{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T07:12:49Z","timestamp":1773213169216,"version":"3.50.1"},"reference-count":62,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2013,1,18]],"date-time":"2013-01-18T00:00:00Z","timestamp":1358467200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Information flow, or information transfer as it may be referred to, is a fundamental notion in general physics which has wide applications in scientific disciplines. Recently, a rigorous formalism has been established with respect to both deterministic and stochastic systems, with flow measures explicitly obtained. These measures possess some important properties, among which is flow or transfer asymmetry. The formalism has been validated and put to application with a variety of benchmark systems, such as the baker transformation, H\u00e9non map, truncated Burgers-Hopf system, Langevin equation, etc. In the chaotic Burgers-Hopf system, all the transfers, save for one, are essentially zero, indicating that the processes underlying a dynamical phenomenon, albeit complex, could be simple. (Truth is simple.) In the Langevin equation case, it is found that there could be no information flowing from one certain time series to another series, though the two are highly correlated. Information flow\/transfer provides a potential measure of the cause\u2013effect relation between dynamical events, a relation usually hidden behind the correlation in a traditional sense.<\/jats:p>","DOI":"10.3390\/e15010327","type":"journal-article","created":{"date-parts":[[2013,1,18]],"date-time":"2013-01-18T11:04:23Z","timestamp":1358507063000},"page":"327-360","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":84,"title":["The Liang-Kleeman Information Flow: Theory and Applications"],"prefix":"10.3390","volume":"15","author":[{"given":"X.","family":"Liang","sequence":"first","affiliation":[{"name":"School of Marine Sciences and School of Mathematics and Statistics, Nanjing University of Information Science and Technology (Nanjing Institute of Meteorology), 219 Ningliu Blvd, Nanjing 210044, China"},{"name":"China Institute for Advanced Study, Central University of Finance and Economics, 39 South CollegeAve, Beijing 100081, China\u00a0"}]}],"member":"1968","published-online":{"date-parts":[[2013,1,18]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1140\/epjst\/e2008-00855-y","article-title":"Transmission of information and synchronization in a pair of coupled chaotic circuits: An experimental overview","volume":"165","author":"Baptista","year":"2008","journal-title":"Eur. Phys. J.-Spec. Top."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"036203","DOI":"10.1103\/PhysRevE.82.036203","article-title":"Combined effect of chemical and electrical synapses in Hindmarsh-Rose neural networks on synchronization and the rate of information","volume":"82","author":"Baptista","year":"2010","journal-title":"Phys. Rev. E"},{"key":"ref_3","unstructured":"Bear, M.F., Connors, B.W., and Paradiso, M.A. (2007). Neuroscience: Exploring the Brain, Lippincott Williams & Wilkins. [3rd ed.]."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"96","DOI":"10.3389\/fnsys.2011.00096","article-title":"Empirical and theoretical aspects of generation and transfer of information in a neuromagnetic source network","volume":"5","author":"Vakorin","year":"2011","journal-title":"Front. Syst. Neurosci."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Ay, N., and Polani, D. (2008). Information flows in causal networks. Advs. Complex Syst., 11.","DOI":"10.1142\/S0219525908001465"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"e28860","DOI":"10.1371\/journal.pone.0028860","article-title":"Directedness of information flow in mobile phone communication networks","volume":"6","author":"Peruani","year":"2011","journal-title":"PLoS One"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"182","DOI":"10.1016\/j.jneumeth.2010.12.019","article-title":"On the estimation of the direction of information flow in networks of dynamical systems","volume":"196","author":"Sommerlade","year":"2011","journal-title":"J. Neurosci. Methods"},{"key":"ref_8","first-page":"1","article-title":"Understanding the earth as a complex system-recent advances in data analysis and modelling in earth sciences","volume":"174","author":"Donner","year":"2009","journal-title":"Eur. Phys. J."