Persistent homology as a universal metric for collective motion

Persistent homology as a universal metric for collective motion | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Article Persistent homology as a universal metric for collective motion Guillermo Legarda Herranz, Emanuele Garone, Mauro Birattari This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8672197/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract From flocking birds to human crowds, collective motion patterns arise from the coordinated motion of individuals in a group. These patterns are notoriously hard to infer from the actions of each individual, yet are clearly distinguishable even by an untrained observer. Established formal approaches towards analysing collective motion patterns rely on the identification of relevant variables that are specific to the system under study. As a result, they fail to address the generality of collective motion as an emergent behaviour across systems. In this paper, we show how topological data analysis allows abstracting collective motion patterns and defining meaningful distances between them. The proposed method uses persistent homology to represent the topological features of velocity fields as sets of persistence diagrams. We show how a distance between these sets can discriminate the various behaviours exhibited by a system, and we demonstrate its generality by applying it to a fish school, a pedestrian crowd, and a robot swarm. We validate our results against those obtained through system-specific abstractions, and illustrate how to use topological representations to explain discrepancies. In short, we show how to quantify and interpret differences between collective motion patterns for any system for which a velocity field can be constructed. This will allow researchers in complex systems to adopt a common framework for collective motion analysis, enabling direct comparison of emergent behaviours across disciplines. Physical sciences/Mathematics and computing/Applied mathematics Physical sciences/Mathematics and computing/Computational science Full Text Additional Declarations There is NO Competing Interest. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-8672197","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":584224327,"identity":"ac3d7b41-81d3-4bc1-932a-566137e75749","order_by":0,"name":"Guillermo Legarda Herranz","email":"data:image/png;base64,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","orcid":"https://orcid.org/0000-0003-4948-9049","institution":"IRIDIA - 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These patterns are notoriously hard to infer from the actions of each individual, yet are clearly distinguishable even by an untrained observer. Established formal approaches towards analysing collective motion patterns rely on the identification of relevant variables that are specific to the system under study. As a result, they fail to address the generality of collective motion as an emergent behaviour across systems. In this paper, we show how topological data analysis allows abstracting collective motion patterns and defining meaningful distances between them. The proposed method uses persistent homology to represent the topological features of velocity fields as sets of persistence diagrams. We show how a distance between these sets can discriminate the various behaviours exhibited by a system, and we demonstrate its generality by applying it to a fish school, a pedestrian crowd, and a robot swarm. 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