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Synergy and Redundancy Measures: Theory and Applications to Characterize Complex Systems, Build Models, and Shape Neural Network Representations

A special issue of Entropy (ISSN 1099-4300). This special issue belongs to the section "Information Theory, Probability and Statistics".

Deadline for manuscript submissions: 20 September 2026 | Viewed by 941

Special Issue Editor


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Guest Editor
Department of Computer Science, School of Science & Technology, City St George’s, University of London, London EC1V 0HB, UK
Interests: data analysis; causal inference; dimensionality reduction; neuroscience; sensitivity analysis; structure learning; information decomposition; information bottlenecks; fairness
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Special Issue Information

Dear Colleagues,

An important aspect of how sources of information are distributed across a set of variables concerns whether different variables provide redundant, unique, or synergistic information when combined with other variables. Intuitively, variables share redundant information if each variable individually carries the same information carried by other variables. Information carried by a certain variable is unique if it is not carried by any other variables or their combination, and a group of variables carries synergistic information if some information arises only when they are combined.

Recent advances have contributed to build an information-theoretic framework to determine the distribution and nature of information extractable from multivariate datasets. Measures of redundant, unique, or synergistic information characterize dependencies between the parts of a multivariate system and can help to understand its function and mechanisms. Furthermore, these measures are also useful to analyze how information is distributed across layers in neural networks or can be used as cost functions to shape the structure of data representations learned by the networks.

This Special Issue welcomes submissions that contribute to advances both in theoretical formulations and in the applications of information-theoretic measures of synergy and redundancy. This Special Issue encompasses the following:

  • Advances in a multivariate formulation of redundancy measures or in the comparison of alternative proposals, addressing their distinctive power to capture relevant structures in both synthetic and experimental datasets.
  • Applications to understand interactions in real complex systems.
  • Advances in the estimation of information-theoretic quantities from high-dimensional datasets.
  • Applications for feature selection and sensitivity analysis.
  • Analyses of the distribution and nature of information across layers in neural networks.
  • Designs of deep learning models to obtain robust or disentangled data representations.
  • Fairness analyses.

Dr. Daniel Chicharro
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 250 words) can be sent to the Editorial Office for assessment.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Entropy is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • mutual information
  • synergy
  • redundancy
  • unique information
  • neural networks
  • disentanglement
  • feature extraction
  • representation learning
  • partial information decomposition
  • fairness

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Published Papers (1 paper)

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Research

15 pages, 252 KB  
Article
What Is Redundancy?
by Clifford Bohm, Christoph Adami and Arend Hintze
Entropy 2026, 28(2), 167; https://doi.org/10.3390/e28020167 - 1 Feb 2026
Viewed by 540
Abstract
Redundancy is a central yet persistently ambiguous concept in multivariate information theory. Across the literature, the same term is used to describe fundamentally distinct phenomena. Operational redundancy concerns how different inputs relate to the prediction of output states, while informational redundancy concerns content [...] Read more.
Redundancy is a central yet persistently ambiguous concept in multivariate information theory. Across the literature, the same term is used to describe fundamentally distinct phenomena. Operational redundancy concerns how different inputs relate to the prediction of output states, while informational redundancy concerns content overlap among inputs relevant to an output. These notions are routinely conflated in decompositions of mutual information, leading to incompatible definitions, contradictory interpretations, and apparent paradoxes—particularly when inputs are statistically independent. We argue that the difficulty in defining redundancy is not primarily technical, but conceptual: the field has not converged on what redundancy is meant to signify. We formalize this distinction by identifying two classes of redundancy. Operational redundancy encompasses task-relative properties and covers conditions when inputs are sufficient or substitutable for prediction. Informational redundancy concerns shared content among inputs, grounded in mutual information between them. Using functional examples and biased input ensembles, we demonstrate the practical distinction between these classes: inputs with no informational overlap can exhibit operational redundancy, while partial observation can induce statistical correlations that create content overlap without reflecting the underlying functional structure. We conclude by proposing a clear separation of these concepts and outlining minimal commitments for each. This separation clarifies why redundancy remains elusive, why no single measure can satisfy all intuitions, and how future work can proceed without redefining information itself. Full article
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