- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Target-density formation in swarms with stochastic sensing and dynamics

Jason Hindes1,*, George Stantchev1, Klimka Szwaykowska Kasraie2, and Ira B. Schwartz1

  • *Contact author: jason.m.hindes.civ@us.navy.mil
PDF

Phys. Rev. Research 7, 043167 – Published 13 November, 2025

DOI: https://doi.org/10.1103/wpgk-hhp7

Abstract

An important goal for swarming research is to create methods for predicting, controlling, and designing swarms, which produce collective dynamics that solve a problem through emergent and stable pattern formation, without the need for constant intervention and with a minimal number of parameters and controls. One such problem involves a swarm collectively producing a desired (target) density through local sensing, motion, and interactions in a domain. Here, we take a statistical physics perspective and develop and analyze a model wherein agents move in a stochastic walk over a networked domain, so as to reduce the error between the swarm density and the target, based on local, random, and uncertain measurements of the current density by the swarming agents. Using a combination of mean-field, small-fluctuation, and finite-number analysis, we are able to quantify how close and how fast a swarm comes to producing a target as a function of sensing uncertainty, stochastic collision rates, numbers of agents, and spatial variation of the target.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (70)

  1. T. Vicsek and A. Zafeiris, Collective motion, Phys. Rep. 517, 71 (2012).
  2. L. Psimen, Active Matter Within and Around Us: From Self-propelled Particles to Flocks and Living Forms, 1st ed. (Springer, Nature Switzerland AG, 2021).
  3. Q. Wang and D. Jin, Active micro/nanoparticles in colloidal microswarms, Nanomaterials 13, 1687 (2023).
  4. J. Zhang, J. Guo, F. Mou, and J. Guan, Light-controlled swarming and assembly of colloidal particles, Micromachines 9, 88 (2018).
  5. M. F. Copeland and D. B. Weibel, Bacterial swarming: A model system for studying dynamic self-assembly. Soft Matter 5, 1174 (2009).
  6. A. Be’er and G. Ariel, A statistical physics view of swarming bacteria, Mov. Ecol. 7, 9 (2019).
  7. G. Theraulaz, E. Bonabeau, S. C. Nicolis, R. V. Solé, V. Fourcassié, S. Blanco, R. Fournier, J. L. Joly, P. Fernández, A. Grimal, P. Dalle, and J. L. Deneubourg, Spatial patterns in ant colonies, Proc. Natl. Acad. Sci. USA 99, 9645 (2002).
  8. M. Sinhuber, K. van der Vaart, Y. Feng, A. M. Reynolds, and N. T. Ouellette, An equation of state for insect swarms, Sci. Rep. 11, 3773 (2021).
  9. C. M. Topaz, M. R. D’Orsogna, L. Edelstein-Keshet, and A. J. Bernoff, Locust dynamics: Behavioral phase change and swarming, PLoS Comput. Biol. 8, e1002642 (2012).
  10. G. F. Young, L. Scardovi, A. Cavagna, I. Giardina, and N. E. Leonard, Starling flock networks manage uncertainty in consensus at low cost, PLoS Comput. Biol. 9, e1002894 (2013).
  11. M. Ballerini, N. Cabibbo, R. Candelier, A. Cavagna, E. Cisbani, I. Giardina, V. Lecomte, A. Orlandi, G. Parisi, A. Procaccini, M. Viale, and V. Zdravkovic, Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study, Proc. Natl. Acad. Sci. USA 105, 1232 (2008).
  12. K. Tunstrøm, Y. Katz, C. C. Ioannou, C. Huepe, M. J. Lutz, and I. D. Couzin, Collective states, multistability and transitional behavior in schooling fish, PLoS Comput. Biol. 9, e1002915 (2013).
