{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,13]],"date-time":"2026-03-13T11:42:19Z","timestamp":1773402139216,"version":"3.50.1"},"reference-count":38,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2024,12,24]],"date-time":"2024-12-24T00:00:00Z","timestamp":1734998400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0"},{"start":{"date-parts":[[2024,12,24]],"date-time":"2024-12-24T00:00:00Z","timestamp":1734998400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0"}],"funder":[{"DOI":"10.13039\/501100001809","name":"Key-Area Research and Development Program of Guangdong Province","doi-asserted-by":"publisher","award":["2024B1111060004"],"award-info":[{"award-number":["2024B1111060004"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Complex Intell. Syst."],"published-print":{"date-parts":[[2025,1]]},"DOI":"10.1007\/s40747-024-01665-z","type":"journal-article","created":{"date-parts":[[2024,12,24]],"date-time":"2024-12-24T07:12:00Z","timestamp":1735024320000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Higher-order topology for collective motions"],"prefix":"10.1007","volume":"11","author":[{"given":"Zijie","family":"Sun","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0587-6752","authenticated-orcid":false,"given":"Tianjiang","family":"Hu","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,12,24]]},"reference":[{"issue":"3","key":"1665_CR1","first-page":"71","volume":"517","author":"V Tam\u00e1s","year":"2012","unstructured":"Tam\u00e1s V, Anna Z (2012) Collective motion. Phys Rep 517(3):71\u2013140","journal-title":"Phys Rep"},{"issue":"1","key":"1665_CR2","doi-asserted-by":"publisher","first-page":"5174","DOI":"10.1038\/s41467-019-13281-4","volume":"10","author":"H Ling","year":"2019","unstructured":"Ling H, Mclvor GE, Westley J, van der Vaart K, Vaughan RT, Thornton A, Ouellette NT (2019) Behavioural plasticity and the transition to order in jackdaw flocks. Nat Commun 10(1):5174","journal-title":"Nat Commun"},{"issue":"6","key":"1665_CR3","doi-asserted-by":"publisher","first-page":"943","DOI":"10.1038\/s41559-019-0891-5","volume":"3","author":"H Ling","year":"2019","unstructured":"Ling H, Mclvor GE, van der Vaart K, Vaughan RT, Thornton A, Ouellette NT (2019) Costs and benefits of social relationships in the collective motion of bird flocks. Nat Ecol Evol 3(6):943\u2013948","journal-title":"Nat Ecol Evol"},{"issue":"26","key":"1665_CR4","doi-asserted-by":"publisher","first-page":"11865","DOI":"10.1073\/pnas.1005766107","volume":"107","author":"A Cavagna","year":"2010","unstructured":"Cavagna A, Cimarelli A, Giardina I, Parisi G, Santagati R, Stefanini F, Viale M (2010) Scale-free correlations in starling flocks. Proc Natl Acad Sci USA 107(26):11865\u201311870","journal-title":"Proc Natl Acad Sci USA"},{"issue":"1","key":"1665_CR5","doi-asserted-by":"publisher","first-page":"5408","DOI":"10.1038\/s41467-020-19086-0","volume":"11","author":"L Li","year":"2020","unstructured":"Li L, Nagy M, Graving JM, Bak-Coleman J, Xie G, Couzin ID (2020) Vortex phase matching as a strategy for schooling in robots and in fish. Nat Commun 11(1):5408","journal-title":"Nat Commun"},{"issue":"41","key":"1665_CR6","doi-asserted-by":"publisher","first-page":"20556","DOI":"10.1073\/pnas.1905585116","volume":"116","author":"MMG Sosna","year":"2019","unstructured":"Sosna MMG, Twomey CR, Bak-Coleman J, Poel W, Daniels BC, Romanczuk P, Couzin ID (2019) Individual and collective encoding of risk in animal groups. Proc Natl Acad Sci USA 116(41):20556\u201320561","journal-title":"Proc Natl Acad Sci USA"},{"issue":"1","key":"1665_CR7","doi-asserted-by":"publisher","first-page":"4779","DOI":"10.1038\/s41467-024-49151-x","volume":"15","author":"Y Xiao","year":"2024","unstructured":"Xiao Y, Lei X, Zheng Z, Xiang Y, Liu YY, Peng X (2024) Perception of motion salience shapes the emergence of collective motions. Nat Commun 15(1):4779","journal-title":"Nat Commun"},{"issue":"4","key":"1665_CR8","doi-asserted-by":"publisher","first-page":"2729","DOI":"10.1109\/TRO.2024.3392078","volume":"40","author":"S