{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,2]],"date-time":"2025-11-02T02:37:25Z","timestamp":1762051045160,"version":"build-2065373602"},"reference-count":28,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2022,4,27]],"date-time":"2022-04-27T00:00:00Z","timestamp":1651017600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Ministerio de Ciencia Innovaci\u00f3n y Universidades of Spain","award":["PGC2018-093854-B-I00"],"award-info":[{"award-number":["PGC2018-093854-B-I00"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this paper, we study the phenomena of collapse and anomalous diffusion in shared mobility systems. In particular, we focus on a fleet of vehicles moving through a stations network and analyse the effect of self-journeys in system stability, using a mathematical simplex under stochastic flows. With a birth-death process approach, we find analytical upper bounds for random walk and we monitor how the system collapses by super diffusing under different randomization conditions. Using the multi-scale entropy metric, we show that real data from a bike-sharing fleet in the city of Salamanca (Spain) present a complex behaviour with more of a 1\/f signal than a disorganized system with a white noise signal.<\/jats:p>","DOI":"10.3390\/e24050606","type":"journal-article","created":{"date-parts":[[2022,4,27]],"date-time":"2022-04-27T13:40:57Z","timestamp":1651066857000},"page":"606","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["A Multi-Scale Entropy Approach to Study Collapse and Anomalous Diffusion in Shared Mobility Systems"],"prefix":"10.3390","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4709-772X","authenticated-orcid":false,"given":"Francisco","family":"Prieto-Castrillo","sequence":"first","affiliation":[{"name":"Facultad de Ciencias, Departamento de Matem\u00e1ticas, Universidad de Oviedo, Calle Federico Garc\u00eda Lorca 18, 33007 Oviedo, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3774-2521","authenticated-orcid":false,"given":"Javier","family":"Borondo","sequence":"additional","affiliation":[{"name":"Facultad de Ciencias Econ\u00f3micas y Empresariales (ICADE), Departamento de Gesti\u00f3n Empresarial, Universidad Pontificia Comillas, Calle Alberto Aguilera 23, 28015 Madrid, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8443-6108","authenticated-orcid":false,"given":"Rub\u00e9n Mart\u00edn","family":"Garc\u00eda","sequence":"additional","affiliation":[{"name":"Facultad de Inform\u00e1tica, Universidad Pontificia de Salamanca, Calle Compa\u00f1\u00eda, 5, 37002 Salamanca, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3949-8232","authenticated-orcid":false,"given":"Rosa M.","family":"Benito","sequence":"additional","affiliation":[{"name":"Grupo de Sistemas Complejos, E.T.S. de Ingenier\u00eda Agron., Alim. y de Biosist., Universidad Polit\u00e9cnica de Madrid, Avda. 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Proceedings of the 2013 IEEE Global Conference on Signal and Information Processing, GlobalSIP 2013\u2014Proceedings, Austin, TX, USA.","DOI":"10.1109\/GlobalSIP.2013.6736952"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Zaltz Austwick, M., O\u2019Brien, O., Strano, E., and Viana, M. (2013). The Structure of Spatial Networks and Communities in Bicycle Sharing Systems. PLoS ONE, 8.","DOI":"10.1371\/journal.pone.0074685"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"13290","DOI":"10.1073\/pnas.1403657111","article-title":"Quantifying the benefits of vehicle pooling with shareability networks","volume":"111","author":"Santi","year":"2013","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Crisostomi, E., Faizrahnemoon, M., Schlote, A., and Shorten, R. (2015, January 19\u201323). A Markov-chain based model for a bike-sharing system. Proceedings of the 2015 International Conference on Connected Vehicles and Expo, ICCVE 2015\u2014Proceedings, Shenzhen, China.","DOI":"10.1109\/ICCVE.2015.12"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1380","DOI":"10.1109\/TASE.2014.2336874","article-title":"Stochastic Petri Net Modeling, Simulation and Analysis of Public Bicycle Sharing Systems","volume":"12","author":"Labadi","year":"2014","journal-title":"IEEE Trans. Autom. Sci. Eng."