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INFORMATION FLOWS IN CAUSAL NETWORKS
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We use a notion of causal independence based on intervention, which is a fundamental concept of the theory of causal networks, to define a measure for the strength of a causal effect. We call this measure "information flow" and compare it with known information flow measures such as transfer entropy.
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“The assumption of conditional independence distinguishes IIT’s causal powers analysis from standard information-theoretic analyses of information flow [10, 27] and corresponds to an assumption that variables are “physical” units”
From Integrated information theory (IIT) 4.0: Formulating the properties of phenomenal existence in physical terms (2023) · cites this paper as [27]
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Digested Information as an Information Theoretic Motivation for Social Interaction
Salge, C., & Polani, D. (2011) · Journal of Artificial Societies and Social Simulation · doi:10.18564/jasss.1699 · Cited by 19
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“…a unified and ’coordinate-free’ framework for identifying limits on control (Touchette and Lloyd 2000, 2004) and for studying how information flows (Ay and Polani 2008) are structured under various scenarios (Klyubin 2007; Lungarella and Sporns 2006), but they also seem to be useful for guiding…”
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Computing the Integrated Information of a Quantum Mechanism
Albantakis, L., Prentner, R., & Durham, I. T. (2023) · Entropy · doi:10.3390/e25030449 · Cited by 26
“with p(st+1|st) = p(st+1|do(st)) [15,16,38,39], where the “do-operator” do(st) indicates that st is imposed by intervention.”
BackgroundProgress on network modeling and analysis of gut microecology: a review
Luo, M., Zhu, J., Jia, J., Zhang, H., & Zhao, J. (2024) · Applied and Environmental Microbiology · doi:10.1128/aem.00092-24 · Cited by 24
“Causal networks delineate the reciprocal causeandeffect connections between multiple microorganisms, where each node signifies a variable and directed arrows between nodes indicate the causeandeffect relationship or data generation process within microecology (72, 73).”
BackgroundInformation Flow in Computational Systems
Venkatesh, P., Dutta, S., & Grover, P. (2020) · IEEE Transactions on Information Theory · doi:10.1109/tit.2020.2987806 · Cited by 20
“There are several works in the literature that discuss measures of information flow in probabilistic graphical models [100], [101], but they are heavily inspired by causality and largely center around an interventionist approach.”
BackgroundHierarchical Behaviours: Getting the Most Bang for Your Bit
van Dijk, S. G., Polani, D., & Nehaniv, C. L. (2011) · Lecture notes in computer science · doi:10.1007/978-3-642-21314-4_43 · Cited by 19
Self-citationExploring Default Mode and Information Flow on the Web
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Causally-Informed Instance-Wise Feature Selection for Explaining Visual Classifiers
Tan, L. (2025) · Entropy · doi:10.3390/e27080814 · Cited by 16
“In contrast to prior methods such as L2X and INVASE, which may conflate statistical association with causation, our approach integrates a principled, information-theoretic measure of causal influence [12,13] into the feature selection process.”
InfluentialTransfer entropies within dynamical effects framework
Smirnov, D. A. (2020) · Physical review. E · doi:10.1103/physreve.102.062139 · Cited by 18
Measuring information transfer in a soft robotic arm
Nakajima, K., Schmidt, N., & Pfeifer, R. (2015) · Bioinspiration & Biomimetics · doi:10.1088/1748-3190/10/3/035007 · Cited by 16
“Similar considerations have also been presented in [32, 33].”
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Woodward, J. (2022) · Jerusalem studies in philosophy and history of science · doi:10.1007/978-3-030-99425-9_14 · Cited by 15
Biehl, M., Ikegami, T., & Polani, D. (2016) · Conference on Artificial Life (ALIFE) · doi:10.7551/978-0-262-33936-0-ch115 · Cited by 26
“As Ay and Polani (2008) note we can then define a partition of V into (V0, V1, V2, .”
