{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,4]],"date-time":"2026-04-04T22:30:05Z","timestamp":1775341805381,"version":"3.50.1"},"reference-count":88,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2023,10,11]],"date-time":"2023-10-11T00:00:00Z","timestamp":1696982400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2023,10,11]],"date-time":"2023-10-11T00:00:00Z","timestamp":1696982400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["IIS-1422400"],"award-info":[{"award-number":["IIS-1422400"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["CCF-1526513"],"award-info":[{"award-number":["CCF-1526513"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1723003"],"award-info":[{"award-number":["DMS-1723003"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["CCF-1740761"],"award-info":[{"award-number":["CCF-1740761"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Discrete Comput Geom"],"published-print":{"date-parts":[[2024,6]]},"DOI":"10.1007\/s00454-023-00590-1","type":"journal-article","created":{"date-parts":[[2023,10,11]],"date-time":"2023-10-11T16:01:50Z","timestamp":1697040110000},"page":"1276-1342","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Extracting Persistent Clusters in Dynamic Data via M\u00f6bius Inversion"],"prefix":"10.1007","volume":"71","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8081-5872","authenticated-orcid":false,"given":"Woojin","family":"Kim","sequence":"first","affiliation":[],"role":[{"role":"author","vocab":"crossref"}]},{"given":"Facundo","family":"M\u00e9moli","sequence":"additional","affiliation":[],"role":[{"role":"author","vocab":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2023,10,11]]},"reference":[{"key":"590_CR1","unstructured":"Adams, H., Ghosh, D., Mask, C., Ott, W., Williams, K.: Efficient evader detection in mobile sensor networks. arXiv preprint arXiv:2101.09813 (2021)"},{"issue":"1","key":"590_CR2","doi-asserted-by":"crossref","first-page":"90","DOI":"10.1177\/0278364914548051","volume":"34","author":"H Adams","year":"2015","unstructured":"Adams, H., Carlsson, G.: Evasion paths in mobile sensor networks. Int. J. Robot. Res. 34(1), 90\u2013104 (2015)","journal-title":"Int. J. Robot. Res."},{"key":"590_CR3","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1017\/S002776300002290X","volume":"1","author":"G Azumaya","year":"1950","unstructured":"Azumaya, G., et al.: Corrections and supplementaries to my paper concerning Krull-Remak-Schmidt\u2019s theorem. Nagoya Math. J. 1, 117\u2013124 (1950)","journal-title":"Nagoya Math. J."},{"key":"590_CR4","doi-asserted-by":"crossref","unstructured":"Bauer, U., Ge, X., Wang, Y.: Measuring distance between Reeb graphs. In: Proceedings of the Thirtieth Annual Symposium on Computational Geometry, pp. 464\u2013473 (2014)","DOI":"10.1145\/2582112.2582169"},{"key":"590_CR5","doi-asserted-by":"publisher","unstructured":"Bauer, U., Munch, E., Wang, Y.: Strong Equivalence of the Interleaving and Functional Distortion Metrics for Reeb Graphs. In: Arge, L., Pach, J. (eds.) 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), vol. 34, pp. 461\u2013475. Schloss Dagstuhl\u2013Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany (2015). https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.461. http:\/\/drops.dagstuhl.de\/opus\/volltexte\/2015\/5146","DOI":"10.4230\/LIPIcs.SOCG.2015.461"},{"issue":"2","key":"590_CR6","first-page":"162","volume":"6","author":"U Bauer","year":"2015","unstructured":"Bauer, U., Lesnick, M.: Induced matchings and the algebraic stability of persistence barcodes. J. Comput. Geom. 6(2), 162\u2013191 (2015)","journal-title":"J. Comput. Geom."