{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,12]],"date-time":"2026-05-12T05:31:02Z","timestamp":1778563862772,"version":"3.51.4"},"reference-count":71,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2017,4,18]],"date-time":"2017-04-18T00:00:00Z","timestamp":1492473600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The emergence of complex datasets permeates versatile research disciplines leading to the necessity to develop methods for tackling complexity through finding the patterns inherent in datasets. The challenge lies in transforming the extracted patterns into pragmatic knowledge. In this paper, new information entropy measures for the characterization of the multidimensional structure extracted from complex datasets are proposed, complementing the conventionally-applied algebraic topology methods. Derived from topological relationships embedded in datasets, multilevel entropy measures are used to track transitions in building the high dimensional structure of datasets captured by the stratified partition of a simplicial complex. The proposed entropies are found suitable for defining and operationalizing the intuitive notions of structural relationships in a cumulative experience of a taxi driver\u2019s cognitive map formed by origins and destinations. The comparison of multilevel integration entropies calculated after each new added ride to the data structure indicates slowing the pace of change over time in the origin-destination structure. The repetitiveness in taxi driver rides, and the stability of origin-destination structure, exhibits the relative invariance of rides in space and time. These results shed light on taxi driver\u2019s ride habits, as well as on the commuting of persons whom he\/she drove.<\/jats:p>","DOI":"10.3390\/e19040172","type":"journal-article","created":{"date-parts":[[2017,4,18]],"date-time":"2017-04-18T11:22:04Z","timestamp":1492514524000},"page":"172","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Multilevel Integration Entropies: The Case of Reconstruction of Structural Quasi-Stability in Building Complex Datasets"],"prefix":"10.3390","volume":"19","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7642-8271","authenticated-orcid":false,"given":"Slobodan","family":"Maleti\u0107","sequence":"first","affiliation":[{"name":"Shenzhen Graduate School, Harbin Institute of Technology, Shenzhen 518055, China"},{"name":"Institute of Nuclear Sciences Vin\u010da, University of Belgrade, Belgrade 11351, Serbia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yi","family":"Zhao","sequence":"additional","affiliation":[{"name":"Shenzhen Graduate School, Harbin Institute of Technology, Shenzhen 518055, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2017,4,18]]},"reference":[{"key":"ref_1","unstructured":"Munkres, J.R. (1984). Elements of Algebraic Topology, Addison-Wesley Publishing."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1002\/j.1538-7305.1948.tb01338.x","article-title":"A mathematical theory of communication","volume":"27","author":"Shannon","year":"1948","journal-title":"Bell Syst. Tech. J."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"3253","DOI":"10.3390\/e17053253","article-title":"The Homological Nature of Entropy","volume":"17","author":"Baudot","year":"2015","journal-title":"Entropy"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"391","DOI":"10.1016\/j.patcog.2014.06.023","article-title":"An entropy-based persistence barcode","volume":"48","author":"Chintakunta","year":"2015","journal-title":"Pattern Recognit."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"6872","DOI":"10.3390\/e17106872","article-title":"Topological Characterization of Complex Systems: Using Persistent Entropy","volume":"17","author":"Merelli","year":"2015","journal-title":"Entropy"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"30","DOI":"10.1166\/jcsmd.2016.1094","article-title":"The influence of architecture of nanoparticle networks on collective charge transport revealed by the fractal time series and topology of phase space manifolds","volume":"4","year":"2016","journal-title":"J. Coupled Syst. Multiscale Dyn."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1140\/epjst\/e2012-01655-6","article-title":"Combinatorial Laplacian and entropy of simplicial complexes associated with complex networks","volume":"212","year":"2012","journal-title":"Eur. Phys. J. ST"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1236","DOI":"10.1038\/srep01236","article-title":"Extracting insights from the shape of complex data using topology","volume":"3","author":"Lum","year":"2013","journal-title":"Sci. Rep."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1090\/S0273-0979-09-01249-X","article-title":"Topology and data","volume":"46","author":"Carlsson","year":"2009","journal-title":"Bull. Am. Math. Soc."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"120201","DOI":"10.1088\/0266-5611\/27\/12\/120201","article-title":"Topological data analysis","volume":"27","author":"Epstein","year":"2011","journal-title":"Inverse Probl."