Self-citationInformation Theoretic Causal Effect Quantification
Wieczorek, A., & Roth, V. (2019) · Entropy · doi:10.3390/e21100975 · Cited by 17
“Example of “vanishing directed information” [28,29,82].”
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van Elteren, C., Quax, R., & Sloot, P. M. A. (2022) · Physica A Statistical Mechanics and its Applications · doi:10.1016/j.physa.2022.126889 · Cited by 15
“…for determining how much information flow between two processes is truly causal; examples include (but not limited to) conditional mutual information under causal intervention [2], causation entropy [52], permutation conditional mutual information [45], time-delayed Shannon mutual information [31].”
BackgroundFoundations and Formalizations of Self-organization
Polani, D. (2007) · Advanced information and knowledge processing · doi:10.1007/978-1-84628-982-8_2 · Cited by 15
“In fact, the concept of information flow has been recently introduced to address exactly this question (Ay and Wennekers 2003; Klyubin et al. 2004; Ay and Krakauer 2006; Ay and Polani 2006), providing an information-theoretic approach to measure causal inference.”
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Data-driven dimensionality reduction and causal inference for spatiotemporal climate fields
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“Answering such questions often relies on quantifying the time-dependent “flow of information” along the underlying causal graph rather than discovering the graph itself [21, 46] (see also [51] in the context of information theory).”
BackgroundSignals That Make a Difference
Calcott, B., Pocheville, A., & Griffiths, P. E. (2017) · The British Journal for the Philosophy of Science · doi:10.1093/bjps/axx022 · Cited by 16
“By definition of information flow (Ay and Polani [2008]), the formula of the information flow from S to A conditional on W 0 reads as follows:”
BackgroundInfluentialDetecting directional couplings from multivariate flows by the joint distance distribution
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Measuring Sample Path Causal Influences With Relative Entropy
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“This example was originally posed in [33] and modified to include noise in [26].”
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Efficient Estimation of Information Transfer
Vicente, R., & Wibral, M. (2014) · Understanding complex systems · doi:10.1007/978-3-642-54474-3_2 · Cited by 14
Transient and equilibrium causal effects in coupled oscillators
Smirnov, D. A. (2018) · Chaos An Interdisciplinary Journal of Nonlinear Science · doi:10.1063/1.5017821 · Cited by 14
A refinement of the common cause principle
Ay, N. (2008) · Discrete Applied Mathematics · doi:10.1016/j.dam.2008.06.032 · Cited by 14
“The intention of this paper is to point out that we can say even more than that by using a quantitative extension of the common cause principle based on the notion of causal information flows [3].”
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Nakajima, K., Schmidt, N. M., & Pfeifer, R. (2016) · Cited by 14
Optimal high-level descriptions of dynamical systemsPreprint
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“Note that many measures of “information flow” in a stochastic dynamical system like causal information flow [4], transfer entropy [73, 65, 66], and Granger causality [46] are motivated by considering the flow between physically distinct subsystems of an overall dynamic system.”
BackgroundA Tale of Two Animats: What Does It Take to Have Goals?
Albantakis, L. (2018) · The frontiers collection · doi:10.1007/978-3-319-75726-1_2 · Cited by 13
“(“Does A always lead to B, or just sometimes?”) [13,15].”
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Sun, C., Shi, C., Jia, Y., & Wu, Y. (2023) · Proceedings of the AAAI Conference on Artificial Intelligence · doi:10.1609/aaai.v37i2.25334 · Cited by 13
“Definition 1 [Information flow from U to V in a directed acyclic graph (Ay and Polani 2008)].”
BackgroundGeometry of Information Integration
Амари, Ш.-И., Tsuchiya, N., & Oizumi, M. (2018) · Springer proceedings in mathematics & statistics · doi:10.1007/978-3-319-97798-0_1 · Cited by 12
“Note that in IIT, “causal” influences are quantified by Pearl’s intervention framework [11, 2, 12] attempting to quantify the “actual” causation.”