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1005","DOI":"10.1175\/JAS3857.1","article-title":"Information flow in ensemble weather prediction","volume":"64","author":"Kleeman","year":"2007","journal-title":"J. Atmos. Sci."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"877","DOI":"10.1016\/j.asr.2010.10.026","article-title":"Predictive space weather: An information theory approach","volume":"47","author":"Materassi","year":"2011","journal-title":"Adv. Space Res."},{"key":"ref_11","unstructured":"Tribbia, J.J. (2005, January 22\u201325). Waves, Information and Local Predictability. Proceedings of the Workshop on Mathematical Issues and Challenges in Data Assimilation for Geophysical Systems: Interdisciplinary Perspectives, IPAM, UCLA."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.rfe.2004.03.001","article-title":"Information flow between the stock and option markets: Where do informed traders trade?","volume":"14","author":"Chen","year":"2005","journal-title":"Rev. Financ. Econ."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"439","DOI":"10.1093\/rfs\/hhr084","article-title":"Jumps and information flow in financial markets","volume":"25","author":"Lee","year":"2012","journal-title":"Rev. Financ. Stud."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"051128","DOI":"10.1103\/PhysRevE.80.051128","article-title":"Estimating causal dependencies in networks of nonlinear stochastic dynamical systems","volume":"80","author":"Sommerlade","year":"2009","journal-title":"Phys. Rev. E"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Zhao, K., Karsai, M., and Bianconi, G. (2011). Entropy of dynamical social networks. PLoS One.","DOI":"10.1371\/journal.pone.0028116"},{"key":"ref_16","first-page":"1","article-title":"The evolution of El Ni\u00f1o, past and future","volume":"164","author":"Cane","year":"2004","journal-title":"Earth Planet. Sci. Lett."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"811","DOI":"10.1175\/1520-0469(1997)054<0811:AEORPF>2.0.CO;2","article-title":"An equatorial ocean recharge paradigm for ENSO. Part I: conceptual model","volume":"54","author":"Jin","year":"1997","journal-title":"J. Atmos. Sci."},{"key":"ref_18","unstructured":"Philander, S.G. (1990). El Ni\u00f1o, La Ni\u00f1a, and the Southern Oscillation, Academic Press."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"2111","DOI":"10.1016\/j.physd.2008.03.036","article-title":"Climate dynamics and fluid mechanics: Natural variability and related uncertainties","volume":"237","author":"Ghil","year":"2008","journal-title":"Physica D"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"L03709","DOI":"10.1029\/2006GL027412","article-title":"A kind of initial errors related to \u201cspring predictability barrier\u201d for El Ni\u00f1o events in Zebiak-Cane model","volume":"34","author":"Mu","year":"2007","journal-title":"Geophys. Res. Lett."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"2262","DOI":"10.1175\/1520-0493(1987)115<2262:AMENO>2.0.CO;2","article-title":"A model El Ni\u00f1o-Southern Oscillation","volume":"115","author":"Zebiak","year":"1987","journal-title":"Mon. Wea. Rev."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"3625","DOI":"10.1016\/j.jcp.2007.05.014","article-title":"El Ni\u00f1o prediction and predictability","volume":"227","author":"Chen","year":"2008","journal-title":"J. Comput. Phys."},{"key":"ref_23","first-page":"1635","article-title":"The allocation of informed trading across related markets: An analysis of the impact of changes in equity-option margin requirements","volume":"50","author":"Mayhew","year":"1995","journal-title":"J. Financ."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"375","DOI":"10.1146\/annurev-conmatphys-062910-140509","article-title":"Life is physics: Evolution as a collective phenomenon far from equilibrium","volume":"2","author":"Goldenfield","year":"2011","journal-title":"Ann. Rev. Condens. Matt. Phys."},{"key":"ref_25","unstructured":"K\u00a8ppers, B. (1990). Information and the Origin of Life, MIT Press."},{"key":"ref_26","unstructured":"Murray, J.D. (2000). Mathematical Biology, Springer-Verlag."