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Edelsbrunner, H., and Harer, J. (2010). Computational Topology: An Introduction, American Mathematical Society.","DOI":"10.1090\/mbk\/069"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1090\/conm\/453\/08802","article-title":"Persistent homology\u2014A survey","volume":"453","author":"Edelsbrunner","year":"2008","journal-title":"Contemp. Math."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1007\/s00454-004-1146-y","article-title":"Computing persistent homology","volume":"33","author":"Zomorodian","year":"2005","journal-title":"Discret. Comput. Geom."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"7265","DOI":"10.1073\/pnas.1102826108","article-title":"Topology based data analysis identifies a subgroup of breast cancers with a unique mutational profile and excellent survival","volume":"108","author":"Nicolau","year":"2001","journal-title":"Proc. Natl Acad. Sci. USA"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"8581","DOI":"10.1038\/ncomms9581","article-title":"Topological data analysis for discovery in preclinical spinal cord injury and traumatic brain injury","volume":"6","author":"Nielson","year":"2015","journal-title":"Nat. Commun."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"20140873","DOI":"10.1098\/rsif.2014.0873","article-title":"Homological scaffolds of brain functional networks","volume":"11","author":"Petri","year":"2014","journal-title":"J. R. Soc. Interface"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"2267","DOI":"10.1109\/TMI.2012.2219590","article-title":"Persistent brain network homology from the perspective of dendrogram","volume":"31","author":"Lee","year":"2012","journal-title":"IEEE Trans. Med. Imaging"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1167\/8.8.11","article-title":"Topological analysis of population activity in visual cortex","volume":"8","author":"Singh","year":"2008","journal-title":"J. Vis."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Dabaghian, Y., M\u00e9moli, F., Frank, L., and Carlsson, G. (2012). A topological paradigm for hippocampal spatial map formation using persistent homology. PLoS Comput. Biol., 8.","DOI":"10.1371\/journal.pcbi.1002581"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Arai, M., Brandt, V., and Dabaghian, Y. (2014). The Effects of Theta Precession on Spatial Learning and Simplicial Complex Dynamics in a Topological Model of the Hippocampal Spatial Map. PLoS Comput. Biol., 10.","DOI":"10.1371\/journal.pcbi.1003651"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"198","DOI":"10.1214\/15-AOAS886","article-title":"Persistent homology analysis of brain artery trees","volume":"10","author":"Bendich","year":"2016","journal-title":"Ann. Appl. Stat."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"144115","DOI":"10.1063\/1.3103496","article-title":"Topological methods for exploring low-density states in biomolecular folding pathways","volume":"130","author":"Yao","year":"2009","journal-title":"J. Chem. Phys."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"431","DOI":"10.1007\/978-0-387-69319-4_22","article-title":"A Topological Characterization of Protein Structure","volume":"7","author":"Krishnamoorthy","year":"2007","journal-title":"Data Min. Biomed. Part IV"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"814","DOI":"10.1002\/cnm.2655","article-title":"Persistent homology analysis of protein structure, flexibility, and folding","volume":"30","author":"Xia","year":"2014","journal-title":"Int. J. Numer. Methods Biomed. Eng."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"18566","DOI":"10.1073\/pnas.1313480110","article-title":"Topology of viral evolution","volume":"110","author":"Chan","year":"2013","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"122","DOI":"10.3389\/fcimb.2014.00122","article-title":"Topological data analysis of Escherichia coli O157: H7 and non-O157 survival in soils","volume":"4","author":"Ibekwe","year":"2014","journal-title":"Front. Cell. Infect. Microbiol."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1205","DOI":"10.1177\/0278364906072252","article-title":"Coordinate-free Coverage in Sensor Networks with Controlled Boundaries via Homology","volume":"25","author":"Ghrist","year":"2006","journal-title":"Int. J. Robot. Res."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"339","DOI":"10.2140\/agt.2007.7.339","article-title":"Coverage in sensor networks via persistent homology","volume":"7","author":"Ghrist","year":"2007","journal-title":"Algebraic Geom. Topol."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"799","DOI":"10.1007\/s10208-014-9206-z","article-title":"Sliding windows and persistence: An application of topological methods to signal analysis","volume":"15","author":"Perea","year":"2015","journal-title":"Found. Comput. Math."