17th International Symposium on
Mathematical Theory of Networks and Systems
Kyoto International Conference Hall, Kyoto, Japan, July 24-28, 2006

MTNS 2006 Paper Abstract

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Paper TuP13.3

Hendrickx, Julien M. (Univ. catholique de Louvain), Blondel, Vincent (Univ. catholique de Louvain)

Convergence of Different Linear and Non-Linear Vicsek Models

Scheduled for presentation during the Regular Session "Linear Systems I" (TuP13), Tuesday, July 25, 2006, 16:10−16:35, Room 101

17th International Symposium on Mathematical Theory of Networks and Systems, July 24-28, 2006, Kyoto, Japan

This information is tentative and subject to change. Compiled on May 19, 2024

Keywords Systems on graphs, Group formation, Linear systems

Abstract

The Vicsek model describes the evolution of a system composed by different agents moving in the plane. Each agent has a constant speed and updates its heading using a local rule depending on the headings of its “neighbors”. Although the original model by Vicsek is non-linear, most of the convergence results obtained so far deal with linearized versions. In this paper, we introduce a new linear model in which the relative importance of each neighbour can vary with the distance and we prove the convergence of all the agents headings. For this purpose, we derive a theorem on the convergence of long products of stochastic products that applies to infinite set of matrices. Using this result we also prove convergence properties for the original non-linear Vicsek model. Moreau obtains similar results to ours but using a proof technique based on convexity and system theory. We present here proofs that are based on elementary linear algebra tools. The results we obtain are somewhat weaker than those of Moreau but have the advantage of being indistinctly applicable to continuous and discontinuous systems.