BackgroundInform: A toolkit for information-theoretic analysis of complex systems
Moore, D. G., Valentini, G., Walker, S. I., & Levin, M. G. (2017) · IEEE Symposium Series on Computational Intelligence (SSCI) · doi:10.1109/ssci.2017.8285197 · Cited by 11
“Block Entropy [42] X Cross Entropy [43] ×* (Multivariate) Mutual Information [43], [44] X Conditional Entropy [43] X Relative Entropy [43], [45] X Entropy Rate [43] X Active Information [39] X Transfer Entropy [37], [38], [46] X Separable Information [47] X Predictive Information [48], [49] X Excess Information [50],…”
BackgroundModel-based and model-free characterization of epidemic outbreaksPreprint
Dehning, J., Spitzner, F. P., Linden, M., et al. (2020) · medRxiv · doi:10.1101/2020.09.16.20187484 · Cited by 11
Information theory for complex systems scientists: What, why, and how
Varley, T. F. (2025) · Physics Reports · doi:10.1016/j.physrep.2025.09.007 · Cited by 38
Graph Neural Network Causal Explanation via Neural Causal Models
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Prediction and modularity in dynamical systems
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“To properly handle the latter, we utilize a notion of causality based on semantics of intervention (Pearl, 2000), recently developed in an information-theoretic direction by Ay and Polani (2008). In Pearl’s treatment, conditional probability distributions represent not only correlations, but also responses of variable…”
MethodOn quantification and maximization of information transfer in network dynamical systems
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“Information-driven self-organisation, by the means of maximising the one-step approximation of the PI has proved to produce a coordinated behaviour among physically coupled but otherwise independent agents [Zahedi et al., 2010, Ay et al., 2008].”
BackgroundSelf-citationChristian Guckelsberger, B., Salge, C., Saunders, R., et al. (2016) · Falmouth University Research Repository (FURR) (Falmouth University) · Cited by 9
Informational architecture across non-living and living collectives
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Fluctuation-response theorem for Kullback-Leibler divergences to quantify causation
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Phase-dynamic causalities within dynamical effects framework
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A Primer on Information Theory with Applications to Neuroscience
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“Thus it is a philosophical question to which extent transfer entropy can be used to infer some form of causality, a question that we will not further pursue here, rather pointing the reader to [86, 47, 67, 7].”
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Schamberg, G., Chapman, W. E., Xie, S., & Coleman, T. P. (2020) · Entropy · doi:10.3390/e22080854 · Cited by 9
“Drawing on the relationship between mutual information and statistical dependence, Ay and Polani [3] define an IT notion of causal independence, which unlike mutual information, is directed.”
BackgroundMethodInfluentialQuantifying Proportionality and the Limits of Higher-Level Causation and Explanation
Gebharter, A., & Eronen, M. I. (2021) · The British Journal for the Philosophy of Science · doi:10.1086/714818 · Cited by 9
Information Flow and Entropy Production on Bayesian Networks
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Information transfers and flows in Markov chains as dynamical causal effects
Smirnov, D. A. (2024) · Chaos An Interdisciplinary Journal of Nonlinear Science · doi:10.1063/5.0189544 · Cited by 8
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“In the causal modelling of brain networks, dynamic causal modelling [15, 16] and transfer entropy [17–20] are also prevalent but essentially based on Bayesian predictions.”
Information Transfer by Particles in Cellular Automata
Lizier, J. T., Prokopenko, M., & Zomaya, A. Y. (2007) · Springer eBooks · doi:10.1007/978-3-540-76931-6_5 · Cited by 8
“Reference [ 18 ] points out that the transfer entropy can be viewed as a conditional mutual information (6), that is the average information in the source about the next state of the destination that was not already contained in the destination’s past.”
