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Topological Data Analysis Meets Information Theory. New Perspectives for the Analysis of Higher-Order Interactions in Complex Systems

A special issue of Entropy (ISSN 1099-4300). This special issue belongs to the section "Multidisciplinary Applications".

Deadline for manuscript submissions: 30 November 2024 | Viewed by 9088

Special Issue Editors

Dr. Tiziano Squartini

E-Mail Website
Guest Editor
IMT School for Advanced Studies, Piazza San Francesco 19, 55100 Lucca, Italy
Interests: complexity; graph theory; information theory; statistical mechanics of networks; pattern detection; network reconstruction; graph combinatorics; systemic risk estimation; (mis)information spreading on social networks; functional brain network analysis; higher-order interactions
Dr. Fernando Antônio Nóbrega Santos

E-Mail Website
Guest Editor
1. Korteweg-de Vries Institute for Mathematics, University of Amsterdam, 1105 AZ Amsterdam, The Netherlands
2. Institute for Advanced Studies (IAS), University of Amsterdam, 1105 AZ Amsterdam, The Netherlands
Interests: statistical mechanics; applied topology and geometry; network science; information theory; neuroscience
Dr. Rossana Mastrandrea

E-Mail Website
Guest Editor
Networks Unit, IMT School for Advanced Studies, Piazza San Francesco 19, 55100 Lucca, Italy
Interests: complex networks; graph theory; statistical physics; randomization techniques for graphs; higher-order interactions; social networks; economics; neuroscience
Special Issues, Collections and Topics in MDPI journals
Dr. Marco Piangerelli

E-Mail Website
Guest Editor
Computer Science Division, School of Science and Technology, University of Camerino, 62032 Camerino, Italy
Interests: complexity; topological data analysis; higher-order interactions; self-adaptive systems; deep learning; information theory; pattern recognition; interpretable machine learning; artificial intelligence; intelligent manufacturing; computer vision; signal processing; robotics
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Complexity lies in the rich variety of interactions taking place among the constituents of a given system. While research has mostly focused on pairwise relationships, recent work has shown that neglecting higher-order interactions can lead to a poor representation of the same systems. Examples of these "group-wise" interactions can be found in many fields, including neuroscience, biology, finance and sociology. Researchers have developed various approaches to quantify and investigate these interactions, including Topological Data Analysis and Information Theory. While the former focuses on defining the structures to be considered and their topological invariants, the latter deals with inferring higher-order interdependencies among the system constituents using, for instance, multivariate information theory. This Special Issue aims to bridge the perspectives of Complex Systems, Topological Data Analysis and Information Theory to better understand higher-order structures. Researchers are encouraged to explore commonalities between these approaches, their integration and the challenges they bring to application domains.

Both theoretical and applied contributions about the following topics fall within the scope of this Special Issue (though well-motivated systematic literature reviews on the same topics may be considered):

- Higher-order representations of interacting systems (e.g., hypergraphs, simplicial complexes);

- Topological data analysis and algebraic topology (e.g., persistent homology, dimension reduction);

- Multivariate information theory for higher-order inference (e.g., higher-order pattern detection);

- Generative models for higher-order interactions;

- Dynamical models of higher-order interactions;

- Entropy (e.g. persistent, Renyi, Shannon, transfer, Tsallis);

- Applications in economics and finance (e.g., cryptocurrencies), neurosciences (e.g., structural and functional brain networks, epilepsy, Alzheimer's disease and dementia), chemistry and biology (protein interactions), cybersecurity, artificial intelligence, machine/deep learning and robotics.

Dr. Tiziano Squartini
Dr. Fernando Antônio Nóbrega Santos
Dr. Rossana Mastrandrea
Dr. Marco Piangerelli
Guest Editors

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 100 words) can be sent to the Editorial Office for announcement on this website.