  13. D. S. Calovi, U. Lopez, S. Ngo, C. Sire, H. Chaté, and G. Theraulaz, Swarming, schooling, milling: Phase diagram of a data-driven fish school model, New J. Phys. 16, 015026 (2014).
  14. D. Helbing, A. Johansson, and H. Z. Al-Abideen, Dynamics of crowd disasters: An empirical study, Phys. Rev. E 75, 046109 (2007).
  15. M. Chraibi, U. Kemloh, A. Schadschneider, and A. Seyfried, Force-based models of pedestrian dynamics, Networks Heterog. Media 6, 425 (2011).
  16. T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen, and O. Shochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett. 75, 1226 (1995).
  17. H. Levine, W.-J. Rappel, and I. Cohen, Self-organization in systems of self-propelled particles, Phys. Rev. E 63, 017101 (2000).
  18. M. R. D’Orsogna, Y. L. Chuang, A. L. Bertozzi, and L. S. Chayes, Self-propelled particles with soft-core interactions: Patterns, stability, and collapse, Phys. Rev. Lett. 96, 104302 (2006).
  19. F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Autom. Control 52, 852 (2007).
  20. Y.-P. Choi, S.-Y. Ha, and Z. Li, Emergent dynamics of the Cucker–Smale flocking model and its variants, in Active Particles: Advances in Theory, Models, and Applications, edited by N. Bellomo, P. Degond, and E. Tadmor (Springer International Publishing, Cham, 2017), Vol. 1, pp. 299–331.
  21. J. Hindes, V. Edwards, K. S. Kasraie, G. Stantchev, and I. B. Schwartz, Swarm shedding in networks of self-propelled agents, Sci. Rep. 11, 13544 (2021).
  22. J. Hindes, V. Edwards, S. Kamimoto, I. Triandaf, and I. B. Schwartz, Unstable modes and bistability in delay-coupled swarms, Phys. Rev. E 101, 042202 (2020).
  23. J. Hindes, K. Daley, G. Stantchev, and I. B Schwartz, Swarming network inference with importance clustering of relative interactions, J. Phys.: Complexity 5, 045009 (2024).
  24. H. Hamann, Swarm Robotics: A Formal Approach (Springer International Publishing, 2018).
  25. M. Dorigo, G. Theraulaz, and V. Trianni, Swarm robotics: Past, present, and future [point of view], Proc. IEEE 109, 1152 (2021).
  26. D. Mezey, R. Bastien, Y. Zheng, N. McKee, D. Stoll, H. Hamann, and P. Romanczuk, Purely vision-based collective movement of robots, npj Robotics 3, 11 (2025).
  27. X. Huang, F. Arvin, C. West, S. Watson, and B. Lennox, Exploration in extreme environments with swarm robotic system, Proceedings of the 2019 IEEE International Conference on Mechatronics (ICM) (IEEE, Piscataway, NJ, 2019), Vol. 1, pp. 193–198.
  28. A. G. Pires, P. A. F. Rezeck, R. A. Chaves, D. G. Macharet, and L. Chaimowicz, Cooperative localization and mapping with robotic swarms, J. Intell. Rob. Syst. 102, 47 (2021).
  29. K. M. Lynch, I. B. Schwartz, P. Yang, and R. A. Freeman, Decentralized environmental modeling by mobile sensor networks, IEEE Trans. Rob. 24, 710 (2008).
  30. B. B. V. L. Deepak and D. R. Parhi, Target seeking behaviour of an intelligent mobile robot using advanced particle swarm optimization, in 2013 International Conference on Control, Automation, Robotics and Embedded Systems (CARE), Jabalpur, India (IEEE, Piscataway, NJ, 2013), pp. 1–6.
  31. R. J. Amala Arokia Nathan, I. Kurmi, and O. Bimber, Drone swarm strategy for the detection and tracking of occluded targets in complex environments, Commun. Eng. 2, 55 (2023).
