Abstract
The remarkable cohesion and coordination of moving animal groups and their collective responsiveness to threats are often attributed to scale-free correlations, where behavioral changes in one animal influence others in the group, regardless of the distance between them. But are these features independent of group size? Here, we investigate group cohesiveness and collective responsiveness in computational models of massive schools of fish of up to 50,000 individuals. We show that as the number of swimmers increases, flow interactions destabilize the school, creating clusters that constantly fragment, disperse, and regroup, much like in natural animal groups. Importantly, while spatial correlations in cohesive and polarized clusters are indeed scale free, fragmentation events are preceded by a decrease in correlation length, weakening the group’s collective responsiveness and leaving it more vulnerable to predation. We further show that information about directional changes propagates linearly in time among group members, thanks to the non-reciprocal nature of visual interactions between individuals. Merging events speed up this information transfer, while fragmentation slows it down. Our findings suggest that flow interactions may have played an important role in group size regulation, behavioral adaptations, and dispersion in living animal groups.
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Data availability
The sample data generated in this study have been deposited in GitHub under accession code https://github.com/ekanso/schooling_extreme. Source data are provided with this paper.
Code availability
Code is openly available on GitHub at https://github.com/ekanso/schooling_extreme.
References
Cavagna, A. et al. Scale-free correlations in starling flocks. Proc. Natl. Acad. Sci. USA 107, 11865–11870 (2010).
Bialek, W. et al. Statistical mechanics for natural flocks of birds. Proc. Natl. Acad. Sci. USA 109, 4786–4791 (2012).
Mora, T. et al. Local equilibrium in bird flocks. Nat. Phys. 12, 1153–1157 (2016).
Couzin, I. D., Krause, J., James, R., Ruxton, G. D. & Franks, N. R. Collective memory and spatial sorting in animal groups. J. Theor. Biol. 218, 1–11 (2002).
Couzin, I. D., Krause, J., Franks, N. R. & Levin, S. A. Effective leadership and decision-making in animal groups on the move. Nature 433, 513–516 (2005).
Gautrais, J. et al. Deciphering interactions in moving animal groups. PLoS Comput. Biol. 8, 1–11 (2012).
Garnier, S., Gautrais, J. & Theraulaz, G. The biological principles of swarm intelligence. Swarm Intell. 1, 3–31 (2007).
Cavagna, A. et al. Dynamic scaling in natural swarms. Nat. Phys. 13, 914–918 (2017).
Sayin, S. et al. The behavioral mechanisms governing collective motion in swarming locusts. Science 387, 995–1000 (2025).
Vicsek, T., Czirók, A., Ben-Jacob, E., Cohen, I. & Shochet, O. Novel type of phase transition in a system of self-driven particles. Phys. Rev. Lett. 75, 1226 (1995).
Toner, J. & Tu, Y. Long-range order in a two-dimensional dynamical xy model: how birds fly together. Phys. Rev. Lett. 75, 4326 (1995).
Toner, J. & Tu, Y. Flocks, herds, and schools: a quantitative theory of flocking. Phys. Rev. E 58, 4828 (1998).
Couzin, I. D., Krause, J. et al. Self-organization and collective behavior in vertebrates. Adv. Study Behav. 32, 10–1016 (2003).
Aranson, I. S. & Tsimring, L. S. Patterns and collective behavior in granular media: theoretical concepts. Rev. Mod. Phys. 78, 641–692 (2006).
Calovi, D. S. et al. Swarming, schooling, milling: phase diagram of a data-driven fish school model. N. J. Phys. 16, 015026 (2014).
Filella, A., Nadal, F., Sire, C., Kanso, E. & Eloy, C. Model of collective fish behavior with hydrodynamic interactions. Phys. Rev. Lett. 120, 198101 (2018).
Huang, C., Ling, F. & Kanso, E. Collective phase transitions in confined fish schools. Proc. Natl. Acad. Sci. USA 121, e2406293121 (2024).
Heydari, S., Hang, H. & Kanso, E. Mapping spatial patterns to energetic benefits in groups of flow-coupled swimmers. Elife 13, RP96129 (2024).
Peterson, A. N., Swanson, N. & McHenry, M. J. Fish communicate with water flow to enhance a school’s social network. J. Exp. Biol. 227, jeb247507 (2024).
Rieucau, G., Fernö, A., Ioannou, C. C. & Handegard, N. O. Towards of a firmer explanation of large shoal formation, maintenance and collective reactions in marine fish. Rev. Fish. Biol. Fish. 25, 21–37 (2015).
Calovi, D. S. et al. Disentangling and modeling interactions in fish with burst-and-coast swimming reveal distinct alignment and attraction behaviors. PLoS Comput. Biol. 14, e1005933 (2018).