BackgroundMethodWhat caused what? An irreducible account of actual causation.Preprint
Albantakis, L., Marshall, W., Hoel, E. P., & Tononi, G. (2017) · arXiv (Cornell University) · Cited by 9
Kvam, P. D., Sokratous, K., Fitch, A. K., & Hintze, A. (2024) · Decision · doi:10.1037/dec0000237 · Cited by 10
“Similarly, the amount of information passing through a system, or how much information needs to be stored in hidden states in order to predict output states, can be quantified (Ay & Polani, 2008).”
Unsupervised Causal Binary Concepts Discovery with VAE for Black-Box Model Explanation
Tran, T. Q., Fukuchi, K., Akimoto, Y., & Sakuma, J. (2022) · Proceedings of the AAAI Conference on Artificial Intelligence · doi:10.1609/aaai.v36i9.21195 · Cited by 8
“Definition 1 (Information flow from U to V in a directed acyclic graph G(Ay and Polani 2008)).”
BackgroundBiological Information as Choice and Construction
Pocheville, A. (2018) · Philosophy of Science · doi:10.1086/699832 · Cited by 7
Coping with dating errors in causality estimation
Smirnov, D. A., Marwan, N., Breitenbach, S. F. M., Lechleitner, F. A., & Kurths, J. (2017) · Europhysics Letters (EPL) · doi:10.1209/0295-5075/117/10004 · Cited by 7
Coping with dating errors in causality estimation
Smirnov, D. A., Marwan, N., Breitenbach, S. F. M., et al. (2017) · Northumbria Research Link (Northumbria University) · Cited by 7
Intrinsic units: identifying a system’s causal grain
Marshall, W., Findlay, G., Albantakis, L., & Tononi, G. (2026) · Neuroscience of Consciousness · doi:10.1093/nc/niag013 · Cited by 12
“…that we can determine the conditional probabilities in (1) for every system state, with p ( u (cid:2) | u ) = p ( u (cid:2) | do ( u )) , where the “do-operator” do ( u ) indicates that u is imposed by intervention (Pearl, 2000; Ay and Polani, 2008; Janzing et al. 2013; Albantakis et al. 2019).”
Design specifications for cellular regulation
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“It is equivalent to potential causal information flow as defined in [21].”
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“1 includes system states that may not be observed during the dynamical evolution of the system, but require system interventions (Pearl, 2000; Ay and Polani, 2008).”
BackgroundInformation theoretic measures of causal influences during transient neural events
Shao, K., Logothetis, N. K., & Besserve, M. (2023) · Frontiers in Network Physiology · doi:10.3389/fnetp.2023.1085347 · Cited by 6
“It can be shown that such change does not preserve the SCM structure, and leads to violations of the Markov properties due to the implicit dependency on the mechanisms relating X(2)p,t and X 1 p,t, which manifest themselves through the p(X(2)p,t|X(1)p,t) term of the marginalization equation (Ay and Polani, 2008; Janzi…”
BackgroundBuilding compressed causal models of the world
Kinney, D. B., & Lombrozo, T. (2024) · Cognitive Psychology · doi:10.1016/j.cogpsych.2024.101682 · Cited by 6
An improved transfer entropy method for establishing causal effects in synchronizing oscillators
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How cognitive and environmental constraints influence the reliability of simulated animats in groups
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“The ^ (”hat”-symbol) above the probability function p indicates that probabilities are interventional (obtained from system perturbations) rather than observational [19,47].”
BackgroundA study for multiscale information transfer measures based on conditional mutual information
Wan, X., & Xu, L. (2018) · PLoS ONE · doi:10.1371/journal.pone.0208423 · Cited by 6
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“X and Y mutually and deterministically cause one another through a copy operation [91, 8]: X(t) copies Y (t 1) and vice versa.”
BackgroundA Synergistic Perspective on Multivariate Computation and Causality in Complex Systems
Varley, T. F. (2024) · Entropy · doi:10.3390/e26100883 · Cited by 6
“The link between information flow, causality, and structure inference is a rich field that we cannot completely survey (for further discussion, see [51–53]), and the problem of true causal information decomposition remains an outstanding issue.”