Zhang","year":"2024","unstructured":"Zhang S, Lei XK, Peng XG, Pan J (2024) Heterogeneous targets trapping with swarm robots by using adaptive density-based interaction. IEEE Trans Rob 40(4):2729\u20132748","journal-title":"IEEE Trans Rob"},{"issue":"5","key":"1665_CR9","doi-asserted-by":"publisher","first-page":"5135","DOI":"10.1007\/s40747-023-01014-6","volume":"9","author":"XK Lei","year":"2023","unstructured":"Lei XK, Zhang S, Xiang YL, Duan MY (2023) Self-organized multi-target trapping of swarm robots with density-based interaction. Complex Intell Sys 9(5):5135\u20135155","journal-title":"Complex Intell Sys"},{"issue":"2","key":"1665_CR10","doi-asserted-by":"publisher","first-page":"2065","DOI":"10.1007\/s40747-022-00900-9","volume":"9","author":"FK Wang","year":"2023","unstructured":"Wang FK, Huang JL, Low KH, Hu TJ (2023) AGDS: adaptive goal-directed strategy for swarm drones flying through unknown environments. Complex Intell Syst 9(2):2065\u20132080","journal-title":"Complex Intell Syst"},{"key":"1665_CR11","doi-asserted-by":"publisher","DOI":"10.1103\/RevModPhys.90.031001","volume":"90","author":"MA Mu\u00f1oz","year":"2017","unstructured":"Mu\u00f1oz MA (2017) Colloquium: Criticality and dynamical scaling in living systems. Rev Mod Phys 90:031001","journal-title":"Rev Mod Phys"},{"key":"1665_CR12","doi-asserted-by":"publisher","DOI":"10.1016\/j.automatica.2023.111046","volume":"153","author":"E Tegling","year":"2023","unstructured":"Tegling E, Bamieh B, Sandberg H (2023) Scale fragilities in localized consensus dynamics. Automatica 153:111046","journal-title":"Automatica"},{"issue":"167","key":"1665_CR13","doi-asserted-by":"publisher","first-page":"20190853","DOI":"10.1098\/rsif.2019.0853","volume":"17","author":"B Bal\u00e1zs","year":"2020","unstructured":"Bal\u00e1zs B, V\u00e1s\u00e1rhelyi G, Vicsek T (2020) Adaptive leadership overcomes persistence-responsivity trade-off in flocking. J R Soc Interface 17(167):20190853","journal-title":"J R Soc Interface"},{"key":"1665_CR14","first-page":"1","volume":"716\u2013717","author":"N Masuda","year":"2016","unstructured":"Masuda N, Porter MA, Lambiotte R (2016) Random walks and diffusion on networks. Phys Rep 716\u2013717:1\u201358","journal-title":"Phys Rep"},{"key":"1665_CR15","volume-title":"Order, disorder and criticality","author":"P Romanczuk","year":"2022","unstructured":"Romanczuk P, Daniels BC (2022) Phase transitions and criticality in the collective behavior of animals\u2014self-organization and biological function. In: Yurij H (ed) Order, disorder and criticality. World Scientific Publishing, Singapore"},{"issue":"13","key":"1665_CR16","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.118.138003","volume":"118","author":"A Cavagna","year":"2017","unstructured":"Cavagna A, Giardina I, Jelic A, Melillo S, Parisi L, Silvestri E, Viale M (2017) Nonsymmetric interactions trigger collective swings in globally ordered systems. Phys Rev Lett 118(13):138003","journal-title":"Phys Rev Lett"},{"key":"1665_CR17","doi-asserted-by":"publisher","first-page":"353","DOI":"10.1137\/18M1201019","volume":"62","author":"MT Schaub","year":"2018","unstructured":"Schaub MT, Benson AR, Horn P, Lippner G, Jadbabaie A (2018) Random walks on simplicial complexes and the normalized Hodge Laplacian. SIAM Rev 62:353\u2013391","journal-title":"SIAM Rev"},{"key":"1665_CR18","doi-asserted-by":"publisher","first-page":"195","DOI":"10.1038\/s42005-021-00695-0","volume":"4","author":"Y Zhang","year":"2020","unstructured":"Zhang Y, Latora V, Motter AE (2020) Unified treatment of synchronization patterns in generalized networks with higher-order, multilayer, and temporal interactions. Commun Phys 4:195","journal-title":"Commun Phys"},{"issue":"10","key":"1665_CR19","doi-asserted-by":"publisher","DOI":"10.1063\/5.0020034","volume":"30","author":"NW Landry","year":"2020","unstructured":"Landry NW, Restrepo JG (2020) The effect of heterogeneity on hypergraph contagion models. Chaos 30(10):103117","journal-title":"Chaos"},{"key":"1665_CR20","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.physrep.2020.05.004","volume":"874","author":"B