},{"issue":"3","key":"590_CR7","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1016\/j.comgeo.2007.10.003","volume":"41","author":"M Benkert","year":"2008","unstructured":"Benkert, M., Gudmundsson, J., H\u00fcbner, F., Wolle, T.: Reporting flock patterns. Comput. Geom. 41(3), 111\u2013125 (2008)","journal-title":"Comput. Geom."},{"key":"590_CR8","volume-title":"Lattice Theory","author":"G Birkhoff","year":"1948","unstructured":"Birkhoff, G.: Lattice Theory, vol. 25. American Mathematical Society, Providence (1948)"},{"issue":"1","key":"590_CR9","doi-asserted-by":"crossref","first-page":"92","DOI":"10.1007\/s00454-021-00298-0","volume":"66","author":"HB Bjerkevik","year":"2021","unstructured":"Bjerkevik, H.B.: On the stability of interval decomposable persistence modules. Discrete Comput. Geom. 66(1), 92\u2013121 (2021)","journal-title":"Discrete Comput. Geom."},{"key":"590_CR10","volume-title":"Graph Theory (Graduate Texts in Mathematics)","author":"J Bondy","year":"2008","unstructured":"Bondy, J., Murty, U.: Graph Theory (Graduate Texts in Mathematics). Springer, New York (2008)"},{"issue":"8","key":"590_CR11","doi-asserted-by":"crossref","first-page":"3571","DOI":"10.1090\/proc\/13465","volume":"145","author":"MB Botnan","year":"2017","unstructured":"Botnan, M.B.: Interval decomposition of infinite zigzag persistence modules. Proc. Am. Math. Soc. 145(8), 3571\u20133577 (2017)","journal-title":"Proc. Am. Math. Soc."},{"issue":"6","key":"590_CR12","doi-asserted-by":"crossref","first-page":"3133","DOI":"10.2140\/agt.2018.18.3133","volume":"18","author":"M Botnan","year":"2018","unstructured":"Botnan, M., Lesnick, M.: Algebraic stability of zigzag persistence modules. Algebraic Geom. Topol. 18(6), 3133\u20133204 (2018)","journal-title":"Algebraic Geom. Topol."},{"key":"590_CR13","volume-title":"Sheaf Theory","author":"GE Bredon","year":"2012","unstructured":"Bredon, G.E.: Sheaf Theory, vol. 170. Springer, New York (2012)"},{"issue":"3","key":"590_CR14","doi-asserted-by":"crossref","first-page":"600","DOI":"10.1007\/s00454-014-9573-x","volume":"51","author":"P Bubenik","year":"2014","unstructured":"Bubenik, P., Scott, J.A.: Categorification of persistent homology. Discrete Comput. Geom. 51(3), 600\u2013627 (2014)","journal-title":"Discrete Comput. Geom."},{"issue":"1","key":"590_CR15","first-page":"75","volume":"6","author":"K Buchin","year":"2015","unstructured":"Buchin, K., Buchin, M., van Kreveld, M.J., Speckmann, B., Staals, F.: Trajectory grouping structure. JoCG 6(1), 75\u201398 (2015)","journal-title":"JoCG"},{"key":"590_CR16","volume-title":"A Course in Metric Geometry. AMS Graduate Studies in Math.","author":"D Burago","year":"2001","unstructured":"Burago, D., Burago, Y., Ivanov, S.: A Course in Metric Geometry. AMS Graduate Studies in Math., vol. 33. American Mathematical Society, Providence (2001)"},{"key":"590_CR17","doi-asserted-by":"crossref","unstructured":"Carlsson, G., M\u00e9moli, F.: Multiparameter hierarchical clustering methods. In: Classification as a Tool for Research, pp. 63\u201370. Springer (2010)","DOI":"10.1007\/978-3-642-10745-0_6"},{"issue":"2","key":"590_CR18","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1090\/S0273-0979-09-01249-X","volume":"46","author":"G Carlsson","year":"2009","unstructured":"Carlsson, G.: Topology and data. Bull. Am. Math. Soc. 46(2), 255\u2013308 (2009)","journal-title":"Bull. Am. Math. Soc."