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Allahverdyan, A.E., Janzing, D., and Mahler, G. (2009). Thermodynamic efficiency of information and heat flow. J. Stat. Mech., PO9011.","DOI":"10.1088\/1742-5468\/2009\/09\/P09011"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"275","DOI":"10.1103\/PhysRevE.59.275","article-title":"Thermodynamic depth of causal states: Objective complexity via minimal representation","volume":"59","author":"Crutchfield","year":"1999","journal-title":"Phys. Rev. E"},{"key":"ref_29","unstructured":"Clayton, P., and Davies, P.C.W. (2006). The Re-emergence of Emergence, Oxford University Press."},{"key":"ref_30","first-page":"126","article-title":"Top-down causation and emergence: Some comments on mechanisms","volume":"2","author":"Ellis","year":"2012","journal-title":"J. R. Soc. Interface"},{"key":"ref_31","first-page":"49","article-title":"Emergence, hierarchy and top-down causation in evolutionary biology","volume":"2","author":"Okasha","year":"2012","journal-title":"J. R. Soc. Interface"},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Walker, S.I., Cisneros, L., and Davies, P.C.W. (2012). Evolutionary transitions and top-down causation.","DOI":"10.1162\/978-0-262-31050-5-ch038"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"e11187","DOI":"10.1371\/journal.pone.0011187","article-title":"Evolution of cooperation on stochastic dynamical networks","volume":"5","author":"Wu","year":"2010","journal-title":"PLoS One"},{"key":"ref_34","unstructured":"Pope, S. (2011). Turbulent Flows, Cambridge University Press. [8th ed.]."},{"key":"ref_35","unstructured":"Faes, L., Nollo, G., Erla, S., Papadelis, C., Braun, C., and Porta, A. (September, January 31). Detecting Nonlinear Causal Interactions between Dynamical Systems by Non-uniform Embedding of Multiple Time Series. Proceedings of the Engineering in Medicine and Biology Society, Buenos Aires, Argentina."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Kantz, H., and Shreiber, T. (2004). Nonlinear Time Series Analysis, Cambridge University Press.","DOI":"10.1017\/CBO9780511755798"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physrep.2006.12.004","article-title":"Causality detection based on information-theoretic approach in time series analysis","volume":"441","author":"Palus","year":"2007","journal-title":"Phys. Rep."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"424","DOI":"10.2307\/1912791","article-title":"Investigating causal relations by econometric models and cross-spectral methods","volume":"37","author":"Granger","year":"1969","journal-title":"Econometrica"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1017\/S0022112084001750","article-title":"The emergence of isolated, coherent vortices in turbulence flows","volume":"146","author":"McWilliams","year":"1984","journal-title":"J. Fluid Mech."},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Salmon, R. (1998). Lectures on Geophysical Fluid Dynamics, Oxford University Press.","DOI":"10.1093\/oso\/9780195108088.001.0001"},{"key":"ref_41","unstructured":"Bar-Yam, Y. (1997). Dynamics of Complex Systems, Addison-Welsley Press."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1016\/0167-2789(94)90273-9","article-title":"The calculi of emergence: computation, dynamics, and induction induction. \u201cSpecial issue on the Proceedings of the Oji International Seminar: Complex Systems-From Complex Dynamics to Artifical Reality\u201d","volume":"75","author":"Crutchfield","year":"1994","journal-title":"Physica D"},{"key":"ref_43","first-page":"49","article-title":"Emergence as a construct: History and issues","volume":"1","author":"Goldstein","year":"1999","journal-title":"Emerg. Complex. Org."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"18","DOI":"10.1002\/cplx.10043","article-title":"The re-emergence of emergence: A venerable concept in search of a theory","volume":"7","author":"Corning","year":"2002","journal-title":"Complexity"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"1773","DOI":"10.1103\/PhysRevLett.60.1773","article-title":"Information transport in sptiotemporal systems","volume":"60","author":"Vastano","year":"1988","journal-title":"Phys. Rev. Lett."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1016\/S0167-2789(02)00432-3","article-title":"Information transfer in continuous processes","volume":"166","author":"Kaiser","year":"2002","journal-title":"Physica D"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"461","DOI":"10.1103\/PhysRevLett.85.461","article-title":"Measuring information transfer","volume":"85","author":"Schreiber","year":"2000","journal-title":"Phys. Rev. Lett."