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

  • complexity
  • information theory
  • topological data analysis
  • higher-order interactions
  • simplicial complexes
  • hypergraphs

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Published Papers (7 papers)

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Research

35 pages, 2268 KiB  
Article
Efficient Search Algorithms for Identifying Synergistic Associations in High-Dimensional Datasets
by Cillian Hourican, Jie Li, Pashupati P. Mishra, Terho Lehtimäki, Binisha H. Mishra, Mika Kähönen, Olli T. Raitakari, Reijo Laaksonen, Liisa Keltikangas-Järvinen, Markus Juonala and Rick Quax
Entropy 2024, 26(11), 968; https://doi.org/10.3390/e26110968 - 11 Nov 2024
Viewed by 522
Abstract
In recent years, there has been a notably increased interest in the study of multivariate interactions and emergent higher-order dependencies. This is particularly evident in the context of identifying synergistic sets, which are defined as combinations of elements whose joint interactions result in [...] Read more.
In recent years, there has been a notably increased interest in the study of multivariate interactions and emergent higher-order dependencies. This is particularly evident in the context of identifying synergistic sets, which are defined as combinations of elements whose joint interactions result in the emergence of information that is not present in any individual subset of those elements. The scalability of frameworks such as partial information decomposition (PID) and those based on multivariate extensions of mutual information, such as O-information, is limited by combinational explosion in the number of sets that must be assessed. In order to address these challenges, we propose a novel approach that utilises stochastic search strategies in order to identify synergistic triplets within datasets. Furthermore, the methodology is extensible to larger sets and various synergy measures. By employing stochastic search, our approach circumvents the constraints of exhaustive enumeration, offering a scalable and efficient means to uncover intricate dependencies. The flexibility of our method is illustrated through its application to two epidemiological datasets: The Young Finns Study and the UK Biobank Nuclear Magnetic Resonance (NMR) data. Additionally, we present a heuristic for reducing the number of synergistic sets to analyse in large datasets by excluding sets with overlapping information. We also illustrate the risks of performing a feature selection before assessing synergistic information in the system. Full article
(This article belongs to the Special Issue Topological Data Analysis Meets Information Theory. New Perspectives for the Analysis of Higher-Order Interactions in Complex Systems)
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24 pages, 1012 KiB  
Article
Bias in O-Information Estimation
by Johanna Gehlen, Jie Li, Cillian Hourican, Stavroula Tassi, Pashupati P. Mishra, Terho Lehtimäki, Mika Kähönen, Olli Raitakari, Jos A. Bosch and Rick Quax
Entropy 2024, 26(10), 837; https://doi.org/10.3390/e26100837 - 30 Sep 2024
Viewed by 1055
Abstract
Higher-order relationships are a central concept in the science of complex systems. A popular method of attempting to estimate the higher-order relationships of synergy and redundancy from data is through the O-information. It is an information–theoretic measure composed of Shannon entropy terms that [...] Read more.
Higher-order relationships are a central concept in the science of complex systems. A popular method of attempting to estimate the higher-order relationships of synergy and redundancy from data is through the O-information. It is an information–theoretic measure composed of Shannon entropy terms that quantifies the balance between redundancy and synergy in a system. However, bias is not yet taken into account in the estimation of the O-information of discrete variables. In this paper, we explain where this bias comes from and explore it for fully synergistic, fully redundant, and fully independent simulated systems of n=3 variables. Specifically, we explore how the sample size and number of bins affect the bias in the O-information estimation. The main finding is that the O-information of independent systems is severely biased towards synergy if the sample size is smaller than the number of jointly possible observations. This could mean that triplets identified as highly synergistic may in fact be close to independent. A bias approximation based on the Miller–Maddow method is derived for the O-information. We find that for systems of n=3 variables the bias approximation can partially correct for the bias. However, simulations of fully independent systems are still required as null models to provide a benchmark of the bias of the O-information. Full article
(This article belongs to the Special Issue Topological Data Analysis Meets Information Theory. New Perspectives for the Analysis of Higher-Order Interactions in Complex Systems)
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24 pages, 1037 KiB  
