{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:17:41Z","timestamp":1760242661000,"version":"build-2065373602"},"reference-count":13,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2016,1,14]],"date-time":"2016-01-14T00:00:00Z","timestamp":1452729600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001711","name":"Swiss National Science Foundation","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100001711","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We consider the dynamics of swarms of scalar Brownian agents subject to local imitation mechanisms implemented using mutual rank-based interactions. For appropriate values of the underlying control parameters, the swarm propagates tightly and the distances separating successive agents are iid exponential random variables. Implicitly, the implementation of rank-based mutual interactions, requires that agents have infinite interaction ranges. Using the probabilistic size of the swarm\u2019s support, we analytically estimate the critical interaction range below that flocked swarms cannot survive. In the second part of the paper, we consider the interactions between two flocked swarms of Brownian agents with finite interaction ranges. Both swarms travel with different barycentric velocities, and agents from both swarms indifferently interact with each other. For appropriate initial configurations, both swarms eventually collide (i.e., all agents interact). Depending on the values of the control parameters, one of the following patterns emerges after collision: (i) Both swarms remain essentially flocked, or (ii) the swarms become ultimately quasi-free and recover their nominal barycentric speeds. We derive a set of analytical flocking conditions based on the generalized rank-based Brownian motion. An extensive set of numerical simulations corroborates our analytical findings.<\/jats:p>","DOI":"10.3390\/e18010027","type":"journal-article","created":{"date-parts":[[2016,1,15]],"date-time":"2016-01-15T09:26:33Z","timestamp":1452849993000},"page":"27","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Interacting Brownian Swarms: Some Analytical Results"],"prefix":"10.3390","volume":"18","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7579-9916","authenticated-orcid":false,"given":"Guillaume","family":"Sartoretti","sequence":"first","affiliation":[{"name":"STI\/IMT\/LPM, Ecole Polytechnique F\u00e9d\u00e9rale de Lausanne, Station 17 (Batiment BM), CH-1015 Lausanne, Switzerland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Max-Olivier","family":"Hongler","sequence":"additional","affiliation":[{"name":"STI\/IMT\/LPM, Ecole Polytechnique F\u00e9d\u00e9rale de Lausanne, Station 17 (Batiment BM), CH-1015 Lausanne, Switzerland"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2016,1,14]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"513","DOI":"10.1038\/nature03236","article-title":"Effective leadership and decision-making in animal groups on the move","volume":"433","author":"Couzin","year":"2005","journal-title":"Nature"},{"key":"ref_2","unstructured":"Sartoretti, G., Hongler, M.O., and Filliger, R. (2014, January 11\u201314). The Estimation Problem and Heterogenous Swarms of Autonomous Agents. Proceedings of Stochastic Modeling Techniques and Data Analysis International Conference (SMTDA 2014), Lisbon, Portugal."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"54","DOI":"10.1007\/s11424-006-0054-z","article-title":"Soft control on collective behavior of a group of autonomous agents by a shill agent","volume":"19","author":"Han","year":"2006","journal-title":"J. Syst. Sci. Complex."},{"key":"ref_4","unstructured":"Sartoretti, G.A., and Hongler, M.O. (2013, January 15\u201318). Soft Control of Swarms: Analytical Approach. Proceedings of the 5th International Conference on Agents and Artificial Intelligence (ICAART 2013), Barcelona, Spain."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Moreno-D\u00edaz, R., Pichler, F., and Quesada-Arencibia, A. (2013). Computer Aided Systems Theory - EUROCAST 2013, Springer.","DOI":"10.1007\/978-3-642-53856-8"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0167-2789(00)00094-4","article-title":"From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators","volume":"143","author":"Strogatz","year":"2000","journal-title":"Physica D"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Acebr\u00f3n, J.A., Bonilla, L.L., Vicente, C.J.P., Ritort, F., and Spigler, R. (2005). The Kuramoto model: A simple paradigm for synchronization phenomena. Rev. Mod. Phys., 77.","DOI":"10.1103\/RevModPhys.77.137"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"2179","DOI":"10.1214\/08-AAP516","article-title":"One dimensional Brownian particle systems with rank-dependent drifts","volume":"18","author":"Pal","year":"2008","journal-title":"Ann. Appl. Probab."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"609","DOI":"10.1214\/10-AAP706","article-title":"Hybrid Atlas Model","volume":"21","author":"Ichiba","year":"2011","journal-title":"Ann. Appl. Probab."},{"key":"ref_10","unstructured":"Hongler, M.O., Gallay, O., and Hashemi, F. (2015). Imitation\u2019s Impact on the Dynamics of Long-Wave Growth. Econ. Model., submited."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1706","DOI":"10.1214\/07-AAP513","article-title":"Propagation of chaos and Poincar\u00e9 inequalities for a system of particles interacting through their CDF","volume":"18","author":"Jourdain","year":"2008","journal-title":"Ann. Appl. Probab."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"303","DOI":"10.3934\/mbe.2014.11.303","article-title":"Local versus nonlocal barycentric interactions in 1D agent dynamics","volume":"11","author":"Hongler","year":"2014","journal-title":"Math. Biosci. Eng."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"2296","DOI":"10.1214\/105051605000000449","article-title":"Atlas Models of Equity Markets","volume":"15","author":"Banner","year":"2005","journal-title":"Ann. Appl. Prob."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/18\/1\/27\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T19:17:42Z","timestamp":1760210262000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/18\/1\/27"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,1,14]]},"references-count":13,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2016,1]]}},"alternative-id":["e18010027"],"URL":"https:\/\/doi.org\/10.3390\/e18010027","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2016,1,14]]}}}