Anderson, P. W. More is different: broken symmetry and the nature of the hierarchical structure of science. Science 177, 393–396 (1972).
Strogatz, S. et al. Fifty years of ‘more is different’. Nat. Rev. Phys. 4, 508–510 (2022).
Partridge, B. L. Internal dynamics and the interrelations of fish in schools. J. Comp. Physiol. 144, 313–325 (1981).
Niwa, H.-S. School size statistics of fish. J. Theor. Biol. 195, 351–361 (1998).
Rieucau, G., De Robertis, A., Boswell, K. & Handegard, N. School density affects the strength of collective avoidance responses in wild-caught Atlantic herring clupea harengus: a simulated predator encounter experiment. J. Fish. Biol. 85, 1650–1664 (2014).
Attanasi, A. et al. Information transfer and behavioural inertia in starling flocks. Nat. Phys. 10, 691–696 (2014).
Nagy, M., Ákos, Z., Biro, D. & Vicsek, T. Hierarchical group dynamics in pigeon flocks. Nature 464, 890–893 (2010).
Ioannou, C. C., Couzin, I. D., James, R., Croft, D. P. & Krause, J. Social organisation and information transfer in schooling fish. Fish. Cogn. Behav. 2, 217–239 (2011).
Handegard, N. O. et al. The dynamics of coordinated group hunting and collective information transfer among schooling prey. Curr. Biol. 22, 1213–1217 (2012).
Cheng Hou Tsang, A. & Kanso, E. Flagella-induced transitions in the collective behavior of confined microswimmers. Phys. Rev. E 90, 021001 (2014).
Tsang, A. C. H. & Kanso, E. Density shock waves in confined microswimmers. Phys. Rev. Lett. 116, 048101 (2016).
Tsang, A. C. H., Shelley, M. J. & Kanso, E. Activity-induced instability of phonons in 1d microfluidic crystals. Soft Matter 14, 945–950 (2018).
Tsang, A. C. H. & Kanso, E. Dipole interactions in doubly periodic domains. J. Nonlinear Sci. 23, 971–991 (2013).
Zhu, X., He, G. & Zhang, X. Flow-mediated interactions between two self-propelled flapping filaments in tandem configuration. Phys. Rev. Lett. 113, 238105 (2014).
Newbolt, J. W. et al. Flow interactions lead to self-organized flight formations disrupted by self-amplifying waves. Nat. Commun. 15, 3462 (2024).
Engelmann, J., Hanke, W., Mogdans, J. & Bleckmann, H. Hydrodynamic stimuli and the fish lateral line. Nature 408, 51–52 (2000).
Ristroph, L., Liao, J. C. & Zhang, J. Lateral line layout correlates with the differential hydrodynamic pressure on swimming fish. Phys. Rev. Lett. 114, 018102 (2015).
Colvert, B. & Kanso, E. Fishlike rheotaxis. J. Fluid Mech. 793, 656–666 (2016).
Ashraf, I. et al. Simple phalanx pattern leads to energy saving in cohesive fish schooling. Proc. Nat. Acad. Sci. USA 114, 9599–9604 (2017).
Zhang, P., Krasner, E., Peterson, S. D. & Porfiri, M. An information-theoretic study of fish swimming in the wake of a pitching airfoil. Phys. D: Nonlinear Phenom. 396, 35–46 (2019).
Li, L. et al. Vortex phase matching as a strategy for schooling in robots and in fish. Nat. Commun. 11, 5408 (2020).
McKee, A., Soto, A. P., Chen, P. & McHenry, M. J. The sensory basis of schooling by intermittent swimming in the rummy-nose tetra (Hemigrammus rhodostomus). Proc. R. Soc. B 287, 20200568 (2020).
Zheng, Z., Tao, Y., Xiang, Y., Lei, X. & Peng, X. Body orientation change of neighbors leads to scale-free correlation in collective motion. Nat. Commun. 15, 8968 (2024).
Avni, Y., Fruchart, M., Martin, D., Seara, D. & Vitelli, V. Nonreciprocalising model. Phys. Rev. Lett., 134, 117103 (2025).
Fruchart, M., Hanai, R., Littlewood, P. B. & Vitelli, V. Non-reciprocal phase transitions. Nature 592, 363–369 (2021).
Kanso, E. & Tsang, A. C. H. Dipole models of self-propelled bodies. Fluid Dyn. Res. 46, 061407 (2014).
Wiener, N. Norbert Wiener: Collected Works, Vol. 1 (MIT Press Cambridge, Mass., 1976).