},{"key":"590_CR19","first-page":"1425","volume":"11","author":"G Carlsson","year":"2010","unstructured":"Carlsson, G., M\u00e9moli, F.: Characterization, stability and convergence of hierarchical clustering methods. J. Mach. Learn. Res. 11, 1425\u20131470 (2010)","journal-title":"J. Mach. Learn. Res."},{"issue":"2","key":"590_CR20","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1007\/s10208-012-9141-9","volume":"13","author":"G Carlsson","year":"2013","unstructured":"Carlsson, G., M\u00e9moli, F.: Classifying clustering schemes. Found. Comput. Math. 13(2), 221\u2013252 (2013)","journal-title":"Found. Comput. Math."},{"issue":"4","key":"590_CR21","doi-asserted-by":"crossref","first-page":"367","DOI":"10.1007\/s10208-010-9066-0","volume":"10","author":"G Carlsson","year":"2010","unstructured":"Carlsson, G., De Silva, V.: Zigzag persistence. Found. Comput. Math. 10(4), 367\u2013405 (2010)","journal-title":"Found. Comput. Math."},{"issue":"1","key":"590_CR22","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1007\/s00454-009-9176-0","volume":"42","author":"G Carlsson","year":"2009","unstructured":"Carlsson, G., Zomorodian, A.: The theory of multidimensional persistence. Discrete Comput. Geom. 42(1), 71\u201393 (2009)","journal-title":"Discrete Comput. Geom."},{"key":"590_CR23","doi-asserted-by":"crossref","unstructured":"Chazal, F., Cohen-Steiner, D., Glisse, M., Guibas, L.J., Oudot, S.: Proximity of persistence modules and their diagrams. In: Proceedings of 25th ACM Symposium on Computational Geometry, pp. 237\u2013246 (2009)","DOI":"10.1145\/1542362.1542407"},{"key":"590_CR24","doi-asserted-by":"crossref","unstructured":"Chazal, F., Fasy, B.T., Lecci, F., Rinaldo, A., Wasserman, L.: Stochastic convergence of persistence landscapes and silhouettes. In: Proceedings of the Thirtieth Annual Symposium on Computational Geometry, pp. 474\u2013483 (2014)","DOI":"10.1145\/2582112.2582128"},{"key":"590_CR25","doi-asserted-by":"crossref","first-page":"48","DOI":"10.3934\/era.2018.25.006","volume":"25","author":"S Chowdhury","year":"2018","unstructured":"Chowdhury, S., M\u00e9moli, F.: Explicit geodesics in Gromov-Hausdorff space. Electron. Res. Announc. 25, 48\u201359 (2018)","journal-title":"Electron. Res. Announc."},{"key":"590_CR26","unstructured":"Clause, N., Kim, W.: Spatiotemporal Persistent Homology Computation Tool. https:\/\/github.com\/ndag\/PHoDMSs (2020)"},{"key":"590_CR27","unstructured":"Clause, N.: Zigzag Persistent Homology and Dynamic Networks. https:\/\/github.com\/ndag\/DynGraphZZ (2021)"},{"issue":"1","key":"590_CR28","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1007\/s00454-006-1276-5","volume":"37","author":"D Cohen-Steiner","year":"2007","unstructured":"Cohen-Steiner, D., Edelsbrunner, H., Harer, J.: Stability of persistence diagrams. Discrete Comput. Geom. 37(1), 103\u2013120 (2007)","journal-title":"Discrete Comput. Geom."},{"key":"590_CR29","unstructured":"Curry, J.M.: Sheaves, cosheaves and applications. PhD thesis, University of Pennsylvania (2014)"},{"issue":"27","key":"590_CR30","first-page":"1012","volume":"35","author":"J Curry","year":"2020","unstructured":"Curry, J., Patel, A.: Classification of constructible cosheaves. Theory Appl. Categories 35(27), 1012\u20131047 (2020)","journal-title":"Theory Appl. Categories"},{"issue":"12","key":"590_CR31","doi-asserted-by":"crossref","first-page":"1205","DOI":"10.1177\/0278364906072252","volume":"25","author":"V De Silva","year":"2006","unstructured":"De Silva, V., Ghrist, R.: Coordinate-free coverage in sensor networks with controlled boundaries via homology. Int. J. Robot. Res. 25(12), 1205\u20131222 (2006)","journal-title":"Int. J. Robot. Res."