Derivation of a Micro-Macro Link for Collective Decision-Making Systems

Uncover Network Features Based on Drift Measurements
  • Heiko Hamann
  • Gabriele Valentini
  • Yara Khaluf
  • Marco Dorigo
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8672)


Relating microscopic features (individual level) to macroscopic features (swarm level) of self-organizing collective systems is challenging. In this paper, we report the mathematical derivation of a macroscopic model starting from a microscopic one for the example of collective decision-making. The collective system is based on the application of a majority rule over groups of variable size which is modeled by chemical reactions (micro-model). From an approximated master equation we derive the drift term of a stochastic differential equation (macro-model) which is applied to predict the expected swarm behavior. We give a recursive definition of the polynomials defining this drift term. Our results are validated by Gillespie simulations and simulations of the locust alignment.


Master Equation Reaction Schema Majority Rule Neighborhood Size Markov Chain Monte Carlo Method 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bonabeau, E., Dorigo, M., Theraulaz, G.: Swarm Intelligence: From Natural to Artificial Systems. Oxford Univ. Press, New York (1999)zbMATHGoogle Scholar
  2. 2.
    Schweitzer, F.: Brownian Agents and Active Particles. On the Emergence of Complex Behavior in the Natural and Social Sciences. Springer, Berlin (2003)Google Scholar
  3. 3.
    Alexander, J.C., Giesen, B., Münch, R., Smelser, N.J. (eds.): The Micro-Macro Link. University of California Press, Berkeley (1987)Google Scholar
  4. 4.
    Schillo, M., Fischer, K., Klein, C.T.: The micro-macro link in DAI and sociology. In: Moss, S., Davidsson, P. (eds.) MABS 2000. LNCS (LNAI), vol. 1979, pp. 133–148. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  5. 5.
    Hamann, H.: Space-Time Continuous Models of Swarm Robotics Systems: Supporting Global-to-Local Programming. Springer, Berlin (2010)Google Scholar
  6. 6.
    Dorigo, M., Birattari, M., Brambilla, M.: Swarm robotics. Scholarpedia 9(1), 1463 (2014)CrossRefGoogle Scholar
  7. 7.
    Prorok, A., Correll, N., Martinoli, A.: Multi-level spatial models for swarm-robotic systems. The International Journal of Robotics Research 30(5), 574–589 (2011)CrossRefGoogle Scholar
  8. 8.
    Berman, S., Kumar, V., Nagpal, R.: Design of control policies for spatially inhomogeneous robot swarms with application to commercial pollination. In: LaValle, S., et al. (eds.) IEEE Int. Conf. on Robotics and Automation, ICRA 2011, pp. 378–385. IEEE Press (2011)Google Scholar
  9. 9.
    Huepe, C., Zschaler, G., Do, A.L., Gross, T.: Adaptive-network models of swarm dynamics. New Journal of Physics 13(7), 073022 (2011)Google Scholar
  10. 10.
    Franks, N.R., Mallon, E.B., Bray, H.E., Hamilton, M.J., Mischler, T.C.: Strategies for choosing between alternatives with different attributes: Exemplified by house-hunting ants. Animal Behavior 65, 215–223 (2003)CrossRefGoogle Scholar
  11. 11.
    Dussutour, A., Beekman, M., Nicolis, S.C., Meyer, B.: Noise improves collective decision-making by ants in dynamic environments. Proceedings of the Royal Society London B 276, 4353–4361 (2009)CrossRefGoogle Scholar
  12. 12.
    Montes de Oca, M., Ferrante, E., Scheidler, A., Pinciroli, C., Birattari, M., Dorigo, M.: Majority-rule opinion dynamics with differential latency: A mechanism for self-organized collective decision-making. Swarm Intelligence 5, 305–327 (2011)CrossRefGoogle Scholar
  13. 13.
    Valentini, G., Hamann, H., Dorigo, M.: Self-organized collective decision making: The weighted voter model. In: Lomuscio, A., et al. (eds.) Proc. of the 13th Int. Conf. on Autonomous Agents and Multiagent Systems, AAMAS 2014, pp. 45–52 (2014)Google Scholar
  14. 14.
    Biancalani, T., Dyson, L., McKane, A.J.: Noise-induced bistable states and their mean switching time in foraging colonies. Phys. Rev. Lett. 112, 038101 (2014)Google Scholar
  15. 15.
    van Kampen, N.G.: Stochastic processes in physics and chemistry. North-Holland, Amsterdam (1981)Google Scholar
  16. 16.
    Buhl, J., Sumpter, D.J.T., Couzin, I.D., Hale, J.J., Despland, E., Miller, E.R., Simpson, S.J.: From disorder to order in marching locusts. Science 312(5778), 1402–1406 (2006)CrossRefGoogle Scholar
  17. 17.
    Czirók, A., Barabási, A.L., Vicsek, T.: Collective motion of self-propelled particles: Kinetic phase transition in one dimension. Phys. Rev. Lett. 82(1), 209–212 (1999)CrossRefGoogle Scholar
  18. 18.
    Gillespie, D.T.: Exact stochastic simulation of coupled chemical reactions. The Journal of Physical Chemistry 81(25), 2340–2361 (1977)CrossRefGoogle Scholar
  19. 19.
    Yates, C.A., Erban, R., Escudero, C., Couzin, I.D., Buhl, J., Kevrekidis, I.G., Maini, P.K., Sumpter, D.J.T.: Inherent noise can facilitate coherence in collective swarm motion. Proc. Natl. Acad. Sci. USA 106(14), 5464–5469 (2009)CrossRefGoogle Scholar
  20. 20.
    Marquardt, D.: An algorithm for least-squares estimation of nonlinear parameters. SIAM Journal on Applied Mathematics 11(2), 431–441 (1963)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Heiko Hamann
    • 1
  • Gabriele Valentini
    • 2
  • Yara Khaluf
    • 1
  • Marco Dorigo
    • 2
  1. 1.Department of Computer ScienceUniversity of PaderbornPaderbornGermany
  2. 2.IRIDIAUniversité Libre de BruxellesBrusselsBelgium

Personalised recommendations