BackgroundA Fluctuation–Dissipation Theorem Perspective on Radiative Responses to Temperature Perturbations
Falasca, F., Basinski‐Ferris, A., Zanna, L., & Zhao, M. (2025) · Journal of Climate · doi:10.1175/jcli-d-24-0479.1 · Cited by 5
Causal Entropy and Information Gain for Measuring Causal Control
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Parametric and Non-parametric Criteria for Causal Inference from Time-Series
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“Consequently, Granger causality measures, and in particular transfer entropy, cannot be used in general as measures of the strength of causal effects [4, 39].”
BackgroundCausal Effects for Prediction and Deliberative Decision Making of Embodied Systems
Ay, N., & Ghazi-Zahedi, K. (2013) · Springer eBooks · doi:10.1007/978-94-007-4792-0_67 · Cited by 5
“We use the formalism of Bayesian networks to study the causal relations in the SML (see previous work [3] and [1] in this context).”
MethodSelf-citationDiversity and similarity of information transfers and flows in continuous-time stochastic systems
Smirnov, D. A. (2025) · Chaos An Interdisciplinary Journal of Nonlinear Science · doi:10.1063/5.0283758 · Cited by 4
Smirnov, D. A. (2024) · Europhysics Letters (EPL) · doi:10.1209/0295-5075/ad7285 · Cited by 4
CiRLExplainer: Causality-Inspired Explainer for Graph Neural Networks via Reinforcement Learning
Hu, W., Wu, J. X., & Qian, Q. (2025) · IEEE Transactions on Neural Networks and Learning Systems · doi:10.1109/tnnls.2025.3543070 · Cited by 4
“OrphicX [44] initially establishes a causal graph, quantifying causal information from latent features to model predictions using measured concepts of information flow [45].”
Cascades Towards Noise-Induced Transitions on Networks Revealed Using Information Flows
van Elteren, C., Quax, R., & Sloot, P. M. A. (2024) · Entropy · doi:10.3390/e26121050 · Cited by 5
Towards automated map updating for mobile robot localization
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“In order to compare the performance of these two structures, we have compared the causal Information Flow parameter described in [9] for the two networks from fig.”
MethodModeling the global ionospheric variations based on complex network
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Quantifying the Autonomy of Structurally Diverse Automata: A Comparison of Candidate Measures
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“tional probability distributions instead of observed distributions [1,20,47,63].”
BackgroundSpecify Robust Causal Representation from Mixed Observations
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Guofu, X., Yang, Y., & Liu, X. (2011) · Lecture notes in computer science · doi:10.1007/978-3-642-23896-3_59 · Cited by 4
Cyclic and multilevel causation in evolutionary processes
Warrell, J., & Gerstein, M. (2020) · Biology & Philosophy · doi:10.1007/s10539-020-09753-3 · Cited by 4
“such as information flow [3], effective information [14], causal specificity [12] and causal strength [18].”
BackgroundInformational Principles of Perception-Action Loops and Collective Behaviours
Capdepuy, P. (2011) · University of Hertfordshire Research Archive (University of Hertfordshire) · Cited by 4
A general framework for characterizing optimal communication in brain networks
Fakhar, K., Hadaeghi, F., Seguin, C., Dixit, S., Messé, A., Zamora‐López, G., Mišić, B., & Hilgetag, C. C. (2024) · eLife · doi:10.7554/elife.101780.3 · Cited by 10
A More Basic Version of Agency? As If!
McGregor, S. (2016) · Lecture notes in computer science · doi:10.1007/978-3-319-43488-9_17 · Cited by 3
Global climate change: Analyzing anthropogenic warming and causality
Stips, A., Macias, D. M., Coughlan, C., et al. (2015) · doi:10.1109/oceans-genova.2015.7271393 · Cited by 3
“It is based on the recently rigorized fundamental physical notion namely information flow, which, for two events, provides not only the magnitude but also the direction of the cause-effect relation lying in between; in this sense, it is an appropriate measure of causality [11].”
