Japanese Journal of Mathematics

, Volume 2, Issue 1, pp 197–227

On the mathematics of emergence

Special Feature: The 1st Takagi Lecture

Abstract.

We describe a setting where convergence to consensus in a population of autonomous agents can be formally addressed and prove some general results establishing conditions under which such convergence occurs. Both continuous and discrete time are considered and a number of particular examples, notably the way in which a population of animals move together, are considered as particular instances of our setting.

Keywords and phrases:

emergence flocking consensus reaching 

Mathematics Subject Classification (2000):

92D50 92D25 91D30 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    R. Agaev and P. Chebotarev, On the spectra of nonsymmetric Laplacian matrices, Linear Algebra Appl., 399 (2005), 157–168.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Y. L. Chuang, Y. R. Huang, M. R. D’Orsogna and A. L. Bertozzi, Multi-vehicle flocking: Scalability of cooperative control algorithms using pairwise potentials, preprint, 2006.Google Scholar
  3. 3.
    F. Chung, Spectral Graph Theory, Amer. Math. Soc., Providence, RI., 1997.Google Scholar
  4. 4.
    F. Chung, Laplacians and the Cheeger inequality for directed graphs, Ann. Comb., 9 (2005), 1–19.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    F. Cucker and S. Smale, Best choices for regularization parameters in learning theory, Found. Comput. Math., 2 (2002), 413–428.MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    F. Cucker and S. Smale, Emergent behavior in flocks, to appear at IEEE Trans. Automat. Control (2007).Google Scholar
  7. 7.
    F. Cucker, S. Smale and D. X. Zhou, Modeling language evolution, Found. Comput. Math., 4 (2004), 315–343.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    G. Flierl, D. Grünbaum, S. Levin and D. Olson, From individuals to aggregations: The interplay between behavior and physics, J. Theoret. Biol., 196 (1999), 397–454.CrossRefGoogle Scholar
  9. 9.
    M. Hirsch and S. Smale, Differential equations, dynamical systems, and linear algebra, Pure Appl. Math., 60, Academic Press, 1974.Google Scholar
  10. 10.
    A. Jadbabaie, J. Lin and A. S. Morse, Coordination of groups of mobile autonomous agents using nearest neighbor rules, IEEE Trans. Automat. Control, 48 (2003), 988–1001.CrossRefMathSciNetGoogle Scholar
  11. 11.
    J. Ke, J. Minett, C.-P. Au and W. S.-Y. Wang, Self-organization and selection in the emergence of vocabulary, Complexity, 7 (2002), 41–54.CrossRefMathSciNetGoogle Scholar
  12. 12.
    B. Mohar, The Laplacian spectrum of graphs, In: Graph Theory, Combinatorics, and Applications, (eds. Y. Alavi, G. Chartrand, O. R. Oellermann and A. J. Schwenk), 2, John Wiley & Sons, 1991, pp. 871–898.Google Scholar
  13. 13.
    P. Niyogi, The Computational Nature of Language Learning and Evolution, The MIT Press, 2006.Google Scholar
  14. 14.
    H. G. Tanner, A. Jadbabaie and G. J. Pappas, Stable flocking of mobile agents, Part I: Fixed topology, In: Proceedings of the 42nd IEEE Conference on Decision and Control, 2003, pp. 2010–2015.Google Scholar
  15. 15.
    H. G. Tanner, A. Jadbabaie and G. J. Pappas, Stable flocking of mobile agents, Part II: Dynamic topology, In: Proceedings of the 42nd IEEE Conference on Decision and Control, 2003, pp. 2016–2021.Google Scholar
  16. 16.
    C. M. Topaz, A. L. Bertozzi and M. A. Lewis, A nonlocal continuum model for biological aggregation, Bull. Math. Biol., 68 (2006), 1601–1623.CrossRefMathSciNetGoogle Scholar
  17. 17.
    J. N. Tsitsiklis, Problems in decentralized decision making and computation, PhD thesis, Department of EECS, MIT, 1984.Google Scholar
  18. 18.
    J. N. Tsitsiklis, D. P. Bertsekas and M. Athans, Distributed asynchronous deterministic and stochastic gradient optimization algorithms, IEEE Trans. Automat. Control, 31 (1986), 803–812.MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    T. Vicsek, A. Czirók, E. Ben-Jacob and O. Shochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226–1229.CrossRefGoogle Scholar

Copyright information

© The Mathematical Society of Japan and Springer 2007

Authors and Affiliations

  1. 1.Department of MathematicsCity University of Hong KongKowloonHong Kong
  2. 2.Toyota Technological Institute at ChicagoChicagoUSA

Personalised recommendations