Ester, M., Kriegel, H.-P., Sander, J. & Xu, X. A Density-Based Algorithm for Discovering Clusters in Large Spatial Databases with Noise. KDD’96, 226–231 (AAAI Press, 1996).
Campello, R. J. G. B., Moulavi, D. & Sander, J. Density-based clustering based on hierarchical density estimates. In Pacific-Asia Conference on Knowledge Discovery and Data Mining https://api.semanticscholar.org/CorpusID:32384865 (2013).
McInnes, L. & Healy, J. Accelerated hierarchical density based clustering. In 2017 IEEE International Conference on Data Mining Workshops (ICDMW) (eds Gottumukkala, R. et al.), 33–42 (IEEE, 2017).
Cavagna, A., Giardina, I. & Grigera, T. S. The physics of flocking: correlation as a compass from experiments to theory. Phys. Rep. 728, 1–62 (2018).
National Academies of Sciences, E. & Medicine. Physics of Life https://nap.nationalacademies.org/catalog/26403/physics-of-life (The National Academies Press, Washington, DC, 2022).
Lecheval, V. et al. Social conformity and propagation of information in collective U-turns of fish schools. Proc. R. Soc. B: Biol. Sci. 285, 20180251 (2018).
Chen, D., Liu, X., Xu, B. & Zhang, H.-T. Intermittence and connectivity of interactions in pigeon flock flights. Sci. Rep. 7, 10452 (2017).
Rosenthal, S. B., Twomey, C. R., Hartnett, A. T., Wu, H. S. & Couzin, I. D. Revealing the hidden networks of interaction in mobile animal groups allows prediction of complex behavioral contagion. Proc. Natl. Acad. Sci. USA 112, 4690–4695 (2015).
Higashi, M. & Yamamura, N. What determines animal group size? insider-outsider conflict and its resolution. Am. Nat. 142, 553–563 (1993).
Bonabeau, E., Dagorn, L. & FREon, P. Scaling in animal group-size distributions. Proc. Natl. Acad. Sci. USA 96, 4472–4477 (1999).
Roberts, G. Why individual vigilance declines as group size increases. Anim. Behav. 51, 1077–1086 (1996).
Bowler, D. E. & Benton, T. G. Causes and consequences of animal dispersal strategies: relating individual behaviour to spatial dynamics. Biol. Rev. 80, 205–225 (2005).
Cardé, R. T. & Mafra-Neto, A. Mechanisms of flight of male moths to pheromone. In InInsect pheromone research: new directions, (pp. 275–290). Boston, MA: Springer US.
Huang, Y., Yen, J. & Kanso, E. Detection and tracking of chemical trails in bio-inspired sensory systems. Eur. J. Comput. Mech. 26, 98–114 (2017).
Hang, H., Jiao, Y., Merel, J. & Kanso, E. Flow currents support simple and versatile trail-tracking strategies. Phys. Rev. Res. https://doi.org/10.1103/qj9q-p7rg (2025).
Jiao, Y., Hang, H., Merel, J. & Kanso, E. Sensing flow gradients is necessary for learning autonomous underwater navigation. Nat. Commun. 16, 3044 (2025).
Kent, M. L. et al. Recommendations for control of pathogens and infectious diseases in fish research facilities. Comp. Biochem. Physiol. Part C: Toxicol. Pharmacol. 149, 240–248 (2009).
Mittal, R., Ni, R. & Seo, J.-H. The flow physics of COVID-19. J. Fluid Mech. 894, F2 (2020).
Pohlmann, K., Grasso, F. W. & Breithaupt, T. Tracking wakes: the nocturnal predatory strategy of piscivorous catfish. Proc. Natl. Acad. Sci. USA 98, 7371–7374 (2001).
Berdahl, A., Torney, C. J., Ioannou, C. C., Faria, J. J. & Couzin, I. D. Emergent sensing of complex environments by mobile animal groups. Science 339, 574–576 (2013).
Basil, J. & Atema, J. Lobster orientation in turbulent odor plumes: simultaneous measurement of tracking behavior and temporal odor patterns. Biol. Bull. 187, 272–273 (1994).
Weissburg, M. J. & Zimmer-Faust, R. K. Odor plumes and how blue crabs use them in finding prey. J. Exp. Biol. 197, 349–375 (1994).
Radakov, D. V. Schooling in the Ecology of Fish (J. Wiley, New York, 1973).
Gerlotto, F., Bertrand, S., Bez, N. & Gutierrez, M. Waves of agitation inside anchovy schools observed with multibeam sonar: a way to transmit information in response to predation. ICES J. Mar. Sci. 63, 1405–1417 (2006).