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s11263-007-0056-x","article-title":"On the local behavior of spaces of natural images","volume":"76","author":"Carlsson","year":"2008","journal-title":"Int. J. Comput. Vis."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1080\/17459737.2013.850597","article-title":"Topology of musical data","volume":"8","author":"Sethares","year":"2014","journal-title":"J. Math. Music"},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Wagner, H., D\u0142otko, P., and Mrozek, M. (2012, January 28\u201330). Computational Topology in Text Mining. Proceedings of the 4th International Workshop Computational Topology in Image Context, CTIC, Bertinoro, Italy.","DOI":"10.1007\/978-3-642-30238-1_8"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"053105","DOI":"10.1063\/1.4949472","article-title":"Persistent topological features of dynamical systems","volume":"26","author":"Zhao","year":"2016","journal-title":"Chaos"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1016\/j.physd.2016.03.006","article-title":"Exploring the Topology of Dynamical Reconstructions","volume":"334","author":"Garland","year":"2016","journal-title":"Physica D"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"7723","DOI":"10.1038\/ncomms8723","article-title":"Topological data analysis of contagion maps for examining spreading processes on networks","volume":"6","author":"Taylor","year":"2015","journal-title":"Nat. Commun."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"815035","DOI":"10.1155\/2013\/815035","article-title":"Persistent Homology of Collaboration Networks","volume":"2013","author":"Carstens","year":"2013","journal-title":"Math. Probl. Eng."},{"key":"ref_37","first-page":"P03034","article-title":"Persistent Homology of Complex Networks","volume":"3","author":"Horak","year":"2009","journal-title":"J. Stat. Mech."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"139","DOI":"10.1016\/S0020-7373(72)80029-4","article-title":"From cohomology in physics to q-connectivity in social sciences","volume":"4","author":"Atkin","year":"1972","journal-title":"Int. J. Man Mach. Stud."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Atkin, R.H. (1977). Combinatorial Connectivities in Social Systems, Birkh\u00e4user Verlag.","DOI":"10.1007\/978-3-0348-5750-5"},{"key":"ref_40","unstructured":"Atkin, R.H. (1974). Mathematical Structure in Human Affairs, Heinemann."},{"key":"ref_41","unstructured":"Gould, P., Johnson, J., and Chapman, G. (1984). The Structure of Television, Pion Limited."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1023\/A:1004255229611","article-title":"Q-Analysis Techniques for Studying Communication Content","volume":"32","author":"Jacobson","year":"1998","journal-title":"Qual. Quant."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"439","DOI":"10.1068\/b100439","article-title":"Rethinking backcloth and traffic: Prespectives from social network analysis and Q-analysis","volume":"10","author":"Seidman","year":"1983","journal-title":"Environ. Plan. B"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"367","DOI":"10.1016\/S0020-7373(80)80021-6","article-title":"Q-analysis and the structure of friendship networks","volume":"12","author":"Freeman","year":"1980","journal-title":"Int. J. Man Mach. Stud."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1016\/0378-8733(81)90009-5","article-title":"Polyhedral Dynamics and Conflict Mobilization in Social Networks","volume":"3","author":"Doreian","year":"1981","journal-title":"Soc. Netw."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1016\/0378-8733(82)90012-0","article-title":"Leveling coalitions as network phenomena","volume":"4","author":"Doreian","year":"1982","journal-title":"Soc. Netw."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1080\/00420987120080421","article-title":"An analysis of urban structure using concepts of algebraic topology","volume":"8","author":"Atkin","year":"1971","journal-title":"Urban Stud."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"531","DOI":"10.1016\/S0020-7373(76)80019-3","article-title":"The q-analysis of road intersections","volume":"8","author":"Johnson","year":"1976","journal-title":"Int. J. Man Mach. Stud."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1007\/BF01030077","article-title":"Geological Similarity by Q-Analysis","volume":"15","author":"Griffiths","year":"1983","journal-title":"Math. Geol."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1016\/0096-3003(83)90036-X","article-title":"Evaluation of the Performance of a Distribution System by Q-Analysis","volume":"13","author":"Duckstein","year":"1983","journal-title":"Appl. Math. Comput."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"411","DOI":"10.1016\/S0377-2217(97)00308-1","article-title":"Q-analysis for modeling and decision making","volume":"103","author":"Duckstein","year":"1997","journal-title":"Eur. J. Oper. Res."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1109\/TSMC.1985.6313367","article-title":"Topological approach to failure diagnosis of large-scale systems","volume":"5","author":"Ishida","year":"1985","journal-title":"IEEE Trans. Syst. Man Cybern."