Federico","year":"2020","unstructured":"Federico B, Giulia C, Iacopo I, Vito L, Maxime L, Alice P, Young JG, Giovanni P (2020) Networks beyond pairwise interactions: structure and dynamics. Phys Rep 874:1\u201392","journal-title":"Phys Rep"},{"issue":"4","key":"1665_CR21","doi-asserted-by":"publisher","first-page":"1232","DOI":"10.1073\/pnas.0711437105","volume":"105","author":"M Ballerini","year":"2008","unstructured":"Ballerini M, Cabibbo N, Candelier R, Cavagna A, Cisbani E, Giardina I, Lecomte V, Orlandi A, Parisi G, Procaccini A, Viale M, Zdravkovic V (2008) Interaction ruling animal collective behavior depends on topological rather than metric distance: evidence from a field study. Proc Natl Acad Sci USA 105(4):1232\u20131237","journal-title":"Proc Natl Acad Sci USA"},{"key":"1665_CR22","doi-asserted-by":"publisher","first-page":"715","DOI":"10.1098\/rsfs.2012.0026","volume":"2","author":"M Camperi","year":"2012","unstructured":"Camperi M, Cavagna A, Giardina I, Parisi G, Silvestri E (2012) Spatially balanced topological interaction grants optimal cohesion in flocking models. Interface Focus 2:715\u2013725","journal-title":"Interface Focus"},{"issue":"7290","key":"1665_CR23","doi-asserted-by":"publisher","first-page":"890","DOI":"10.1038\/nature08891","volume":"464","author":"M Nagy","year":"2010","unstructured":"Nagy M, Akos Z, Biro D, Vicsek T (2010) Hierarchical group dynamics in pigeon flocks. Nature 464(7290):890\u2013893","journal-title":"Nature"},{"issue":"1518","key":"1665_CR24","doi-asserted-by":"publisher","first-page":"781","DOI":"10.1098\/rstb.2008.0233","volume":"364","author":"JR Dyer","year":"2009","unstructured":"Dyer JR, Johansson A, Helbing D, Couzin ID, Krause J (2009) Leadership, consensus decision making and collective behaviour in humans. Philos Trans R Soc Lond B Biol Sci 364(1518):781\u2013789","journal-title":"Philos Trans R Soc Lond B Biol Sci"},{"issue":"6","key":"1665_CR25","doi-asserted-by":"publisher","first-page":"1226","DOI":"10.1103\/PhysRevLett.75.1226","volume":"75","author":"T Vicsek","year":"1995","unstructured":"Vicsek T, Czir\u00f3k A, Ben-Jacob E, Cohen II, Shochet O (1995) Novel type of phase transition in a system of self-driven particles. Phys Rev Lett 75(6):1226\u20131229","journal-title":"Phys Rev Lett"},{"key":"1665_CR26","doi-asserted-by":"publisher","first-page":"601","DOI":"10.1007\/s10955-014-1119-3","volume":"158","author":"A Cavagna","year":"2014","unstructured":"Cavagna A, Castello LD, Giardina I, Grigera TS, Jelic A, Melillo S, Mora T, Parisi L, Silvestri E, Viale M, Walczak AM (2014) Flocking and turning: a new model for self-organized collective motion. J Stat Phys 158:601\u2013627","journal-title":"J Stat Phys"},{"issue":"13","key":"1665_CR27","doi-asserted-by":"publisher","first-page":"5263","DOI":"10.1073\/pnas.1217513110","volume":"110","author":"N Miller","year":"2013","unstructured":"Miller N, Garnier S, Hartnett AT, Couzin ID (2013) Both information and social cohesion determine collective decisions in animal groups. Proc Natl Acad Sci USA 110(13):5263\u20135268","journal-title":"Proc Natl Acad Sci USA"},{"key":"1665_CR28","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-84800-015-5","volume-title":"Distributed consensus in multi-vehicle cooperative control","author":"W Ren","year":"2008","unstructured":"Ren W, Beard RW (2008) Distributed consensus in multi-vehicle cooperative control. Springer, London"},{"key":"1665_CR29","doi-asserted-by":"publisher","first-page":"165","DOI":"10.1080\/00150199108014063","volume":"113","author":"F Gouda","year":"1991","unstructured":"Gouda F, Skarp K, Lagerwall S (1991) Dielectric studies of the soft mode and Goldstone mode in ferroelectric liquid crystals. Ferroelectrics 113:165\u2013206","journal-title":"Ferroelectrics"},{"key":"1665_CR30","doi-asserted-by":"publisher","first-page":"943","DOI":"10.1038\/nphys4187","volume":"13","author":"S Hoinka","year":"2017","unstructured":"Hoinka S, Dyke P, Lingham M, Kinnunen JJ, Bruun GM, Vale CJ (2017) Goldstone mode and pair-breaking excitations