},{"issue":"4","key":"590_CR32","doi-asserted-by":"crossref","first-page":"854","DOI":"10.1007\/s00454-016-9763-9","volume":"55","author":"V De Silva","year":"2016","unstructured":"De Silva, V., Munch, E., Patel, A.: Categorified Reeb graphs. Discrete Comput. Geom. 55(4), 854\u2013906 (2016)","journal-title":"Discrete Comput. Geom."},{"key":"590_CR33","unstructured":"De\u00a0Silva, V., Ghrist, R., et al.: Homological sensor networks. Notices of the American mathematical society 54(1) (2007)"},{"key":"590_CR34","doi-asserted-by":"publisher","unstructured":"Dey, T.K., Hou, T.: Computing zigzag persistence on graphs in near-linear time. In: 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), vol. 189, pp. 30\u201313015. Schloss Dagstuhl\u2014Leibniz-Zentrum f\u00fcr Informatik, (2021). https:\/\/doi.org\/10.4230\/LIPIcs.SoCG.2021.30","DOI":"10.4230\/LIPIcs.SoCG.2021.30"},{"key":"590_CR35","unstructured":"Dey, T.K., Hou, T.: Updating zigzag persistence and maintaining representatives over changing filtrations. arXiv preprint arXiv:2112.02352 (2021)"},{"issue":"2","key":"590_CR36","doi-asserted-by":"crossref","first-page":"423","DOI":"10.1007\/s00454-016-9758-6","volume":"55","author":"B Di Fabio","year":"2016","unstructured":"Di Fabio, B., Landi, C.: The edit distance for Reeb graphs of surfaces. Discrete Comput. Geom. 55(2), 423\u2013461 (2016)","journal-title":"Discrete Comput. Geom."},{"key":"590_CR37","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1090\/conm\/453\/08802","volume":"453","author":"H Edelsbrunner","year":"2008","unstructured":"Edelsbrunner, H., Harer, J.: Persistent homology\u2014a survey. Contemp. Math. 453, 257\u2013282 (2008)","journal-title":"Contemp. Math."},{"key":"590_CR38","doi-asserted-by":"crossref","first-page":"511","DOI":"10.1007\/s00454-002-2885-2","volume":"28","author":"H Edelsbrunner","year":"2002","unstructured":"Edelsbrunner, H., Letscher, D., Zomorodian, A.: Topological persistence and simplification. Discrete Comput. Geom. 28, 511\u2013533 (2002)","journal-title":"Discrete Comput. Geom."},{"issue":"1","key":"590_CR39","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1007\/BF01298413","volume":"6","author":"P Gabriel","year":"1972","unstructured":"Gabriel, P.: Unzerlegbare darstellungen i. Manuscr. Math. 6(1), 71\u2013103 (1972)","journal-title":"Manuscr. Math."},{"key":"590_CR40","doi-asserted-by":"crossref","unstructured":"Gamble, J., Chintakunta, H., Krim, H.: Applied topology in static and dynamic sensor networks. In: 2012 International Conference on Signal Processing and Communications (SPCOM), pp. 1\u20135 (2012). IEEE","DOI":"10.1109\/SPCOM.2012.6290237"},{"issue":"1","key":"590_CR41","doi-asserted-by":"crossref","first-page":"325","DOI":"10.4310\/HHA.2022.v24.n1.a16","volume":"24","author":"R Ghrist","year":"2022","unstructured":"Ghrist, R., Riess, H.: Cellular sheaves of lattices and the Tarski Laplacian. Homol. Homotopys. Appl. 24(1), 325\u2013345 (2022)","journal-title":"Homol. Homotopys. Appl."},{"key":"590_CR42","doi-asserted-by":"crossref","unstructured":"Gonzalez-Diaz, R., Jimenez, M.