The Role of Largest Connected Components in Collective Motion

  • Heiko HamannEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11172)


Systems showing collective motion are partially described by a distribution of positions and a distribution of velocities. While models of collective motion often focus on system features governed mostly by velocity distributions, the model presented in this paper also incorporates features influenced by positional distributions. A significant feature, the size of the largest connected component of the graph induced by the particle positions and their perception range, is identified using a 1-d self-propelled particle model (SPP). Based on largest connected components, properties of the system dynamics are found that are time-invariant. A simplified macroscopic model can be defined based on this time-invariance, which may allow for simple, concise, and precise predictions of systems showing collective motion.


  1. 1.
    Chazelle, B.: An algorithmic approach to collective behavior. J. Stat. Phys. 158(3), 514–548 (2015)MathSciNetCrossRefGoogle Scholar
  2. 2.
    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
  3. 3.
    Czirók, A., Vicsek, T.: Collective behavior of interacting self-propelled particles. Physica A 281, 17–29 (2000)CrossRefGoogle Scholar
  4. 4.
    Degond, P., Yang, T.: Diffusion in a continuum model of self-propelled particles with alignment interaction. Math. Models Methods Appl. Sci. 20, 1459–1490 (2010)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Hamann, H.: Space-Time Continuous Models of Swarm Robotics Systems: Supporting Global-to-Local Programming. Springer, Berlin (2010). Scholar
  6. 6.
    Hamann, H.: Towards swarm calculus: universal properties of swarm performance and collective decisions. In: Dorigo, M., Birattari, M., Blum, C., Christensen, A.L., Engelbrecht, A.P., Groß, R., Stützle, T. (eds.) ANTS 2012. LNCS, vol. 7461, pp. 168–179. Springer, Heidelberg (2012). Scholar
  7. 7.
    Hamann, H.: Towards swarm calculus: urn models of collective decisions and universal properties of swarm performance. Swarm Intell. 7(2–3), 145–172 (2013). Scholar
  8. 8.
    Hamann, H.: Swarm Robotics: A Formal Approach. Springer, Cham (2018). Scholar
  9. 9.
    Hamann, H., Meyer, B., Schmickl, T., Crailsheim, K.: A model of symmetry breaking in collective decision-making. In: Doncieux, S., Girard, B., Guillot, A., Hallam, J., Meyer, J.-A., Mouret, J.-B. (eds.) SAB 2010. LNCS (LNAI), vol. 6226, pp. 639–648. Springer, Heidelberg (2010). Scholar
  10. 10.
    Hamann, H., Valentini, G.: Swarm in a fly bottle: feedback-based analysis of self-organizing temporary lock-ins. In: Dorigo, M., Birattari, M., Garnier, S., Hamann, H., Montes de Oca, M., Solnon, C., Stützle, T. (eds.) ANTS 2014. LNCS, vol. 8667, pp. 170–181. Springer, Cham (2014). Scholar
  11. 11.
    Hamann, H., Wörn, H.: A framework of space-time continuous models for algorithm design in swarm robotics. Swarm Intell. 2(2–4), 209–239 (2008). Scholar
  12. 12.
    Helbing, D., Schweitzer, F., Keltsch, J., Molnár, P.: Active walker model for the formation of human and animal trail systems. Physical Review E 56(3), 2527–2539 (1997)CrossRefGoogle Scholar
  13. 13.
    Hillen, T., Painter, K.J.: A user’s guide to PDE models for chemotaxis. Math. Biol. 58, 183–217 (2009). Scholar
  14. 14.
    Khaluf, Y., Pinciroli, C., Valentini, G., Hamann, H.: The impact of agent density on scalability in collective systems: noise-induced versus majority-based bistability. Swarm Intell. 11(2), 155–179 (2017). Scholar
  15. 15.
    Levine, H., Rappel, W.J., Cohen, I.: Self-organization in systems of self-propelled particles. Phys. Rev. E 63(1), 17101 (2000)CrossRefGoogle Scholar
  16. 16.
    Milutinovic, D., Lima, P.: Cells and Robots: Modeling and Control of Large-Size Agent Populations. Springer, Berlin (2007). Scholar
  17. 17.
    Okubo, A.: Dynamical aspects of animal grouping: swarms, schools, flocks, and herds. Adv. Biophys. 22, 1–94 (1986)CrossRefGoogle Scholar
  18. 18.
    Prorok, A., Correll, N., Martinoli, A.: Multi-level spatial models for swarm-robotic systems. Int. J. Robot. Res. 30(5), 574–589 (2011)CrossRefGoogle Scholar
  19. 19.
    Reina, A., Marshall, J.A.R., Trianni, V., Bose, T.: Model of the best-of-\(n\) nest-site selection process in honeybees. Phys. Rev. E: 95, 052411 (2017). Scholar
  20. 20.
    Reina, A., Valentini, G., Fernández-Oto, C., Dorigo, M., Trianni, V.: A design pattern for decentralised decision making. PLOS ONE 10(10), 1–18 (2015). Scholar
  21. 21.
    Schimansky-Geier, L., Mieth, M., Rosé, H., Malchow, H.: Structure formation by active Brownian particles. Phys. Lett. A 207, 140–146 (1995)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Schweitzer, F.: Brownian Agents and Active Particles: On the Emergence of Complex Behavior in the Natural and Social Sciences. Springer, Berlin (2003)zbMATHGoogle Scholar
  23. 23.
    Valentini, G., Hamann, H.: Time-variant feedback processes in collective decision-making systems: influence and effect of dynamic neighborhood sizes. Swarm Intelligence 9(2–3), 153–176 (2015). Scholar
  24. 24.
    Vicsek, T., Czirók, A., Ben-Jacob, E., Cohen, I., Shochet, O.: Novel type of phase transition in a system of self-driven particles. Phys. Rev. Lett. 6(75), 1226–1229 (1995)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Vicsek, T., Zafeiris, A.: Collective motion. Phys. Rep. 517(3–4), 71–140 (2012)CrossRefGoogle Scholar
  26. 26.
    Yates, C.A., et al.: Inherent noise can facilitate coherence in collective swarm motion. Proc. Natl. Acad. Sci. USA 106(14), 5464–5469 (2009). Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Institute of Computer EngineeringUniversity of LübeckLübeckGermany

Personalised recommendations