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    Flocking dynamics and mean-field limit in the Cucker-Smale-type model with topological interactions

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    Type
    Article
    Authors
    Haskovec, Jan cc
    KAUST Department
    Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
    Date
    2013-10
    Permanent link to this record
    http://hdl.handle.net/10754/563025
    
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    Abstract
    We introduce a Cucker-Smale-type model for flocking, where the strength of interaction between two agents depends on their relative separation (called "topological distance" in previous works), which is the number of intermediate individuals separating them. This makes the model scale-free and is motivated by recent extensive observations of starling flocks, suggesting that the interaction ruling animal collective behavior depends on topological rather than the metric distance. We study the conditions leading to asymptotic flocking in the topological model, defined as the convergence of the agents' velocities to a common vector. The shift from metric to topological interactions requires development of new analytical methods, taking into account the graph-theoretical nature of the problem. Moreover, we provide a rigorous derivation of the mean-field limit of large populations, recovering kinetic and hydrodynamic descriptions. In particular, we introduce the novel concept of relative separation in continuum descriptions, which is applicable to a broad variety of models of collective behavior. As an example, we shortly discuss a topological modification of the attraction-repulsion model and illustrate with numerical simulations that the modified model produces interesting new pattern dynamics. © 2013 Elsevier B.V. All rights reserved.
    Citation
    Haskovec, J. (2013). Flocking dynamics and mean-field limit in the Cucker–Smale-type model with topological interactions. Physica D: Nonlinear Phenomena, 261, 42–51. doi:10.1016/j.physd.2013.06.006
    Publisher
    Elsevier BV
    Journal
    Physica D: Nonlinear Phenomena
    DOI
    10.1016/j.physd.2013.06.006
    arXiv
    1301.0925
    ae974a485f413a2113503eed53cd6c53
    10.1016/j.physd.2013.06.006
    Scopus Count
    Collections
    Articles; Computer, Electrical and Mathematical Science and Engineering (CEMSE) Division

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