-J., Medrano, B.: Spatiotemporal barcodes for image sequence analysis. In: International Workshop on Combinatorial Image Analysis, pp. 61\u201370 (2015). Springer","DOI":"10.1007\/978-3-319-26145-4_5"},{"key":"590_CR43","doi-asserted-by":"crossref","unstructured":"Gudmundsson, J., van Kreveld, M.: Computing longest duration flocks in trajectory data. In: Proceedings of the 14th Annual ACM International Symposium on Advances in Geographic Information Systems, pp. 35\u201342 (2006). ACM","DOI":"10.1145\/1183471.1183479"},{"issue":"2","key":"590_CR44","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1007\/s10707-006-0002-z","volume":"11","author":"J Gudmundsson","year":"2007","unstructured":"Gudmundsson, J., van Kreveld, M., Speckmann, B.: Efficient detection of patterns in 2d trajectories of moving points. Geoinformatica 11(2), 195\u2013215 (2007)","journal-title":"Geoinformatica"},{"key":"590_CR45","doi-asserted-by":"crossref","unstructured":"Hajij, M., Wang, B., Scheidegger, C., Rosen, P.: Visual detection of structural changes in time-varying graphs using persistent homology, 125\u2013134 (2018). IEEE","DOI":"10.1109\/PacificVis.2018.00024"},{"key":"590_CR46","doi-asserted-by":"crossref","unstructured":"Huang, Y., Chen, C., Dong, P.: Modeling herds and their evolvements from trajectory data. In: International Conference on Geographic Information Science, pp. 90\u2013105 (2008). Springer","DOI":"10.1007\/978-3-540-87473-7_6"},{"key":"590_CR47","doi-asserted-by":"crossref","unstructured":"Hwang, S.-Y., Liu, Y.-H., Chiu, J.-K., Lim, E.-P.: Mining mobile group patterns: A trajectory-based approach. In: PAKDD, vol. 3518, pp. 713\u2013718 (2005). Springer","DOI":"10.1007\/11430919_82"},{"key":"590_CR48","first-page":"286","volume-title":"Mathematical Taxonomy","author":"N Jardine","year":"1971","unstructured":"Jardine, N., Sibson, R.: Mathematical Taxonomy, p. 286. Wiley, London (1971). (Wiley Series in Probability and Mathematical Statistics)"},{"issue":"1","key":"590_CR49","doi-asserted-by":"crossref","first-page":"1068","DOI":"10.14778\/1453856.1453971","volume":"1","author":"H Jeung","year":"2008","unstructured":"Jeung, H., Yiu, M.L., Zhou, X., Jensen, C.S., Shen, H.T.: Discovery of convoys in trajectory databases. Proc. VLDB Endow. 1(1), 1068\u20131080 (2008)","journal-title":"Proc. VLDB Endow."},{"key":"590_CR50","doi-asserted-by":"crossref","unstructured":"Kalnis, P., Mamoulis, N., Bakiras, S.: On discovering moving clusters in spatio-temporal data. In: SSTD, vol. 3633, pp. 364\u2013381 (2005). Springer","DOI":"10.1007\/11535331_21"},{"key":"590_CR51","volume-title":"Geometry Helps to Compare Persistence Diagrams","author":"M Kerber","year":"2017","unstructured":"Kerber, M., Morozov, D., Nigmetov, A.: Geometry Helps to Compare Persistence Diagrams. ACM, New York (2017)"},{"key":"590_CR52","doi-asserted-by":"crossref","unstructured":"Kim, W., M\u00e9moli, F., Smith, Z.: Analysis of dynamic graphs and dynamic metric spaces via zigzag persistence. In: Topological Data Analysis, pp. 371\u2013389. Springer, (2020)","DOI":"10.1007\/978-3-030-43408-3_14"},{"key":"590_CR53","unstructured":"Kim, W., M\u00e9moli, F., Smith, Z.: Clustering behavior summary of dynamic metric data (2017). https:\/\/research.math.osu.edu\/networks\/formigrams"},{"key":"590_CR54","unstructured":"Kim, W., M\u00e9moli, F., Stefanou, A.: Interleaving by parts for persistence in a poset. arXiv preprint arXiv:1912.04366 (2019)"},{"key":"590_CR55","unstructured":"Kim, W., M\u00e9moli, F.: Formigrams: Clustering summaries of dynamic data. In: Proceedings of 30th Canadian Conference on Computational Geometry (CCCG18) (2018)"},{"key":"590_CR56","unstructured":"Kim, W., Memoli, F.: Stable signatures for dynamic graphs and dynamic metric spaces via zigzag persistence. arXiv preprint arXiv:1712.04064v4 (2017)"},{"issue":"4","key":"590_CR57","doi-asserted-by":"crossref","first-page":"533","DOI":"10.1007\/s41468-021-00075-1","volume":"5","author":"W Kim","year":"2021","unstructured":"Kim, W., M\u00e9moli, F.: Generalized persistence diagrams for persistence modules over posets. J. Appl. Comput. Topol. 