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Entropy-Based Applications in Sociophysics II

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

Deadline for manuscript submissions: 30 September 2025 | Viewed by 3880

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Special Issue Editors

Prof. Dr. Marcel Ausloos

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1.School of Business, University of Leicester, Brookfield, Leicester, LE2 1RQ, UK 2.Babeş-Bolyai University, 400084, Cluj-Napoca, Romania 3.Department of Statistics and Econometrics, Bucharest University of Economic Studies, 010552, Bucharest, Romania 4.Group of Researchers Applying Physics in Economy and Sociology (GRAPES), Beauvallon, Sart Tilman, B-4031, Angleur, Liège, Belgium
Interests: econophysics; sociophysics; nonlinear dynamics; nonequilibrium systems; networks; phase transitions; growth (and decay) models; fractals; scientometrics; statistical physics; applied mathematics
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Dr. Francisco W. De Sousa Lima

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Department of Physics, Federal University of Piauí (UFPI), Teresina 64049550, Brazil
Interests: Monte Carlo simulation; networks; critical exponents; disorder and Ising models
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Special Issue Information

Dear Colleagues,

The study of sociophysics has greatly increased in the last two decades. The models used in sociophysics mainly envisage the macroscopic dynamics of social systems or/and networks. Then, the statistical physics tools successfully applied in treating diverse systems in the physical world are used to find extensive applications in problems related to such topics. Stauffer, in 2012, raised an interesting and rather fundamental question: does sociophysics have any practical applications? The answer came in 2017 from Galam with a model that uses local-majority-rule arguments and obeys threshold dynamics. Within this perspective, the dynamics of opinions obey discoverable universal quantitative laws and can be modeled in the same way that scientists model the physical world. As a consequence, inspiring very active practitioners, opinion-dynamics models have become a mainstream of research in sociophysics. In these models, opinion entropy, based on Shannon entropy, is a useful tool to evaluate the uncertainty of opinions, whence for exploring the dynamics of opinion entropy and “controlling” the formation of public opinion.

As a result of cross-fertilization, mixed research fields use knowledge, methodologies, methods and tools of (statistical) physics (and thermodynamics) for modelling, explaining and forecasting social phenomena. No need to say that there are many concepts which have not been used or objectives which have not yet been considered.

Evolving, we wish to give opportunities to researchers to present studies which provide not only standard statistical physics modelling techniques, but also rather novel ways of analyses, sourced from entropy, whence complexity, theoretical and practical ideas.

Thus, this Special Issue is intended to contain articles of prominent and creative researchers in the field of sociophysics. We hope that this Special Issue will be an inspiration in these fascinating areas of broadly-understood modern and challenging applications of (statistical) physics and thermodynamics ideas and concepts.

Prof. Dr. Marcel Ausloos
Dr. Francisco W. De Sousa Lima
Guest Editors

Manuscript Submission Information

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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

opinion dynamics
votes
consensus
econophysics
networks
based agents models
phase transitions
equilibrium and non-equilibrium concepts

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Published Papers (7 papers)

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Research

14 pages, 617 KiB

Open AccessArticle

Iterative Forecasting of Financial Time Series: The Greek Stock Market from 2019 to 2024

by Evangelos Bakalis and Francesco Zerbetto

Entropy 2025, 27(5), 497; https://doi.org/10.3390/e27050497 - 4 May 2025

Viewed by 387

Abstract

Predicting the evolution of financial data, if at all possible, would be very beneficial in revealing the ways in which different aspects of a global environment can impact local economies. We employ an iterative stochastic differential equation that accurately forecasts an economic time [...] Read more.

