A Survey on Cucker–Smale Model and Its Variant

DOI

10.2991/978-94-6239-774-3_8How to use a DOI?

Keywords

Cucker-Smale model; flocking dynamics; collective movement

Abstract

Collective motion is a phenomenon widely observed in biological systems and engineering multi-agent networks, where a large number of individuals achieve coordinated movement through dispersed local interactions. Typical examples include flocks of birds, schools of fish, and swarms of robots. Although experience tells us that collective patterns can be generated in simulations, these cannot explain the mechanisms or conditions under which local interactions produce global coordination, so people began to seek rigorous mathematical models. Understanding how this global order emerges from simple interaction rules has become a central problem in applied mathematics, control theory, and complex systems.

The Cucker-Smale (CS) model provides a fundamental mathematical framework for this phenomenon, describing velocity alignment through distance-dependent pairwise interactions. Since its introduction in 2007, the CS model has inspired numerous theoretical developments, ranging from sophisticated swarm analysis techniques to extended models incorporating stochastic effects, finite sensing range, complex network structures, and geometric constraints.

This article presents a structured survey of the Cucker-Smale model and its major extensions. We first review the classical flocking theory and key analytical approaches, including Lyapunov methods and mean-field limits. We then discuss model refinements motivated by realism, such as noise, cut-off interactions, leader-follower networks, signed graphs, and normalized alignment. Finally, we highlight recent advances in adaptive dynamics and flocking on non-Euclidean manifolds. This article provides a reference for readers seeking an introduction to CS models. By organizing existing results within a unified framework, the article provides a solid foundation for beginners interested in the further study of collective motion.

Copyright

© 2026 The Author(s)

Open Access

Open Access This chapter is licensed under the terms of the Creative Commons Attribution-NonCommercial 4.0 International License (http://creativecommons.org/licenses/by-nc/4.0/), which permits any noncommercial use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

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TY  - CONF
AU  - Haolin He
PY  - 2026
DA  - 2026/09/11
TI  - A Survey on Cucker–Smale Model and Its Variant
BT  - Proceedings of the 2026 5th International Conference on Mathematical Statistics and Economic Analysis (MSEA 2026
PB  - Atlantis Press
SP  - 83
EP  - 92
SN  - 2352-5428
UR  - https://doi.org/10.2991/978-94-6239-774-3_8
DO  - 10.2991/978-94-6239-774-3_8
ID  - He2026
ER  -