  32. L. M. Gladence, V. M. Anu, A. Anderson, I. Stanley, J. A. Fernando J, and S. Revathy, Swarm intelligence in disaster recovery, 2021 5th International Conference on Intelligent Computing and Control Systems (ICICCS), Madurai, India (IEEE, Piscataway, NJ, 2021), pp. 1–8.
  33. H. Li, C. Feng, H. Ehrhard, Y. Shen, B. Cobos, F. Zhang, K. Elamvazhuthi, S. Berman, M. Haberland, and A. L. Bertozzi, Decentralized stochastic control of robotic swarm density: Theory, simulation, and experiment, 2017 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), Vancouver, BC, Canada (IEEE, Piscataway, NJ, 2017), pp. 4341–4347.
  34. C. Walton, I. Kaminer, Q. Gong, A. H. Clark, and T. Tsatsanifos, Defense against adversarial swarms with parameter uncertainty, Sensors 22, 4773 (2022).
  35. J. Hindes, V. Edwards, M. A. Hsieh, and I. B. Schwartz, Critical transition for colliding swarms, Phys. Rev. E 103, 062602 (2021).
  36. A. T. Liu, M. Hempel, J. F. Yang, A. M. Brooks, A. Pervan, V. B. Koman, G. Zhang, D. Kozawa, S. Yang, D. I. Goldman, M. Z. Miskin, A. W. Richa, D. Randall, T. D. Murphey, T. Palacios, and M. S. Strano, Colloidal robotics, Nat. Mater. 22, 1453 (2023).
  37. W. Ren, Consensus strategies for cooperative control of vehicle formations, IET Control Theory Appl. 1, 505 (2007).
  38. W. Ren, Formation keeping and attitude alignment for multiple spacecraft through local interactions, J. Guid. Control. Dyn. 30, 633 (2007).
  39. X. Li and M. F. Ercan, Decentralized coordination control for a network of mobile robotic sensors, Wireless Pers. Commun. 102, 2429 (2018).
  40. J. Cortes, S. Martinez, T. Karatas, and F. Bullo, Coverage control for mobile sensing networks, IEEE Trans. Robot. Autom. 20, 243 (2004).
  41. K. Elamvazhuthi, C. Adams and S. Berman, Coverage and field estimation on bounded domains by diffusive swarms, in 2016 IEEE 55th Conference on Decision and Control (CDC), Las Vegas, NV (IEEE, Piscataway, NJ, 2016), pp. 2867–2874.
  42. K. Elamvazhuthi, H. Kuiper, and S. Berman, Pde-based optimization for stochastic mapping and coverage strategies using robotic ensembles, Automatica 95, 356 (2018).
  43. S. Zhao, S. Ramakrishnan and M. Kumar, Density-based control of multiple robots, in Proceedings of the 2011 American Control Conference, San Francisco, CA (IEEE, Piscataway, NJ, 2011), pp. 481–486.
  44. U. Eren and B. Acikmese, Velocity field generation for density control of swarms using heat equation and smoothing kernels, IFAC-PapersOnLine 50, 9405 (2017).
  45. T. Zheng and H. Lin, Distributed density filtering for large-scale systems using mean-filed models, 2021 American Control Conference (ACC), New Orleans, LA (IEEE, Piscataway, NJ, 2021), pp. 334–339.
  46. T. Zheng, Q. Han, and H. Lin, Transporting robotic swarms via mean-field feedback control, IEEE Trans. Autom. Control 67, 4170 (2022).
  47. C. Sinigaglia, A. Manzoni, F. Braghin, and S. Berman, Robust optimal density control of robotic swarms, Automatica 176, 112218 (2025).
  48. I. Chattopadhyay and A. Ray, Supervised self-organization of homogeneous swarms using ergodic projections of Markov chains, IEEE Trans. Syst. Man Cybern. Part B 39, 1505 (2009).
  49. B. Açikmeşe and D. S. Bayard, A Markov chain approach to probabilistic swarm guidance, 2012 American Control Conference (ACC), Montreal, QC, Canada (IEEE, Piscataway, NJ, 2012), pp. 6300–6307.
  50. D. Cui and H. Li, Density regulation of large-scale robotic swarm using robust model predictive mean-field control, Automatica 169, 111832 (2024).