BackgroundLocal Information Maximisation creates Emergent Flocking Behaviour
Salge, C., & Polani, D. (2011) · University of Hertfordshire Research Archive (University of Hertfordshire) · Cited by 3
Self-citationAdding causality to the information-theoretic perspective on individuality
Bourrat, P. (2024) · European Journal for Philosophy of Science · doi:10.1007/s13194-023-00566-1 · Cited by 3
Models of Information Processing in the Sensorimotor Loop
Polani, D., & Möller, M. (2008) · Information Theory and Statistical Learning · doi:10.1007/978-0-387-84816-7_12 · Cited by 3
“for general CBNs [3].”
Self-citationExact Algorithms for MRE Inference
Zhu, X., & Yuan, C. (2016) · Journal of Artificial Intelligence Research · doi:10.1613/jair.4867 · Cited by 3
Estimating causal effects by bounding confounding
Geiger, P., Janzing, D., & Schölkopf, B. (2014) · MPG.PuRe (Max Planck Society) · Cited by 3
Domain Generalization Study of Empirical Risk Minimization From Causal Perspectives
Mo, Z., Zhang, Z., & Tsui, K. (2025) · IEEE Transactions on Multimedia · doi:10.1109/tmm.2025.3535276 · Cited by 2
Lee, J. (2025) · Physical review. E · doi:10.1103/physreve.111.024308 · Cited by 3
Falasca, F. (2025) · Physical Review Research · doi:10.1103/2f4r-k8lr · Cited by 8
Information flow in learning a coin-tossing game
Sato, Y., & Ay, N. (2016) · Nonlinear Theory and Its Applications IEICE · doi:10.1587/nolta.7.118 · Cited by 2
“A measure of causal information flow [2] was applied to various of these structures, verifying its consistency with the prescribed causal connections.”
BackgroundMethodSelf-citation
Metadata
- Authors
- AY, N., & POLANI, D.
- Affiliations
- Santa Fe Institute (US)
- University of Hertfordshire (GB)
- Venue
- Advances in Complex Systems
- Publisher
- World Scientific Pub Co Pte Lt
- Published
- 2008-02
- Volume
- 11
- Issue
- 01
- Pages
- 17-41
- ISSN
- 0219-5259 (ISSN-L)
- 1793-6802
- OpenAlex
- W2060841334
- Semantic Scholar
- cea9940860abc3dea72d9ebdbb699a9d885c0724
- Cited by
- 287
- References
- 18
- Citations by year
- 2026: 12 · 2025: 20 · 2024: 19 · 2023: 23 · 2022: 16 · 2021: 20 · 2020: 14 · 2019: 9 · 2018: 16 · 2017: 22 · 2016: 21 · 2015: 15 · 2014: 18 · 2013: 20 · 2012: 9
- FWCI
- 12.16
- Type
- journal article
- Language
- en
- Open access
- green
- uhra.herts.ac.uk
- Locations
- Advances in Complex Systems (publishedVersion)
- University of Hertfordshire Research Archive (University of Hertfordshire) (submittedVersion)
- RePEc: Research Papers in Economics (submittedVersion)
- Max Planck Institute for Plasma Physics (submittedVersion)
- Funders
- Santa Fe Institute
- Topics
- Physical Sciences › Computer Science › Artificial Intelligence › Bayesian Modeling and Causal Inference
- Social Sciences › Arts and Humanities › History and Philosophy of Science › Philosophy of Biology and Scientific Explanation
- Social Sciences › Economics, Econometrics and Finance › Economics and Econometrics › Complex Systems and Time Series Analysis
- Keywords
- information flow, causal networks, causal independence, intervention, transfer entropy, causal inference
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