatial distribution 



This work was partially supported by the European Union’s Horizon 2020 research and innovation program under the FET Grant “flora robotica,” No. 640959.

Supplementary material

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Supplementary material 1 (gif 9787 KB)


  1. Angluin, D., Aspnes, J., & Eisenstat, D. (2008). A simple population protocol for fast robust approximate majority. Distributed Computing, 21(2), 87–102.CrossRefzbMATHGoogle Scholar
  2. Arnold, L. (2003). Random dynamical systems. Berlin: Springer.Google Scholar
  3. Beckers, R., Deneubourg, J.-L., Goss, S., & Pasteels, J. M. (1990). Collective decision making through food recruitment. Insectes Sociaux, 37(3), 258–267.CrossRefGoogle Scholar
  4. Biancalani, T., Dyson, L., & McKane, A. J. (2014). Noise-induced bistable states and their mean switching time in foraging colonies. Physical Review Letters, 112, 038101.CrossRefGoogle Scholar
  5. Dorigo, M., Birattari, M., & Brambilla, M. (2014). Swarm robotics. Scholarpedia, 9(1), 1463.CrossRefGoogle Scholar
  6. 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 London B, 276, 4353–4361.CrossRefGoogle Scholar
  7. Dyson, L., Yates, C., Buhl, J., & McKane, A. (2015). Onset of collective motion in locusts is captured by a minimal model. Physical Review E, 92(5), 052708.CrossRefGoogle Scholar
  8. Galam, S. (2000). Real space renormalization group and totalitarian paradox of majority rule voting. Physica A: Statistical Mechanics and its Applications, 285(1–2), 66–76.MathSciNetCrossRefzbMATHGoogle Scholar
  9. Gardiner, C. W. (1985). Handbook of stochastic methods for physics, chemistry and the natural sciences. Berlin: Springer.CrossRefzbMATHGoogle Scholar
  10. Grüter, C., Schürch, R., Czaczkes, T., Taylor, K., Durance, T., Jones, S., et al. (2012). Negative feedback enables fast and flexible collective decision-making in ants. PLoS ONE, 7(9), e44501.CrossRefGoogle Scholar
  11. Gutiérrez, Á., Campo, A., Monasterio-Huelin, F., Magdalena, L., & Dorigo, M. (2010). Collective decision-making based on social odometry. Neural Computing and Applications, 19(6), 807–823.CrossRefGoogle Scholar
  12. Halloy, J., Sempo, G., Caprari, G., Rivault, C., Asadpour, M., Tâche, F., et al. (2007). Social integration of robots into groups of cockroaches to control self-organized choices. Science, 318(5853), 1155–1158.CrossRefGoogle Scholar
  13. Hamann, H., Karsai, I., & Schmickl, T. (2013). Time delay implies cost on task switching: A model to investigate the efficiency of task partitioning. Bulletin of Mathematical Biology, 75(7), 1181–1206.MathSciNetCrossRefzbMATHGoogle Scholar
  14. Hamann, H., Valentini, G., Khaluf, Y., & Dorigo, M. (2014). Derivation of a micro-macro link for collective decision-making systems: Uncover network features based on drift measurements. In T. Bartz-Beielstein, J. Branke, B. Filipič, & J. Smith (Eds.), 13th International conference on parallel problem solving from nature (PPSN 2014), volume 8672 of LNCS (pp. 181–190). Berlin: Springer.Google Scholar
  15. Hamann, H., & Wörn, H. (2008). A framework of space-time continuous models for algorithm design in swarm robotics. Swarm Intelligence, 2(2–4), 209–239.CrossRefGoogle Scholar
  16. Houchmandzadeh, B., & Vallade, M. (2015). Exact results for a noise-induced bistable system. Physical Review E, 91(2), 022115.MathSciNetCrossRefGoogle Scholar
  17. Huepe, C., Zschaler, G., Do, A.-L., & Gross, T. (2011). Adaptive-network models of swarm dynamics. New Journal of Physics, 13(7), 073022.CrossRefGoogle Scholar
  18. Hunter, J. J. (2005). Stationary distributions and mean first passage times of perturbed Markov chains. Linear Algebra and its Applications, 410, 217–243.MathSciNetCrossRefzbMATHGoogle Scholar
  19. Hunter, J. J. (2007). Simple procedures for finding mean first passage times in Markov chains. Asia-Pacific Journal of Operational Research, 24(06), 813–829.MathSciNetCrossRefzbMATHGoogle Scholar
  20. Ijspeert, A. J., Martinoli, A., Billard, A., & Gambardella, L. M. (2001). Collaboration through the exploitation of local interactions in autonomous collective robotics: The stick pulling experiment. Autonomous Robots, 11, 149–171.CrossRefzbMATHGoogle Scholar
  21. Jeanne, R. L. (1986). The organization of work in Polybia occidentalis: Costs and benefits of specialization in a social wasp. Behavioral Ecology and Sociobiology, 19(5), 333–341.CrossRefGoogle Scholar
  22. Khaluf, Y., & Dorigo, M. (2016). Modeling robot swarms using integrals of birth-death processes. ACM Transactions on Autonomous and Adaptive Systems (TAAS), 11(2), 8.Google Scholar
  23. Khaluf, Y., & Hamann, H. (2016). On the definition of self-organizing systems: Relevance of positive/negative feedback and fluctuations. In ANTS 2016, volume 9882 of LNCS (p. 298). Berlin: Springer.Google Scholar
