{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,10]],"date-time":"2026-04-10T14:36:59Z","timestamp":1775831819134,"version":"3.50.1"},"reference-count":56,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2020,8,6]],"date-time":"2020-08-06T00:00:00Z","timestamp":1596672000000},"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>Recently, it has been argued that entropy can be a direct measure of complexity, where the smaller value of entropy indicates lower system complexity, while its larger value indicates higher system complexity. We dispute this view and propose a universal measure of complexity that is based on Gell-Mann\u2019s view of complexity. Our universal measure of complexity is based on a non-linear transformation of time-dependent entropy, where the system state with the highest complexity is the most distant from all the states of the system of lesser or no complexity. We have shown that the most complex is the optimally mixed state consisting of pure states, i.e., of the most regular and most disordered which the space of states of a given system allows. A parsimonious paradigmatic example of the simplest system with a small and a large number of degrees of freedom is shown to support this methodology. Several important features of this universal measure are pointed out, especially its flexibility (i.e., its openness to extensions), suitability to the analysis of system critical behaviour, and suitability to study the dynamic complexity.<\/jats:p>","DOI":"10.3390\/e22080866","type":"journal-article","created":{"date-parts":[[2020,8,7]],"date-time":"2020-08-07T09:30:54Z","timestamp":1596792654000},"page":"866","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":14,"title":["Towards a Universal Measure of Complexity"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5164-6931","authenticated-orcid":false,"given":"Jaros\u0142aw","family":"Klamut","sequence":"first","affiliation":[{"name":"Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland"}]},{"given":"Ryszard","family":"Kutner","sequence":"additional","affiliation":[{"name":"Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1352-8897","authenticated-orcid":false,"given":"Zbigniew R.","family":"Struzik","sequence":"additional","affiliation":[{"name":"Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland"},{"name":"Graduate School of Education, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan"},{"name":"Advanced Center for Computing and Communication, RIKEN, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan"}]}],"member":"1968","published-online":{"date-parts":[[2020,8,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Nicolis, G., and Nicolis, C. (2012). Foundations of Complex Systems. Emergence, Information and Prediction, World Science Publication. [2nd ed.].","DOI":"10.1142\/9789814366618"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1016\/j.physrep.2012.01.007","article-title":"Physical approach to complex systems","volume":"515","year":"2012","journal-title":"Phys. Rep."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1275","DOI":"10.1103\/RevModPhys.80.1275","article-title":"Critical phenomena in complex networks","volume":"80","author":"Dorogovtsev","year":"2008","journal-title":"Rev. Mod. Phys."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1103\/RevModPhys.74.47","article-title":"Statistical mechanics of complex networks","volume":"74","author":"Albert","year":"2002","journal-title":"Rev. Mod. Phys."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"2297","DOI":"10.1073\/pnas.88.6.2297","article-title":"Approximate entropy as a measure of system complexity","volume":"88","author":"Pincus","year":"1991","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Dorogovtsev, S.N. (2010). Lectures on Complex Networks, Clarendon Press.","DOI":"10.1093\/acprof:oso\/9780199548927.001.0001"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"346","DOI":"10.1103\/PhysRevLett.50.346","article-title":"On the characterization of strange attractors","volume":"50","author":"Grassberger","year":"1983","journal-title":"Phys. Rev. Lett."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"2039","DOI":"10.1152\/ajpheart.2000.278.6.H2039","article-title":"Physiological time-series analysis using approximate entropy and sample entropy","volume":"278","author":"Richman","year":"2000","journal-title":"Am. J. Physiol. Heart Circ. Physiol."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2989","DOI":"10.3390\/e15082989","article-title":"Time evolution of relative entropies for anomalous diffusion","volume":"15","author":"Prehl","year":"2013","journal-title":"Entropy"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Thurner, S., Hanel, R., and Klimek, P. (2018). Introduction to the Theory of Complex Systems, Oxford Univ. Press.","DOI":"10.1093\/oso\/9780198821939.001.0001"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Popiel, N.J.M., Khajehabdollahi, S., Abeyasinghe, P.M., Riganello, F., Nichols, E., Owen, A.M., and Soddu, A. (2020). The Emergence of Integrated Information, Complexity, and \u2018Consciousness\u2019 at Criticality. Entropy, 22.","DOI":"10.3390\/e22030339"},{"key":"ref_12","unstructured":"(2020, August 05). Available online: https:\/\/en.wikipedia.org\/wiki\/Complexity."