5(4), 533\u2013581 (2021)","journal-title":"J. Appl. Comput. Topol."},{"issue":"3","key":"590_CR58","doi-asserted-by":"crossref","first-page":"831","DOI":"10.1007\/s00454-019-00168-w","volume":"66","author":"W Kim","year":"2021","unstructured":"Kim, W., M\u00e9moli, F.: Spatiotemporal persistent homology for dynamic metric spaces. Discrete Comput. Geom. 66(3), 831\u2013875 (2021)","journal-title":"Discrete Comput. Geom."},{"issue":"1\u20132","key":"590_CR59","doi-asserted-by":"crossref","first-page":"723","DOI":"10.14778\/1920841.1920934","volume":"3","author":"Z Li","year":"2010","unstructured":"Li, Z., Ding, B., Han, J., Kays, R.: Swarm: mining relaxed temporal moving object clusters. Proc. VLDB Endow. 3(1\u20132), 723\u2013734 (2010)","journal-title":"Proc. VLDB Endow."},{"key":"590_CR60","volume-title":"Categories for the Working Mathematician","author":"S Mac Lane","year":"2013","unstructured":"Mac Lane, S.: Categories for the Working Mathematician, vol. 5. Springer, New York (2013)"},{"issue":"7","key":"590_CR61","doi-asserted-by":"crossref","first-page":"3149","DOI":"10.1090\/proc\/14929","volume":"148","author":"A McCleary","year":"2020","unstructured":"McCleary, A., Patel, A.: Bottleneck stability for generalized persistence diagrams. Proc. Am. Math. Soc. U.S.A. 148(7), 3149\u20133161 (2020)","journal-title":"Proc. Am. Math. Soc. U.S.A."},{"issue":"2","key":"590_CR62","doi-asserted-by":"crossref","first-page":"134","DOI":"10.1137\/20M1373700","volume":"6","author":"A McCleary","year":"2022","unstructured":"McCleary, A., Patel, A.: Edit distance and persistence diagrams over lattices. SIAM J. Appl. Algebra Geom. 6(2), 134\u2013155 (2022)","journal-title":"SIAM J. Appl. Algebra Geom."},{"key":"590_CR63","unstructured":"M\u00e9moli, F.: A distance between filtered spaces via tripods. arXiv preprint arXiv:1704.03965 (2017)"},{"key":"590_CR64","volume-title":"Theory of Categories","author":"B Mitchell","year":"1965","unstructured":"Mitchell, B.: Theory of Categories, vol. 17. Academic Press, Washington, DC (1965)"},{"key":"590_CR65","doi-asserted-by":"crossref","first-page":"22","DOI":"10.1007\/s00454-012-9465-x","volume":"49","author":"D Morozov","year":"2013","unstructured":"Morozov, D., Beketayev, K., Weber, G.: Interleaving distance between merge trees. Discrete Comput. Geom. 49, 22\u201345 (2013)","journal-title":"Discrete Comput. Geom."},{"key":"590_CR66","unstructured":"Munch, E.: Applications of persistent homology to time varying systems. PhD thesis, Duke University (2013)"},{"key":"590_CR67","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511601156","volume-title":"Animal Groups in Three Dimensions: How Species Aggregate","author":"JK Parrish","year":"1997","unstructured":"Parrish, J.K., Hamner, W.M.: Animal Groups in Three Dimensions: How Species Aggregate. Cambridge University Press, Cambridge (1997)"},{"key":"590_CR68","unstructured":"Patel, A.: Reeb spaces and the robustness of preimages. PhD thesis, Duke University (2010)"},{"issue":"3","key":"590_CR69","doi-asserted-by":"crossref","first-page":"397","DOI":"10.1007\/s41468-018-0012-6","volume":"1","author":"A Patel","year":"2018","unstructured":"Patel, A.: Generalized persistence diagrams. J. Appl. Comput. Topol. 1(3), 397\u2013419 (2018)","journal-title":"J. Appl. Comput. Topol."},{"issue":"1","key":"590_CR70","doi-asserted-by":"crossref","first-page":"233","DOI":"10.4310\/HHA.2020.v22.n1.a14","volume":"22","author":"V Puuska","year":"2020","unstructured":"Puuska, V.: Erosion distance for generalized persistence modules. Homol. Homotopye Appl. 22(1), 233\u2013254 (2020)","journal-title":"Homol. Homotopye Appl."