IDEAS home Printed from https://ideas.repec.org/p/tse/wpaper/31577.html
   My bibliography  Save this paper

Kinetic models for topological nearest-neighbor interactions

Author

Listed:
  • Blanchet, Adrien
  • Degond, Pierre

Abstract

We consider systems of agents interacting through topological interactions. These have been shown to play an important part in animal and human behavior. Precisely, the system consists of a finite number of particles characterized by their positions and velocities. At random times a randomly chosen particle, the follower adopts the velocity of its closest neighbor, the leader. We study the limit of a system size going to infinity and, under the assumption of propagation of chaos, show that the limit kinetic equation is a non-standard spatial diffusion equation for the particle distribution function. We also study the case wherein the particles interact with their K closest neighbors and show that the corresponding kinetic equation is the same. Finally, we prove that these models can be seen as a singular limit of the smooth rank-based model previously studied in [10]. The proofs are based on a combinatorial interpretation of the rank as well as some concentration of measure arguments.

Suggested Citation

  • Blanchet, Adrien & Degond, Pierre, 2017. "Kinetic models for topological nearest-neighbor interactions," TSE Working Papers 17-786, Toulouse School of Economics (TSE).
    • Handle: RePEc:tse:wpaper:31577
    as

    Download full text from publisher

    File URL: https://www.tse-fr.eu/sites/default/files/TSE/documents/doc/wp/2017/wp_tse_786.pdf
    File Function: Full text
    Download Restriction: no
    ---><---

    Other versions of this item:

    References listed on IDEAS

    as
    1. Anders Eriksson & Martin Nilsson Jacobi & Johan Nyström & Kolbjørn Tunstrøm, 2010. "Determining interaction rules in animal swarms," Behavioral Ecology, International Society for Behavioral Ecology, vol. 21(5), pages 1106-1111.
    2. Tomoyuki Ichiba & Vassilios Papathanakos & Adrian Banner & Ioannis Karatzas & Robert Fernholz, 2009. "Hybrid Atlas models," Papers 0909.0065, arXiv.org, revised Apr 2011.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. David Itkin & Martin Larsson, 2021. "On A Class Of Rank-Based Continuous Semimartingales," Papers 2104.04396, arXiv.org.
    2. Ricardo T. Fernholz & Robert Fernholz, 2022. "Permutation-weighted portfolios and the efficiency of commodity futures markets," Annals of Finance, Springer, vol. 18(1), pages 81-108, March.
    3. Aditya Maheshwari & Andrey Sarantsev, 2017. "Modeling Financial System with Interbank Flows, Borrowing, and Investing," Papers 1707.03542, arXiv.org, revised Oct 2018.
    4. Brandon Flores & Blessing Ofori-Atta & Andrey Sarantsev, 2021. "A stock market model based on CAPM and market size," Annals of Finance, Springer, vol. 17(3), pages 405-424, September.
    5. Fernholz, Ricardo T., 2016. "A Model of economic mobility and the distribution of wealth," Journal of Macroeconomics, Elsevier, vol. 50(C), pages 168-192.
    6. Fernholz, Ricardo & Fernholz, Robert, 2014. "Instability and concentration in the distribution of wealth," Journal of Economic Dynamics and Control, Elsevier, vol. 44(C), pages 251-269.
    7. David Itkin & Martin Larsson, 2021. "Open Markets and Hybrid Jacobi Processes," Papers 2110.14046, arXiv.org, revised Mar 2024.
    8. Hernández, Freddy & Jara, Milton & Valentim, Fábio Júlio, 2017. "Equilibrium fluctuations for a discrete Atlas model," Stochastic Processes and their Applications, Elsevier, vol. 127(3), pages 783-802.
    9. Alexander Vervuurt & Ioannis Karatzas, 2015. "Diversity-weighted portfolios with negative parameter," Annals of Finance, Springer, vol. 11(3), pages 411-432, November.
    10. Vassilios Papathanakos, 2016. "Portfolio Optimization in the Stochastic Portfolio Theory Framework," Papers 1601.07628, arXiv.org.
    11. Robert Fernholz & Tomoyuki Ichiba & Ioannis Karatzas, 2013. "A second-order stock market model," Annals of Finance, Springer, vol. 9(3), pages 439-454, August.
    12. Benjamin Jourdain & Julien Reygner, 2015. "Capital distribution and portfolio performance in the mean-field Atlas model," Post-Print hal-00921151, HAL.
    13. Benjamin Jourdain & Julien Reygner, 2013. "Capital distribution and portfolio performance in the mean-field Atlas model," Papers 1312.5660, arXiv.org, revised Aug 2014.
    14. Robert Fernholz, 2017. "Stratonovich representation of semimartingale rank processes," Papers 1705.00336, arXiv.org.
    15. Christa Cuchiero & Walter Schachermayer & Ting‐Kam Leonard Wong, 2019. "Cover's universal portfolio, stochastic portfolio theory, and the numéraire portfolio," Mathematical Finance, Wiley Blackwell, vol. 29(3), pages 773-803, July.
    16. Aditya Maheshwari & Andrey Sarantsev, 2018. "Modeling Financial System with Interbank Flows, Borrowing, and Investing," Risks, MDPI, vol. 6(4), pages 1-26, November.
    17. Ricardo T. Fernholz & Robert Fernholz, 2017. "Zipf's Law for Atlas Models," Papers 1707.04285, arXiv.org, revised Jun 2020.
    18. Ricardo T. Fernholz & Christoffer Koch, 2018. "The Rank Effect," Papers 1812.06000, arXiv.org.
    19. Alexander Vervuurt, 2015. "Topics in Stochastic Portfolio Theory," Papers 1504.02988, arXiv.org.
    20. Ioannis Karatzas & Soumik Pal & Mykhaylo Shkolnikov, 2012. "Systems of Brownian particles with asymmetric collisions," Papers 1210.0259, arXiv.org.

    More about this item

    Keywords

    rank-based interaction; spatial diffusion equation; continuity equation; concentration of measure;
    All these keywords.

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:tse:wpaper:31577. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: the person in charge (email available below). General contact details of provider: https://edirc.repec.org/data/tsetofr.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.