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Purnama, I.B.I., Bergmann, N., Jurdak, R., and Zhao, K. (2015, January 10\u201314). Characterising and predicting urban mobility dynamics by mining bike sharing system data. Proceedings of the 2015 IEEE 12th International Conference on Ubiquitous Intelligence and Computing, 2015 IEEE 12th International Conference on Advanced and Trusted Computing, 2015 IEEE 15th International Conference on Scalable Computing and Communications, Beijing, China.","DOI":"10.1109\/UIC-ATC-ScalCom-CBDCom-IoP.2015.46"},{"key":"ref_12","first-page":"1","article-title":"Understanding congested travel in urban areas","volume":"7","author":"Lima","year":"2016","journal-title":"Nat. Commun."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Preisler, T., Dethlefs, T., and Renz, W. (2016, January 11\u201314). Self-Organizing Redistribution of Bicycles in a Bike-Sharing System based on Decentralized Control. Proceedings of the 2016 Federated Conference on Computer Science and Information Systems (FedCSIS), Gdansk, Poland.","DOI":"10.15439\/2016F126"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1038\/srep42868","article-title":"Scaling law of urban ride sharing","volume":"7","author":"Tachet","year":"2017","journal-title":"Sci. Rep."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"534","DOI":"10.1038\/s41586-018-0095-1","article-title":"Addressing the minimum fleet problem in on-demand urban mobility","volume":"557","author":"Vazifeh","year":"2018","journal-title":"Nature"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Chiariotti, F., Pielli, C., Zanella, A., and Zorzi, M. (2018). A dynamic approach to rebalancing bike-sharing systems. Sensors, 18.","DOI":"10.3390\/s18020512"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Benito, R.M., Cherifi, C., Cherifi, H., Moro, E., Rocha, L.M., and Sales-Pardo, M. (2022). Understanding Imbalance Mechanisms in Shared Mobility Systems. Complex Networks & Their Applications X, Springer International Publishing.","DOI":"10.1007\/978-3-030-93409-5"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Atmanspacher, H. (2016). On macrostates in complex multi-scale systems. Entropy, 18.","DOI":"10.3390\/e18120426"},{"key":"ref_19","first-page":"2297","article-title":"Approximate entropy as a measure of system complexity","volume":"88","author":"Pincus","year":"1991","journal-title":"Mathematics"},{"key":"ref_20","first-page":"H1643","article-title":"Physiological time-series analysis: What does regularity quantify?","volume":"266","author":"Pincus","year":"1994","journal-title":"Am. J. Physiol."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"172","DOI":"10.1016\/S0076-6879(04)84011-4","article-title":"Sample Entropy","volume":"384","author":"Richman","year":"2004","journal-title":"Methods Enzymol."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"6","DOI":"10.1103\/PhysRevLett.89.068102","article-title":"Multiscale Entropy Analysis of Complex Physiologic Time Series","volume":"89","author":"Costa","year":"2002","journal-title":"Phys. Rev. Lett."},{"key":"ref_23","first-page":"1","article-title":"Multiscale entropy analysis of biological signals","volume":"71","author":"Costa","year":"2005","journal-title":"Phys. Rev. Stat. Nonlinear Soft Matter Phys."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"3110","DOI":"10.3390\/e17053110","article-title":"The multiscale entropy algorithm and its variants: A review","volume":"17","year":"2015","journal-title":"Entropy"},{"key":"ref_25","first-page":"533","article-title":"Complexity 1\/f noise. A phase space approach","volume":"1","author":"Zhang","year":"1991","journal-title":"J. Phys. Arch."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"411","DOI":"10.1007\/BF01053799","article-title":"On the phase space approach to complexity","volume":"69","author":"Fogedby","year":"1992","journal-title":"J. Stat. Phys."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1002\/cplx.20014","article-title":"Multiscale variety in complex systems","volume":"9","year":"2004","journal-title":"Complexity"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1511","DOI":"10.1088\/0305-4470\/29\/7\/020","article-title":"Analytical inversion of symmetric tridiagonal matrices","volume":"29","author":"Hu","year":"1996","journal-title":"J. Phys. Math. 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