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"6905","DOI":"10.1073\/pnas.1406071111","article-title":"How multiplicity of random processes determines entropy and the derivation of the maximum entropy principle for complex systems","volume":"111","author":"Thurner","year":"2014","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"907","DOI":"10.1007\/BF00668821","article-title":"Toward a quantitative theory of self-generated complexity","volume":"25","author":"Grassbereger","year":"1986","journal-title":"Int. J. Theor. Phys."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1016\/S0378-4371(01)00444-7","article-title":"Complexity through nonextensivity","volume":"302","author":"Bialek","year":"2001","journal-title":"Phys. A"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1002\/cplx.20249","article-title":"An information-theoretic primer on complexity, self-organization, and emergence","volume":"15","author":"Prokopenko","year":"2008","journal-title":"Complexity"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Borda, M. (2011). Fundamentals in Information Theory and Coding, Springer.","DOI":"10.1007\/978-3-642-20347-3"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1038\/nphys2190","article-title":"Between order and chaos","volume":"8","author":"Crutchfield","year":"2012","journal-title":"Nat. Phys."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Baker, L.G., and Gollub, J.P. (1996). Chaotic Dynamics: An Introduction, Cambridge University Press. [2nd ed.]. Section 3.","DOI":"10.1017\/CBO9781139170864"},{"key":"ref_20","unstructured":"Schuster, H.G. (1988). Deterministic Chaos. An Introduction, VCH. [2nd ed.]."},{"key":"ref_21","unstructured":"Henkel, M., Hinrichsen, H., and L\u00fcbeck, S. (2008). Non-Equilibrium Phase Transitions. Volume I: Absorbing Phase Transitions, Springer. Section 4.1.7."},{"key":"ref_22","unstructured":"Wolfram, S. (2002). A New Kind of Science, Wolfram Media Inc."},{"key":"ref_23","first-page":"17","article-title":"Plectics: The study of simplicity and complexity","volume":"1","year":"2002","journal-title":"Europhysicsnews"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"44","DOI":"10.1002\/(SICI)1099-0526(199609\/10)2:1<44::AID-CPLX10>3.0.CO;2-X","article-title":"Information measures, effective complexity and total information","volume":"2","author":"Lloyd","year":"1996","journal-title":"Complexity"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Gell-Mann, M., and Tsallis, C. (2004). Effective complexity. Nonextensive Entropy: Interdisciplinary Applications, Oxford University Press.","DOI":"10.1093\/oso\/9780195159769.003.0028"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"4593","DOI":"10.1109\/TIT.2010.2053892","article-title":"Effective complexity and its relation to logical depth","volume":"56","author":"Ay","year":"2010","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_27","unstructured":"Freeman, W.H. (1994). The Quark and the Jaguar: Adventures in the Simple and the Complex, W.H. Freeman. [8th ed.]."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"50","DOI":"10.1140\/epjb\/e2016-70578-3","article-title":"The continuous time random walk still trendy: Fifty-year history, state of art, and outlook","volume":"90","author":"Kutner","year":"2017","journal-title":"Eur. Phys. J."},{"key":"ref_29","first-page":"1","article-title":"Dragon-kings, black swans and the prediction of crises","volume":"1","author":"Sornette","year":"2009","journal-title":"Int. J. Terraspace Sci. Eng."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1140\/epjst\/e2012-01559-5","article-title":"Dragon-kings: Mechanisms, statistical methods and empirical evidence","volume":"205","author":"Sornette","year":"2012","journal-title":"Eur. Phys. J. Spec. Top."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"5963","DOI":"10.1016\/j.physa.2013.07.064","article-title":"Structural and topological phase transition on the german stock exchange","volume":"392","author":"Sienkiewicz","year":"2013","journal-title":"Phys. A"},{"key":"ref_32","first-page":"1","article-title":"Temporal condensation and dynamic \u03bb-transition within the complex network: An application to real-life market evolution","volume":"34","author":"Szewczak","year":"2015","journal-title":"Eur. Phys. J. B"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"126","DOI":"10.1016\/j.chaos.2016.03.005","article-title":"Dynamic bifurcations on financial markets","volume":"88","author":"Denys","year":"2016","journal-title":"Chaos Solitons Fractals"},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Jakimowicz, A. (2020). The role of entropy in the development of economics. Entropy, 22.","DOI":"10.3390\/e22040452"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1515\/ijnsns-2014-0085","article-title":"Fundamental sources of economic complexity","volume":"17","author":"Jakimowicz","year":"2016","journal-title":"Int. J. Nonlinear Sci. Numer."