Article
Inferring Dealer Networks in the Foreign Exchange Market Using Conditional Transfer Entropy: Analysis of a Central Bank Announcement
by Aleksander Janczewski, Ioannis Anagnostou and Drona Kandhai
Entropy 2024, 26(9), 738; https://doi.org/10.3390/e26090738 - 29 Aug 2024
Viewed by 909
Abstract
The foreign exchange (FX) market has evolved into a complex system where locally generated information percolates through the dealer network via high-frequency interactions. Information related to major events, such as economic announcements, spreads rapidly through this network, potentially inducing volatility, liquidity disruptions, and [...] Read more.
The foreign exchange (FX) market has evolved into a complex system where locally generated information percolates through the dealer network via high-frequency interactions. Information related to major events, such as economic announcements, spreads rapidly through this network, potentially inducing volatility, liquidity disruptions, and contagion effects across financial markets. Yet, research on the mechanics of information flows in the FX market is limited. In this paper, we introduce a novel approach employing conditional transfer entropy to construct networks of information flows. Leveraging a unique, high-resolution dataset of bid and ask prices, we investigate the impact of an announcement by the European Central Bank on the information transfer within the market. During the announcement, we identify key dealers as information sources, conduits, and sinks, and, through comparison to a baseline, uncover shifts in the network topology. Full article
(This article belongs to the Special Issue Topological Data Analysis Meets Information Theory. New Perspectives for the Analysis of Higher-Order Interactions in Complex Systems)
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15 pages, 1641 KiB  
Article
Singular-Value-Decomposition-Based Matrix Surgery
by Jehan Ghafuri and Sabah Jassim
Entropy 2024, 26(8), 701; https://doi.org/10.3390/e26080701 - 17 Aug 2024
Viewed by 903
Abstract
This paper is motivated by the need to stabilise the impact of deep learning (DL) training for medical image analysis on the conditioning of convolution filters in relation to model overfitting and robustness. We present a simple strategy to reduce square matrix condition [...] Read more.
This paper is motivated by the need to stabilise the impact of deep learning (DL) training for medical image analysis on the conditioning of convolution filters in relation to model overfitting and robustness. We present a simple strategy to reduce square matrix condition numbers and investigate its effect on the spatial distributions of point clouds of well- and ill-conditioned matrices. For a square matrix, the SVD surgery strategy works by: (1) computing its singular value decomposition (SVD), (2) changing a few of the smaller singular values relative to the largest one, and (3) reconstructing the matrix by reverse SVD. Applying SVD surgery on CNN convolution filters during training acts as spectral regularisation of the DL model without requiring the learning of extra parameters. The fact that the further away a matrix is from the non-invertible matrices, the higher its condition number is suggests that the spatial distributions of square matrices and those of their inverses are correlated to their condition number distributions. We shall examine this assertion empirically by showing that applying various versions of SVD surgery on point clouds of matrices leads to bringing their persistent diagrams (PDs) closer to the matrices of the point clouds of their inverses. Full article
(This article belongs to the Special Issue Topological Data Analysis Meets Information Theory. New Perspectives for the Analysis of Higher-Order Interactions in Complex Systems)
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23 pages, 7837 KiB  
Article
Understanding Higher-Order Interactions in Information Space
by Herbert Edelsbrunner, Katharina Ölsböck and Hubert Wagner
Entropy 2024, 26(8), 637; https://doi.org/10.3390/e26080637 - 27 Jul 2024
Viewed by 1317
Abstract
Methods used in topological data analysis naturally capture higher-order interactions in point cloud data embedded in a metric space. This methodology was recently extended to data living in an information space, by which we mean a space measured with an information theoretical distance. [...] Read more.