},{"issue":"4","key":"590_CR71","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1145\/37402.37406","volume":"21","author":"CW Reynolds","year":"1987","unstructured":"Reynolds, C.W.: Flocks, herds and schools: a distributed behavioral model. ACM SIGGRAPH Comput. Graph. 21(4), 25\u201334 (1987)","journal-title":"ACM SIGGRAPH Comput. Graph."},{"key":"590_CR72","unstructured":"Rolle, A., Scoccola, L.: Stable and consistent density-based clustering. arXiv preprint arXiv:2005.09048 (2020)"},{"issue":"2","key":"590_CR73","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1145\/3172867","volume":"51","author":"G Rossetti","year":"2018","unstructured":"Rossetti, G., Cazabet, R.: Community discovery in dynamic networks: a survey. ACM Comput. Surv. 51(2), 1\u201337 (2018)","journal-title":"ACM Comput. Surv."},{"issue":"4","key":"590_CR74","first-page":"340","volume":"2","author":"G-C Rota","year":"1964","unstructured":"Rota, G.-C.: On the foundations of combinatorial theory i. theory of M\u00f6bius functions. Probab. Theory Relat. Fields 2(4), 340\u2013368 (1964)","journal-title":"Probab. Theory Relat. Fields"},{"key":"590_CR75","doi-asserted-by":"crossref","unstructured":"Rubenstein, M., Ahler, C., Nagpal, R.: Kilobot: A low cost scalable robot system for collective behaviors. In: 2012 IEEE International Conference on Robotics and Automation, pp. 3293\u20133298 (2012). IEEE","DOI":"10.1109\/ICRA.2012.6224638"},{"key":"590_CR76","unstructured":"Schmiedl, F.: Shape matching and mesh segmentation. PhD thesis, Technische Universit\u00e4t M\u00fcnchen (2015)"},{"issue":"4","key":"590_CR77","doi-asserted-by":"crossref","first-page":"854","DOI":"10.1007\/s00454-017-9889-4","volume":"57","author":"F Schmiedl","year":"2017","unstructured":"Schmiedl, F.: Computational aspects of the Gromov-Hausdorff distance and its application in non-rigid shape matching. Discrete Comput. Geom. 57(4), 854\u2013880 (2017)","journal-title":"Discrete Comput. Geom."},{"issue":"17","key":"590_CR78","doi-asserted-by":"crossref","DOI":"10.1103\/PhysRevLett.119.178003","volume":"119","author":"M Sinhuber","year":"2017","unstructured":"Sinhuber, M., Ouellette, N.T.: Phase coexistence in insect swarms. Phys. Rev. Lett. 119(17), 178003 (2017)","journal-title":"Phys. Rev. Lett."},{"key":"590_CR79","doi-asserted-by":"crossref","DOI":"10.1515\/9781400837106","volume-title":"Collective Animal Behavior","author":"DJ Sumpter","year":"2010","unstructured":"Sumpter, D.J.: Collective Animal Behavior. Princeton University Press, Princeton (2010)"},{"issue":"5","key":"590_CR80","doi-asserted-by":"crossref","first-page":"0126383","DOI":"10.1371\/journal.pone.0126383","volume":"10","author":"CM Topaz","year":"2015","unstructured":"Topaz, C.M., Ziegelmeier, L., Halverson, T.: Topological data analysis of biological aggregation models. PLoS ONE 10(5), 0126383 (2015)","journal-title":"PLoS ONE"},{"key":"590_CR81","doi-asserted-by":"publisher","unstructured":"van Goethem, A., van Kreveld, M., L\u00f6ffler, M., Speckmann, B., Staals, F.: Grouping Time-Varying Data for Interactive Exploration. In: 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), vol. 51, pp. 61\u201316116. Schloss Dagstuhl\u2013Leibniz-Zentrum fuer Informatik (2016). https:\/\/doi.org\/10.4230\/LIPIcs.SoCG.2016.61.http:\/\/drops.dagstuhl.de\/opus\/volltexte\/2016\/5953","DOI":"10.4230\/LIPIcs.SoCG.2016.61"},{"issue":"02","key":"590_CR82","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1142\/S0218195918600051","volume":"28","author":"M van Kreveld","year":"2018","unstructured":"van Kreveld, M., L\u00f6ffler, M., Staals, F., Wiratma, L.: A refined definition for groups of moving entities and its computation. Int. J. Comput. Geom. Appl. 28(02), 181\u2013196 (2018)","journal-title":"Int. J. Comput. Geom. Appl."