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"334","DOI":"10.1016\/0022-247X(79)90121-5","article-title":"Polyhedral Dynamics and the Controllability of Dynamical Systems","volume":"68","author":"Casti","year":"1979","journal-title":"J. Math. Anal. Appl."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1016\/S0020-7373(72)80008-7","article-title":"Multi-dimensional Structure in the Game of Chess","volume":"4","author":"Atkin","year":"1972","journal-title":"Int. J. Man Mach. Stud."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"568","DOI":"10.1007\/978-3-540-69387-1_65","article-title":"Simplicial Complexes of Networks and Their Statistical Properties","volume":"5102","year":"2008","journal-title":"Lect. Notes Comput. Sci."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1037\/h0061626","article-title":"Cognitive maps in rats and men","volume":"55","author":"Tolman","year":"1948","journal-title":"Psychol. Rev."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0272-4944(05)80194-X","article-title":"Cognitive maps: What are they and why study them?","volume":"14","author":"Kitchin","year":"1994","journal-title":"J. Environ. Psychol."},{"key":"ref_58","unstructured":"Downs, R.M., and Stea, D. (1973). Cognitive maps in perception and thought. Image and Environment, Aldine."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1016\/0016-7185(92)90011-R","article-title":"Distortions in cognitive maps","volume":"23","author":"Tversky","year":"1992","journal-title":"Geoforum"},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"353","DOI":"10.1016\/j.sbspro.2011.07.030","article-title":"The Use of Geospatial Information and Spatial Cognition of Taxi Drivers in Tokyo","volume":"21","author":"Wakabayashia","year":"2011","journal-title":"Procedia Soc. Behav. Sci."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1016\/S0272-4944(88)80020-3","article-title":"Spatio-temporal aspects of the mental representation of urban space","volume":"8","author":"Giraudo","year":"1988","journal-title":"J. Environ. Psychol."},{"key":"ref_62","doi-asserted-by":"crossref","unstructured":"Peng, C., Jin, X., Wong, K.-C., Shi, M., and Li\u00f2, P. (2012). Collective Human Mobility Pattern from Taxi Trips in Urban Area. PLoS ONE, 7.","DOI":"10.1371\/annotation\/f0d48839-ed4b-4cb2-822a-d449a6b4fa5d"},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"462","DOI":"10.1038\/nature04292","article-title":"The scaling laws of human travel","volume":"439","author":"Brockmann","year":"2006","journal-title":"Nature"},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"779","DOI":"10.1038\/nature06958","article-title":"Understanding individual human mobility patterns","volume":"453","author":"Hidalgo","year":"2008","journal-title":"Nature"},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"1018","DOI":"10.1126\/science.1177170","article-title":"Limits of Predictability in Human Mobility","volume":"327","author":"Song","year":"2010","journal-title":"Science"},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1037\/h0060810","article-title":"The discrimination of stimulus configurations and the hypothesis of afferent neural interaction","volume":"52","author":"Hull","year":"1945","journal-title":"Psychol. Rev."},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"258","DOI":"10.1037\/h0055756","article-title":"Forming Impressions of Personality","volume":"41","author":"Asch","year":"1946","journal-title":"J. Abnorm. Soc. Psychol."},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1068\/b080073","article-title":"Some structures and notation of Q-analysis","volume":"8","author":"Johnson","year":"1981","journal-title":"Envniron. Plan. B"},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"84","DOI":"10.2307\/1969768","article-title":"Homology groups of relations","volume":"56","author":"Dowker","year":"1952","journal-title":"Ann. Math."},{"key":"ref_70","doi-asserted-by":"crossref","first-page":"565","DOI":"10.1016\/j.jenvp.2010.03.003","article-title":"The effect of navigational expertise on wayfinding in new environment","volume":"30","author":"Woollett","year":"2010","journal-title":"J. Environ. Psychol."},{"key":"ref_71","unstructured":"Frank, A.U., and Campari, I. (1993). Cognitive maps, cognitive collages, and spatial mental models. Spatial Information Theory: A Theoretical Basis for GIS, Springer-Verlag."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/19\/4\/172\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T18:32:51Z","timestamp":1760207571000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/19\/4\/172"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,4,18]]},"references-count":71,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2017,4]]}},"alternative-id":["e19040172"],"URL":"https:\/\/doi.org\/10.3390\/e19040172","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,4,18]]}}}