in atomic Fermi superfluids. Nat Phys 13:943\u2013946","journal-title":"Nat Phys"},{"issue":"18","key":"1665_CR31","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.83.184206","volume":"83","author":"NC Murphy","year":"2011","unstructured":"Murphy NC, Wortis R, Atkinson WA (2011) Generalized inverse participation ratio as a possible measure of localization for interacting systems. Phys Rev B 83(18):184206","journal-title":"Phys Rev B"},{"key":"1665_CR32","doi-asserted-by":"publisher","DOI":"10.1016\/j.chaos.2024.115302","volume":"186","author":"YJ Zhou","year":"2024","unstructured":"Zhou YJ, Zheng ZC, Wang T, Peng XG (2024) Swarm dynamics of delayed self-propelled particles with non-reciprocal interactions. Chaos Solitons Fract 186:115302","journal-title":"Chaos Solitons Fract"},{"key":"1665_CR33","doi-asserted-by":"publisher","DOI":"10.1016\/j.chaos.2024.114596","volume":"180","author":"YJ Zhou","year":"2024","unstructured":"Zhou YJ, Wang TH, Lei XK, Peng XG (2024) Collective behavior of self-propelled particles with heterogeneity in both dynamics and delays Chaos. Solitons Fractals 180:114596","journal-title":"Solitons Fractals"},{"issue":"1","key":"1665_CR34","doi-asserted-by":"publisher","first-page":"1605","DOI":"10.1038\/s41467-023-37190-9","volume":"14","author":"YZ Zhang","year":"2023","unstructured":"Zhang YZ, Lucas M, Battiston F (2023) Higher-order interactions shape collective dynamics differently in hypergraphs and simplicial complexes. Nat Commun 14(1):1605","journal-title":"Nat Commun"},{"issue":"1","key":"1665_CR35","doi-asserted-by":"publisher","first-page":"129","DOI":"10.1038\/s42005-024-01614-9","volume":"7","author":"JY Li","year":"2024","unstructured":"Li JY, Wu XQ, L\u00fc JH, Lei L (2024) Enhancing predictive accuracy in social contagion dynamics via directed hypergraph structures. Commun Phys 7(1):129","journal-title":"Commun Phys"},{"issue":"3","key":"1665_CR36","doi-asserted-by":"publisher","first-page":"0062","DOI":"10.1038\/s41559-016-0062","volume":"1","author":"MM Mayfield","year":"2017","unstructured":"Mayfield MM, Stouffer DB (2017) Higher-order interactions capture unexplained complexity in diverse communities. Nat Ecol Evol 1(3):0062","journal-title":"Nat Ecol Evol"},{"key":"1665_CR37","doi-asserted-by":"publisher","first-page":"117","DOI":"10.1007\/s00265-010-1111-0","volume":"65","author":"NW Bode","year":"2011","unstructured":"Bode NW, Wood AJ, Franks DW (2011) Social networks and models for collective motion in animals. Behav Ecol Sociobiol 65:117\u2013130","journal-title":"Behav Ecol Sociobiol"},{"issue":"3","key":"1665_CR38","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevResearch.2.033410","volume":"2","author":"M Lucas","year":"2020","unstructured":"Lucas M, Cencetti G, Battiston F (2020) Multiorder Laplacian for synchronization in higher-order networks. Phys Rev Res 2(3):033410","journal-title":"Phys Rev Res"}],"container-title":["Complex &amp; Intelligent Systems"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40747-024-01665-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40747-024-01665-z\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40747-024-01665-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,1,30]],"date-time":"2025-01-30T20:18:20Z","timestamp":1738268300000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40747-024-01665-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,12,24]]},"references-count":38,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2025,1]]}},"alternative-id":["1665"],"URL":"https:\/\/doi.org\/10.1007\/s40747-024-01665-z","relation":{},"ISSN":["2199-4536","2198-6053"],"issn-type":[{"value":"2199-4536","type":"print"},{"value":"2198-6053","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,12,24]]},"assertion":[{"value":"27 June 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"2 October 2024","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 December 2024","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"All authors state that there is no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"91"}}