},{"issue":"1","key":"590_CR83","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1111\/cgf.12512","volume":"34","author":"C Vehlow","year":"2015","unstructured":"Vehlow, C., Beck, F., Auw\u00e4rter, P., Weiskopf, D.: Visualizing the evolution of communities in dynamic graphs. Comput. Graph. Forum 34(1), 277\u2013288 (2015). (Wiley Online Library)","journal-title":"Comput. Graph. Forum"},{"key":"590_CR84","doi-asserted-by":"crossref","unstructured":"Vieira, M.R., Bakalov, P., Tsotras, V.J.: On-line discovery of flock patterns in spatio-temporal data. In: Proceedings of the 17th ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems, pp. 286\u2013295 (2009). ACM","DOI":"10.1145\/1653771.1653812"},{"issue":"3","key":"590_CR85","doi-asserted-by":"crossref","first-page":"515","DOI":"10.1007\/s00778-006-0019-9","volume":"17","author":"Y Wang","year":"2008","unstructured":"Wang, Y., Lim, E.-P., Hwang, S.-Y.: Efficient algorithms for mining maximal valid groups. VLDB J. 17(3), 515\u2013535 (2008)","journal-title":"VLDB J."},{"key":"590_CR86","unstructured":"Wikipedia: Formicarium\u2014Wikipedia, The Free Encyclopedia. https:\/\/en.wikipedia.org\/wiki\/Formicarium. Accessed 12 Dec 2021"},{"key":"590_CR87","doi-asserted-by":"crossref","unstructured":"Wiratma, L., van Kreveld, M., L\u00f6ffler, M., Staals, F.: An experimental evaluation of grouping definitions for moving entities. In: Proceedings of the 27th ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems, pp. 89\u201398 (2019)","DOI":"10.1145\/3347146.3359346"},{"issue":"1","key":"590_CR88","doi-asserted-by":"crossref","first-page":"1","DOI":"10.3934\/fods.2021033","volume":"4","author":"L Xian","year":"2022","unstructured":"Xian, L., Adams, H., Topaz, C.M., Ziegelmeier, L.: Capturing dynamics of time-varying data via topology. Found. Data Sci. 4(1), 1\u201336 (2022)","journal-title":"Found. Data Sci."}],"container-title":["Discrete &amp; Computational Geometry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-023-00590-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00454-023-00590-1\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-023-00590-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,5,6]],"date-time":"2024-05-06T20:21:24Z","timestamp":1715026884000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00454-023-00590-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,10,11]]},"references-count":88,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2024,6]]}},"alternative-id":["590"],"URL":"https:\/\/doi.org\/10.1007\/s00454-023-00590-1","relation":{},"ISSN":["0179-5376","1432-0444"],"issn-type":[{"value":"0179-5376","type":"print"},{"value":"1432-0444","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,10,11]]},"assertion":[{"value":"18 February 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 July 2023","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 July 2023","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 October 2023","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors have no competing interests to declare that are relevant to the content of this article.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest\/Competing interests"}}]}}