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"745","DOI":"10.1142\/S0219525908001957","article-title":"Econophysics and economic complexity","volume":"11","author":"Rossler","year":"2008","journal-title":"Adv. Complex Syst."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"3091","DOI":"10.1140\/epjst\/e2016-60166-y","article-title":"Entropy and econophysics","volume":"225","author":"Rossler","year":"2016","journal-title":"Eur. Phys. J. Spec. Top."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Zambelli, S., and George, D.A.R. (2012). Nonlinearity, Complexity, and Randomness in Economics: Towards Algorithmic Foundations for Economic, Wiley\u2013Blackwell.","DOI":"10.1002\/9781118300442"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"240","DOI":"10.1016\/j.physa.2018.10.019","article-title":"Econophysics and sociophysics: Their milestones & challenges","volume":"516","author":"Kutner","year":"2019","journal-title":"Physica A"},{"key":"ref_40","first-page":"16","article-title":"What is Complexity?","volume":"1","year":"1995","journal-title":"Complexity"},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Bertin, E. (2012). A Concise Introduction to the Statistical Physics of Complex Systems, Springer.","DOI":"10.1007\/978-3-642-23923-6"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"391","DOI":"10.1119\/1.1971557","article-title":"Gibbs vs boltzmann entropies","volume":"33","author":"Jaynes","year":"1965","journal-title":"Am. J. Phys."},{"key":"ref_43","unstructured":"Beck, C.H., and Schl\u00f6gl, F. (1995). Thermodynamics of Chaotic Systems. An Introduction, Cambridge University Press."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1007\/BF01016429","article-title":"Possible generalization of boltzmann-gibbs statistics","volume":"52","author":"Tsallis","year":"1988","journal-title":"J. Stat. Phys."},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Van Kampen, G. (2007). Stochastic Processes in Physics and Chemistry, Elsevier. [3rd ed.].","DOI":"10.1016\/B978-044452965-7\/50006-4"},{"key":"ref_46","unstructured":"Bahcall, S. (2019). Loonshots: How to Nurture the Crazy Ideas that Win Wars, Cure Diseases, and Transform Industries, St. Martin\u2019s Press."},{"key":"ref_47","unstructured":"West, G. (2017). Scale: The Universal Laws of Life, Growth, and Death in Organisms, Cities, and Companies, Penguin Books."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"178","DOI":"10.1002\/(SICI)1520-6505(1998)6:5<178::AID-EVAN5>3.0.CO;2-8","article-title":"The social brain hypothesis","volume":"6","author":"Dunbar","year":"1998","journal-title":"Evol. Antropol."},{"key":"ref_49","doi-asserted-by":"crossref","unstructured":"Acharjee, S., Bora, B., and Dunbar, R.I.M. (2020). On M-polynomials of dunbar graphs in social networks. Symmetry, 12.","DOI":"10.3390\/sym12060932"},{"key":"ref_50","doi-asserted-by":"crossref","unstructured":"Kubo, R., Toda, M., and Hashitsume, N. (1985). Statistical Physics II. Nonequilibrium Statistical Mechanics, Springer.","DOI":"10.1007\/978-3-642-96701-6"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1007\/s11517-014-1216-0","article-title":"Assessing the complexity of short-term heartbeat interval series by distribution entropy","volume":"53","author":"Li","year":"2015","journal-title":"Med. Biol. Eng. Comput."},{"key":"ref_52","doi-asserted-by":"crossref","unstructured":"Sornette, D. (2000). Critical Phenomena in Natural Sciences. Chaos, Fractals, Selforganization and Disorder: Concepts and Tools, Springer.","DOI":"10.1007\/978-3-662-04174-1"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"18","DOI":"10.1103\/PhysRevA.1.18","article-title":"Decay of the velocity autocorrelation function","volume":"1","author":"Alder","year":"1970","journal-title":"Phys. Rev. A"},{"key":"ref_54","doi-asserted-by":"crossref","unstructured":"Bunde, A., and Havlin, S. (1996). Fractals and Disordered Systems, Second Revised and Enlarged Edition, Springer.","DOI":"10.1007\/978-3-642-84868-1"},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"063507","DOI":"10.1063\/1.3152608","article-title":"Solution of the fractional Langevin equation and the Mittag\u2013Leffler functions","volume":"50","author":"Camargo","year":"2009","journal-title":"J. Math. Phys."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1103\/RevModPhys.50.221","article-title":"General properties of entropy","volume":"50","author":"Wehrl","year":"1978","journal-title":"Rev. Mod. Phys."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/8\/866\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:57:22Z","timestamp":1760176642000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/8\/866"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,8,6]]},"references-count":56,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2020,8]]}},"alternative-id":["e22080866"],"URL":"https:\/\/doi.org\/10.3390\/e22080866","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,8,6]]}}}