  51. S. Bandyopadhyay, S.-J. Chung, and F. Y. Hadaegh, Probabilistic and distributed control of a large-scale swarm of autonomous agents, IEEE Trans. Robot. 33, 1103 (2017).
  52. N. G. van Kampen, Stochastic Processes in Physics and Chemistry, 3rd ed. (Elsevier Science Publishers, Amsterdam, 2007).
  53. N. Masuda and C. L. Vestergaard, Gillespie Algorithms for Stochastic Multiagent Dynamics in Populations and Networks (Cambridge University Press, 2023).
  54. If an agent at patch i moves to j, the change in the squared Euclidean distance to the target is 2ẑij/N for large N, as perceived by the agent.
  55. Note that in order for β to remain finite as N→∞ for fixed M and w, V should scale linearly with N. In addition, if ejection occurs only when collisions are near a patch boundary, one can reduce β by the appropriate fraction of the patch volume that is close enough to a boundary to result in ejection.
  56. M. I. Dykman, E. Mori, J. Ross, and P. M. Hunt, Large fluctuations and optimal paths in chemical kinetics, J. Chem. Phys. 100, 5735 (1994).
  57. M. Newman, Networks (Oxford University Press, Oxford, 2018).
  58. P. van Mieghem, Graph Spectra for Complex Networks (Cambridge University Press, New York, 2011).
  59. T. Ishida, A model of octopus epidermis pattern mimicry mechanisms using inverse operation of the turing reaction model, PLoS ONE 16, e0256025 (2021).
  60. Given a random sample, the target densities are renormalized, y¯s→y¯s/∑iy¯i, so that ∑sy¯s=1.
  61. V. Belik, T. Geisel, and D. Brockmann, Natural human mobility patterns and spatial spread of infectious diseases, Phys. Rev. X 1, 011001 (2011).
  62. F. Heylighen, Stigmergy as a universal coordination mechanism I: Definition and components, Cognit. Syst. Res. 38, 4 (2016).
  63. B. A Camley, Collective gradient sensing and chemotaxis: Modeling and recent developments, J. Phys.: Condens. Matter 30, 223001 (2018).
  64. I. Ullah, D. Adhikari, H. Khan, M. S. Anwar, S. Ahmad, and X. Bai, Mobile robot localization: Current challenges and future prospective, Comput. Sci. Rev. 53, 100651 (2024).
  65. R. Visvanathan, S. M. Mamduh, K. Kamarudin, A. S. A. Yeon, A. Zakaria, A. Y. M. Shakaff, L. M. Kamarudin, and F. S. A. Saad, Mobile robot localization system using multiple ceiling mounted cameras, Proceedings of the 2015 IEEE SENSORS (IEEE, Piscataway, NJ, 2015), pp. 1–4.
  66. E. P. Ipiña and B. A. Camley, Collective gradient sensing with limited positional information, Phys. Rev. E 105, 044410 (2022).
  67. A. Hopkins and B. A. Camley, Leader cells in collective chemotaxis: Optimality and trade-offs, Phys. Rev. E 100, 032417 (2019).
  68. L. J. S. Allen, An Introduction to Stochastic Processes with Applications to Biology , 2nd ed. (Chapman & Hall/CRC, Boca Raton, FL, 2011).
  69. A. K. Bali and A. Kumar, Object counting from images using deep learning technique, in Innovative Computing and Communications, edited by A. E. Hassanien, S. Anand, A. Jaiswal, and P. Kumar (Springer Nature, Singapore, 2025), pp. 223–237.
  70. N. Masuda, M. A. Porter, and R. Lambiotte, Random walks and diffusion on networks, Phys. Rep. 716–717, 1 (2017).

Outline

Information

Phys. Rev. Research 7, 043167– Published 13 November, 2025
  • Received 3 July 2025
  • Accepted 12 October 2025
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation