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  • Open Access

Topology and geometry of clusters in the Vicsek model across scales

Ashiqur Rahman*

Michael Napoli* and Ofek Lauber Bonomo

Maurizio Porfiri†

Nicole Abaid‡

  • Department of Mechanical Engineering, Virginia Tech, Blacksburg, Virginia, USA

  • Department of Mechanical and Aerospace Engineering and Center for Urban Science and Progress, Tandon School of Engineering, New York University, Brooklyn, New York, USA

  • Department of Mechanical and Aerospace Engineering, Center for Urban Science and Progress, Department of Civil, Urban, and Environmental Engineering, and Department of Biomedical Engineering, Tandon School of Engineering, New York University, Brooklyn, New York, USA

  • Department of Mathematics and the Center for the Mathematics of Biosystems, Virginia Tech, Blacksburg, Virginia, USA

  • *These authors contributed equally to this work.
  • †Contact author: mporfiri@nyu.edu
  • ‡Contact author: nabaid@vt.edu
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Phys. Rev. Research 8, 033039 – Published 10 July, 2026

DOI: https://doi.org/10.1103/hkrb-wpdr

Abstract

Collective motion in active matter can arise from interaction rules that generate group-level order. A seminal example is the Vicsek model, which describes flocking using a simple local alignment rule. Despite three decades of research, the network topology induced by the particles’ alignment remains understudied. Here, we explore clusters—defined as connected subgraphs of the system-wide interaction network—through topological and geometric measures at the critical noise and across various speeds and densities. We find that the average degree increases polylogarithmically with cluster mass while the variations from the average degree remain fixed across scales. The average path length scaling reveals a two-dimensional network structure, and the average clustering coefficient converges to a fixed value. Geometric analyses reveal that cluster area grows linearly with mass, with the perimeter growing more slowly. We discuss the findings using the degree distribution and positional roles of particles within a cluster. Our results indicate that clusters in the Vicsek model have uniform density with a self-similar network topology across scales, despite the lack of an explicit mechanism to form such interaction networks.

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Phys. Rev. Research 8, 033039– Published 10 July, 2026
  • Received 1 December 2025
  • Accepted 21 May 2026
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