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physd.2007.04.002","article-title":"A rigorous formalism of information transfer between dynamical system components. I. Discrete mapping","volume":"231","author":"Liang","year":"2007","journal-title":"Physica D"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"244101","DOI":"10.1103\/PhysRevLett.95.244101","article-title":"Information transfer between dynamical system components","volume":"95","author":"Liang","year":"2005","journal-title":"Phys. Rev. Lett."},{"key":"ref_50","doi-asserted-by":"crossref","unstructured":"Lasota, A., and Mackey, M.C. (1994). Chaos, Fractals, and Noise: Stochastic Aspects of Dynamics, Springer.","DOI":"10.1007\/978-1-4612-4286-4"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1016\/j.physd.2006.12.012","article-title":"A rigorous formalism of information transfer between dynamical system components. II. Continuous flow","volume":"227","author":"Liang","year":"2007","journal-title":"Physica D"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1016\/j.dynatmoce.2011.03.003","article-title":"Uncertainty generation in deterministic fluid flows: Theory and applications with an atmospheric stability model","volume":"52","author":"Liang","year":"2011","journal-title":"Dyn. Atmos. Oceans"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"9558","DOI":"10.1073\/pnas.0703499104","article-title":"Information flow between subspaces of complex dynamical systems","volume":"104","author":"Majda","year":"2007","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"031113","DOI":"10.1103\/PhysRevE.78.031113","article-title":"Information flow within stochastic dynamical systems","volume":"78","author":"Liang","year":"2008","journal-title":"Phys. Rev. E"},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"12413","DOI":"10.1073\/pnas.230433997","article-title":"Remarkable statistical behavior for truncated Burgers-Hopf dynamics","volume":"97","author":"Majda","year":"2000","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_56","doi-asserted-by":"crossref","unstructured":"Gardiner, C.W. (1985). Handbook of Stochastic Methods for Physics, Chemistry, and the Natural Sciences, Springer-Verlag.","DOI":"10.1007\/978-3-662-02452-2"},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"437","DOI":"10.1137\/06066895X","article-title":"Multiscale window transform","volume":"6","author":"Liang","year":"2007","journal-title":"SIAM J. Multiscale Model. Simul."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"290","DOI":"10.1175\/2008JPO3950.1","article-title":"Multiscale processes and nonlinear dynamics of the circulation and upwelling events off Monterey Bay","volume":"39","author":"Liang","year":"2009","journal-title":"J. Phys. Oceanogr."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"2057","DOI":"10.1175\/1520-0469(2002)059<2057:MDPUUR>2.0.CO;2","article-title":"Measuring dynamical prediction utility using relative entropy","volume":"59","author":"Kleeman","year":"2002","journal-title":"J. Atmos. Sci."},{"key":"ref_60","unstructured":"Cover, T.M., and Thomas, J.A. (1991). Elements of Information Theory, Wiley."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"1073","DOI":"10.1088\/0253-6102\/49\/5\/01","article-title":"Emerging of stochastic dynamical equalities and steady state thermodynamics from Darwinian dynamics","volume":"49","author":"Ao","year":"2008","journal-title":"Commun. Theor. Phys."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"4811","DOI":"10.1175\/2008JCLI2193.1","article-title":"Interdecadal variation of ENSO predictability in multiple models","volume":"21","author":"Tang","year":"2008","journal-title":"J. Clim."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/15\/1\/327\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T21:44:23Z","timestamp":1760219063000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/15\/1\/327"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,1,18]]},"references-count":62,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2013,1]]}},"alternative-id":["e15010327"],"URL":"https:\/\/doi.org\/10.3390\/e15010327","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2013,1,18]]}}}