  24. Lerman, K., Martinoli, A., & Galstyan, A. (2005). A review of probabilistic macroscopic models for swarm robotic systems. In E. Şahin, & W. M. Spears (Eds.), Swarm robotics—SAB 2004 International workshop, volume 3342 of LNCS (pp. 143–152). Berlin: Springer.Google Scholar
  25. Mallon, E., Pratt, S., & Franks, N. (2001). Individual and collective decision-making during nest site selection by the ant Leptothorax albipennis. Behavioral Ecology and Sociobiology, 50(4), 352–359.CrossRefGoogle Scholar
  26. Martinoli, A., Easton, K., & Agassounon, W. (2004). Modeling swarm robotic systems: A case study in collaborative distributed manipulation. International Journal of Robotics Research, 23(4), 415–436.CrossRefGoogle Scholar
  27. Meyer, B., Beekman, M., & Dussutour, A. (2008). Noise-induced adaptive decision-making in ant-foraging. In Simulation of adaptive behavior (SAB), number 5040 in LNCS (pp. 415–425). Berlin: Springer.Google Scholar
  28. Montes de Oca, M., Ferrante, E., Scheidler, A., Pinciroli, C., Birattari, M., & Dorigo, M. (2011). Majority-rule opinion dynamics with differential latency: A mechanism for self-organized collective decision-making. Swarm Intelligence, 5(3–4), 305–327.CrossRefGoogle Scholar
  29. Ohkubo, J., Shnerb, N., & Kessler, D. A. (2008). Transition phenomena induced by internal noise and quasi-absorbing state. Journal of the Physical Society of Japan, 77(4), 044002.CrossRefGoogle Scholar
  30. Olfati-Saber, R., Fax, A., & Murray, R. M. (2007). Consensus and cooperation in networked multi-agent systems. Proceedings of the IEEE, 95(1), 215–233.CrossRefGoogle Scholar
  31. Reina, A., Miletitch, R., Dorigo, M., & Trianni, V. (2015a). A quantitative micro-macro link for collective decisions: The shortest path discovery/selection example. Swarm Intelligence, 9(2–3), 75–102.CrossRefGoogle Scholar
  32. Reina, A., Valentini, G., Fernández-Oto, C., Dorigo, M., & Trianni, V. (2015b). A design pattern for decentralised decision making. PLoS ONE, 10(10), e0140950.CrossRefGoogle Scholar
  33. Rubenstein, M., Cornejo, A., & Nagpal, R. (2014). Programmable self-assembly in a thousand-robot swarm. Science, 345(6198), 795–799.CrossRefGoogle Scholar
  34. 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
  35. 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
  36. 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
  37. Seeley, T. D., Visscher, P., Schlegel, T., Hogan, P., Franks, N., & Marshall, J. (2012). Stop signals provide cross inhibition in collective decision-making by honeybee swarms. Science, 335(6064), 108–111.CrossRefGoogle Scholar
  38. Streit, R. (2010). Poisson point processes: Imaging, tracking, and sensing. New York: Springer.CrossRefGoogle Scholar
  39. Szopek, M., Schmickl, T., Thenius, R., Radspieler, G., & Crailsheim, K. (2013). Dynamics of collective decision making of honeybees in complex temperature fields. PLoS ONE, 8(10), e76250.CrossRefGoogle Scholar
  40. Togashi, Y., & Kaneko, K. (2001). Transitions induced by the discreteness of molecules in a small autocatalytic system. Physical Review Letters, 86, 2459–2462.CrossRefGoogle Scholar
  41. Valentini, G., Ferrante, E., & Dorigo, M. (2017). The best-of-n problem in robot swarms: Formalization, state of the art, and novel perspectives. Frontiers in Robotics and AI, 4, 9.CrossRefGoogle Scholar
  42. Valentini, G., Ferrante, E., Hamann, H., & Dorigo, M. (2015). Collective decision with 100 Kilobots: Speed versus accuracy in binary discrimination problems. Autonomous Agents and Multi-Agent Systems, 30(3), 553–580.CrossRefGoogle Scholar
  43. Valentini, G., & Hamann, H. (2015). Time-variant feedback processes in collective decision-making systems: Influence and effect of dynamic neighborhood sizes. Swarm Intelligence, 9(2–3), 153–176.CrossRefGoogle Scholar
  44. Valentini, G., Hamann, H., & Dorigo, M. (2014). Self-organized collective decision making: The weighted voter model. In Lomuscio, A., Scerri, P., Bazzan, A., & Huhns, M., (eds), Proceedings of the 13th International Conference on Autonomous Agents and Multiagent Systems, AAMAS ’14, (pp. 45–52). IFAAMAS.Google Scholar
  45. Yates, C., Erban, R., Escudero, C., Couzin, I., Buhl, J., Kevrekidis, I., et al. (2009). Inherent noise can facilitate coherence in collective swarm motion. PNAS, 106(14), 5464–5469.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Yara Khaluf
    • 1
  • Carlo Pinciroli
    • 2
  • Gabriele Valentini
    • 3
  • Heiko Hamann
    • 4
  1. 1.Department of Information TechnologyGhent UniversityGhentBelgium
  2. 2.Robotics Engineering and Computer ScienceWorcester Polytechnic InstituteWorcesterUSA
  3. 3.School of Earth and Space ExplorationArizona State UniversityTempeUSA
  4. 4.Institute of Computer EngineeringUniversity of LübeckLübeckGermany

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