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Flocking Behavior of the Cucker–Smale Model on Infinite Graphs with a Central Vertex Group

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Abstract

This article investigates the flocking behavior of the Cucker–Smale (CS) model on infinite graphs, considering both standard and cut-off interactions. We introduce the concept of connected infinite graphs with a central vertex group and then derive sufficient conditions for the CS model to produce flocking behavior. For standard interaction, we find that the CS model will exhibit flocking behavior exponentially when the connected infinite graph is equipped with a central vertex group. However, for cut-off interaction, we need the time-varying graph induced by interparticle distance to have a fixed central vertex group and the coupling strength to be above a certain threshold to produce the flocking behavior. Our theoretical analysis shows that if a connected infinite graph has a central vertex group, the second eigenvalue of the corresponding Laplacian is positive, which is crucial for the proof of flocking behavior. The consistent convergence towards flocking may well reveal the advantages and necessities of having a central vertex group in an infinite-particle complex system with sufficient intelligence.

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Acknowledgements

The authors are grateful to the anonymous reviewers whose comments enriched the paper. Meanwhile, the authors are also very thankful to Qifan Mao for his suggestions on properly conveying the results.

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  1. School of Mathematics, Harbin Institute of Technology, Harbin, 150001, People’s Republic of China

    Xinyu Wang & Xiaoping Xue

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  1. Xinyu Wang
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  2. Xiaoping Xue
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Correspondence to Xiaoping Xue.

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Communicated by Hal Tasaki.

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Wang, X., Xue, X. Flocking Behavior of the Cucker–Smale Model on Infinite Graphs with a Central Vertex Group. J Stat Phys 191, 47 (2024). https://doi.org/10.1007/s10955-024-03255-2

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