Methods used in topological data analysis naturally capture higher-order interactions in point cloud data embedded in a metric space. This methodology was recently extended to data living in an information space, by which we mean a space measured with an information theoretical distance. One such setting is a finite collection of discrete probability distributions embedded in the probability simplex measured with the relative entropy (Kullback–Leibler divergence). More generally, one can work with a Bregman divergence parameterized by a different notion of entropy. While theoretical algorithms exist for this setup, there is a paucity of implementations for exploring and comparing geometric-topological properties of various information spaces. The interest of this work is therefore twofold. First, we propose the first robust algorithms and software for geometric and topological data analysis in information space. Perhaps surprisingly, despite working with Bregman divergences, our design reuses robust libraries for the Euclidean case. Second, using the new software, we take the first steps towards understanding the geometric-topological structure of these spaces. In particular, we compare them with the more familiar spaces equipped with the Euclidean and Fisher metrics. Full article
(This article belongs to the Special Issue Topological Data Analysis Meets Information Theory. New Perspectives for the Analysis of Higher-Order Interactions in Complex Systems)
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15 pages, 2895 KiB  
Article
Patterns in Temporal Networks with Higher-Order Egocentric Structures
by Beatriz Arregui-García, Antonio Longa, Quintino Francesco Lotito, Sandro Meloni and Giulia Cencetti
Entropy 2024, 26(3), 256; https://doi.org/10.3390/e26030256 - 13 Mar 2024
Cited by 4 | Viewed by 1580
Abstract
The analysis of complex and time-evolving interactions, such as those within social dynamics, represents a current challenge in the science of complex systems. Temporal networks stand as a suitable tool for schematizing such systems, encoding all the interactions appearing between pairs of individuals [...] Read more.
The analysis of complex and time-evolving interactions, such as those within social dynamics, represents a current challenge in the science of complex systems. Temporal networks stand as a suitable tool for schematizing such systems, encoding all the interactions appearing between pairs of individuals in discrete time. Over the years, network science has developed many measures to analyze and compare temporal networks. Some of them imply a decomposition of the network into small pieces of interactions; i.e., only involving a few nodes for a short time range. Along this line, a possible way to decompose a network is to assume an egocentric perspective; i.e., to consider for each node the time evolution of its neighborhood. This was proposed by Longa et al. by defining the “egocentric temporal neighborhood”, which has proven to be a useful tool for characterizing temporal networks relative to social interactions. However, this definition neglects group interactions (quite common in social domains), as they are always decomposed into pairwise connections. A more general framework that also allows considering larger interactions is represented by higher-order networks. Here, we generalize the description of social interactions to hypergraphs. Consequently, we generalize their decomposition into “hyper egocentric temporal neighborhoods”. This enables the analysis of social interactions, facilitating comparisons between different datasets or nodes within a dataset, while considering the intrinsic complexity presented by higher-order interactions. Even if we limit the order of interactions to the second order (triplets of nodes), our results reveal the importance of a higher-order representation.In fact, our analyses show that second-order structures are responsible for the majority of the variability at all scales: between datasets, amongst nodes, and over time. Full article
(This article belongs to the Special Issue Topological Data Analysis Meets Information Theory. New Perspectives for the Analysis of Higher-Order Interactions in Complex Systems)
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28 pages, 6348 KiB  
Article
A Probabilistic Result on Impulsive Noise Reduction in Topological Data Analysis through Group Equivariant Non-Expansive Operators
by Patrizio Frosini, Ivan Gridelli and Andrea Pascucci
Entropy 2023, 25(8), 1150; https://doi.org/10.3390/e25081150 - 31 Jul 2023
Cited by 1 | Viewed by 1136
Abstract
In recent years, group equivariant non-expansive operators (GENEOs) have started to find applications in the fields of Topological Data Analysis and Machine Learning. In this paper we show how these operators can be of use also for the removal of impulsive noise and [...] Read more.
In recent years, group equivariant non-expansive operators (GENEOs) have started to find applications in the fields of Topological Data Analysis and Machine Learning. In this paper we show how these operators can be of use also for the removal of impulsive noise and to increase the stability of TDA in the presence of noisy data. In particular, we prove that GENEOs can control the expected value of the perturbation of persistence diagrams caused by uniformly distributed impulsive noise, when data are represented by L-Lipschitz functions from R to R. Full article
(This article belongs to the Special Issue Topological Data Analysis Meets Information Theory. New Perspectives for the Analysis of Higher-Order Interactions in Complex Systems)
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