{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,3]],"date-time":"2026-04-03T08:15:28Z","timestamp":1775204128787,"version":"3.50.1"},"update-to":[{"DOI":"10.1371\/journal.pcbi.1007811","type":"new_version","label":"New version","source":"publisher","updated":{"date-parts":[[2021,2,25]],"date-time":"2021-02-25T00:00:00Z","timestamp":1614211200000}}],"reference-count":37,"publisher":"Public Library of Science (PLoS)","issue":"2","license":[{"start":{"date-parts":[[2021,2,12]],"date-time":"2021-02-12T00:00:00Z","timestamp":1613088000000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"DARPA Young Faculty Award","award":["N66001-17-1-4038"],"award-info":[{"award-number":["N66001-17-1-4038"]}]},{"name":"DARPA Young Faculty Award","award":["N66001-17-1-4038"],"award-info":[{"award-number":["N66001-17-1-4038"]}]},{"name":"DARPA Young Faculty Award","award":["N66001-17-1-4038"],"award-info":[{"award-number":["N66001-17-1-4038"]}]},{"DOI":"10.13039\/100000192","name":"National Oceanic and Atmospheric Administration","doi-asserted-by":"crossref","award":["NOAA-AWD100582"],"award-info":[{"award-number":["NOAA-AWD100582"]}],"id":[{"id":"10.13039\/100000192","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/100000893","name":"Simons Foundation","doi-asserted-by":"publisher","award":["395890"],"award-info":[{"award-number":["395890"]}],"id":[{"id":"10.13039\/100000893","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100008982","name":"National Science Foundation","doi-asserted-by":"publisher","award":["OCE-184857"],"award-info":[{"award-number":["OCE-184857"]}],"id":[{"id":"10.13039\/501100008982","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["www.ploscompbiol.org"],"crossmark-restriction":false},"short-container-title":["PLoS Comput Biol"],"abstract":"<jats:p>\n                    Collective behavior is an emergent property of numerous complex systems, from financial markets to cancer cells to predator-prey ecological systems. Characterizing modes of collective behavior is often done through human observation, training generative models, or other supervised learning techniques. Each of these cases requires knowledge of and a method for characterizing the macro-state(s) of the system. This presents a challenge for studying novel systems where there may be little prior knowledge. Here, we present a new unsupervised method of detecting emergent behavior in complex systems, and discerning between distinct collective behaviors. We require only metrics,\n                    <jats:italic>d<\/jats:italic>\n                    <jats:sup>(1)<\/jats:sup>\n                    ,\n                    <jats:italic>d<\/jats:italic>\n                    <jats:sup>(2)<\/jats:sup>\n                    , defined on the set of agents,\n                    <jats:italic>X<\/jats:italic>\n                    , which measure agents\u2019 nearness in variables of interest. We apply the method of diffusion maps to the systems (\n                    <jats:italic>X<\/jats:italic>\n                    ,\n                    <jats:italic>d<\/jats:italic>\n                    <jats:sup>\n                      (\n                      <jats:italic>i<\/jats:italic>\n                      )\n                    <\/jats:sup>\n                    ) to recover efficient embeddings of their interaction networks. Comparing these geometries, we formulate a measure of similarity between two networks, called the map alignment statistic (MAS). A large MAS is evidence that the two networks are codetermined in some fashion, indicating an emergent relationship between the metrics\n                    <jats:italic>d<\/jats:italic>\n                    <jats:sup>(1)<\/jats:sup>\n                    and\n                    <jats:italic>d<\/jats:italic>\n                    <jats:sup>(2)<\/jats:sup>\n                    . Additionally, the form of the macro-scale organization is encoded in the covariances among the two sets of diffusion map components. Using these covariances we discern between different modes of collective behavior in a data-driven, unsupervised manner. This method is demonstrated on a synthetic flocking model as well as empirical fish schooling data. We show that our state classification subdivides the known behaviors of the school in a meaningful manner, leading to a finer description of the system\u2019s behavior.\n                  <\/jats:p>","DOI":"10.1371\/journal.pcbi.1007811","type":"journal-article","created":{"date-parts":[[2021,2,12]],"date-time":"2021-02-12T16:54:18Z","timestamp":1613148858000},"page":"e1007811","update-policy":"https:\/\/doi.org\/10.1371\/journal.pcbi.corrections_policy","source":"Crossref","is-referenced-by-count":6,"title":["Unsupervised manifold learning of collective 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