Jiao, Y., Colvert, B., Man, Y., McHenry, M. J. & Kanso, E. Evaluating evasion strategies in zebrafish larvae. Proc. Natl. Acad. Sci. USA 120, e2218909120 (2023).
Humphries, D. & Driver, P. Protean defence by prey animals. Oecologia 5, 285–302 (1970).
Angelani, L. Collective predation and escape strategies. Phys. Rev. Lett. 109, 118104 (2012).
Papadopoulou, M., Hildenbrandt, H., Sankey, D. W., Portugal, S. J. & Hemelrijk, C. K. Self-organization of collective escape in pigeon flocks. PLoS Comput. Biol. 18, e1009772 (2022).
Lam, S. K., Pitrou, A. & Seibert, S. Numba: a llvm-based python jit compiler. In Proc. Second Workshop on the LLVM Compiler Infrastructure in HPC (ed. Finkel, H.), 1–6 (ACM: NewYork, NY, USA, 2015).
Greengard, L. & Rokhlin, V. A fast algorithm for particle simulations. J. Comput. Phys. 73, 325–348 (1987).
Ying, L., Biros, G. & Zorin, D. A kernel-independent adaptive fast multipole algorithm in two and three dimensions. J. Comput. Phys. 196, 591–626 (2004).
Barber, C. B., Dobkin, D. P. & Huhdanpaa, H. The quickhull algorithm for convex hulls. ACM Trans. Math. Softw. (TOMS) 22, 469–483 (1996).
Virtanen, P. et al. SciPy 1.0: fundamental algorithms for scientific computing in Python. Nat. Methods 17, 261–272 (2020).
Kloeden, P. E., Platen, E., Kloeden, P. E. & Platen, E. Stochastic Differential Equations (Springer, 1992).
Miyahara, H., Yoneki, H., Mizohata, T. & Roychowdhury, V. Vicsek model meets DBSCAN: Cluster phases in the Vicsek model. J. Phys. Soc. Japan 94, 084002 (2025).
Dempster, A. P., Laird, N. M. & Rubin, D. B. Maximum likelihood from incomplete data via the em algorithm. J. R. Stat. Soc.: Ser. B (Methodol.) 39, 1–22 (1977).
MacQueen, J. Some methods for classification and analysis of multivariate observations. In Proc. 5-th Berkeley Symposium on Mathematical Statistics and Probability/University of California Press (eds Lucien, M. et al.) (1967).
Duda, R. O. & Hart, P. E. Pattern classification and scene analysis. In A Wiley-Interscience Publication https://api.semanticscholar.org/CorpusID:12946615 (1974).
Schubert, E., Sander, J., Ester, M., Kriegel, H. P. & Xu, X. Dbscan revisited, revisited: why and how you should (still) use DBSCAN. ACM Trans. Database Syst. 42, 1–21 (2017).
Kang, C.-P. et al. An automatic method to calculate heart rate from zebrafish larval cardiac videos. BMC Bioinforma. 19, 1–10 (2018).
Pedregosa, F. et al. Scikit-learn: machine learning in Python. J. Mach. Learn. Res. 12, 2825–2830 (2011).
Attanasi, A. et al. Emergence of collective changes in travel direction of starling flocks from individual birds’ fluctuations. J. R. Soc. Interface 12, 20150319 (2015).
Cavagna, A. & Giardina, I. Bird flocks as condensed matter. Annu. Rev. Condens. Matter Phys. 5, 183–207 (2014).
Student. The probable error of a mean. Biometrika 6 1–25 (1908).
Fisher, R. A. On the interpretation of χ 2 from contingency tables, and the calculation of p. J. R. Stat. Soc. 85, 87–94 (1922).
Acknowledgements
Funding support provided by the NSF grants RAISE IOS-2034043 and CBET-2100209, ONR grants N00014-22-1-2655 and N00014-19-1-2035, and NIH grant R01-HL153622 (all to E.K.). E.K. is grateful to Andrea Cavagna for a helpful discussion. The work of E.K. was supported in part by grant NSF PHY-2309135 to the Kavli Institute for Theoretical Physics (KITP) and by Princeton University through the William R. Kenan, Jr., Visiting Professorship.
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E.K. conceptualized and supervised the research; H.H. and C.H. wrote code with input from A.B.; H.H. performed simulations and collected data; H.H. and E.K. analyzed the data and prepared figures; E.K. wrote the manuscript and all authors edited and approved it.
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Hang, H., Huang, C., Barnett, A. et al. Self-reorganization and information transfer in large-scale models of fish schools. Nat Commun (2026). https://doi.org/10.1038/s41467-026-70569-y
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DOI: https://doi.org/10.1038/s41467-026-70569-y