α

-stable Lévy motion. The computation of the scaling exponent and the value of

α

, which characterises the type of the

α

-stable Lévy motion, are crucial for the iterative scheme. These two indices can be determined at each iteration from the form of the structure function, for the computation of which we use the method of generalised moments. Their values are used for the creation of the corresponding

α

-stable Lévy noise, which acts as a seed for the stochastic component. Furthermore, the drift and diffusion terms are calculated at each iteration. The proposed model is general, allowing the kind of stochastic process to vary from one iterative step to another, and its applicability is not restricted to financial data. As a case study, we consider Greece’s stock market general index over a period of five years, from September 2019 to September 2024, after the completion of bailout programmes. Greece’s economy changed from a restricted to a free market over the chosen era, and its stock market trading increments are likely to be describable by an

α

-stable L’evy motion. We find that

α = 2

and the scaling exponent H varies over time for every iterative step we perform. The forecasting points follow the same trend, are in good agreement with the actual data, and for most of the forecasts, the percentage error is less than 2%. Full article

(This article belongs to the Special Issue Entropy-Based Applications in Sociophysics II)

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16 pages, 5283 KiB

Open AccessArticle

Does Excellence Correspond to Universal Inequality Level?

by Soumyajyoti Biswas, Bikas K. Chakrabarti, Asim Ghosh, Sourav Ghosh, Máté Józsa and Zoltán Néda

Entropy 2025, 27(5), 495; https://doi.org/10.3390/e27050495 - 2 May 2025

Viewed by 275

Abstract

We study the inequality of citations received for different publications of various researchers and Nobel laureates in Physics, Chemistry, Medicine and Economics using Google Scholar data from 2012 to 2024. Citation distributions are found to be highly unequal, with even greater disparity among Nobel laureates. Measures of inequality, such as the Gini and Kolkata indices, emerge as useful indicators for distinguishing Nobel laureates from others. Such high inequality corresponds to growing critical fluctuations, suggesting that excellence aligns with an imminent (self-organized dynamical) critical point. Additionally, Nobel laureates exhibit systematically lower values of the Tsallis–Pareto parameter b and Shannon entropy, indicating more structured citation distributions. We also analyze the inequality in Olympic medal tallies across countries and find similar levels of disparity. Our results suggest that inequality measures can serve as proxies for competitiveness and excellence. Full article

(This article belongs to the Special Issue Entropy-Based Applications in Sociophysics II)

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19 pages, 862 KiB

Open AccessArticle

Empirical Study on Fluctuation Theorem for Volatility Cascade Processes in Stock Markets

by Jun-ichi Maskawa

Entropy 2025, 27(4), 435; https://doi.org/10.3390/e27040435 - 17 Apr 2025

Viewed by 479

Abstract

This study investigates the properties of financial markets that arise from the multi-scale structure of volatility, particularly intermittency, by employing robust theoretical tools from nonequilibrium thermodynamics. Intermittency in velocity fields along spatial and temporal axes is a well-known phenomenon in developed turbulence, with extensive research dedicated to its structures and underlying mechanisms. In turbulence, such intermittency is explained through energy cascades, where energy injected at macroscopic scales is transferred to microscopic scales. Similarly, analogous cascade processes have been proposed to explain the intermittency observed in financial time series. In this work, we model volatility cascade processes in the stock market by applying the framework of stochastic thermodynamics to a Langevin system that describes the dynamics. We introduce thermodynamic concepts such as temperature, heat, work, and entropy into the analysis of financial markets. This framework allows for a detailed investigation of individual trajectories of volatility cascades across longer to shorter time scales. Further, we conduct an empirical study primarily using the normalized average of intraday logarithmic stock prices of the constituent stocks in the FTSE 100 Index listed on the London Stock Exchange (LSE), along with two additional data sets from the Tokyo Stock Exchange (TSE). Our Langevin-based model successfully reproduces the empirical distribution of volatility—defined as the absolute value of the wavelet coefficients across time scales—and the cascade trajectories satisfy the Integral Fluctuation Theorem associated with entropy production. A detailed analysis of the cascade trajectories reveals that, for the LSE data set, volatility cascades from larger to smaller time scales occur in a causal manner along the temporal axis, consistent with known stylized facts of financial time series. In contrast, for the two data sets from the TSE, while similar behavior is observed at smaller time scales, anti-causal behavior emerges at longer time scales. Full article

(This article belongs to the Special Issue Entropy-Based Applications in Sociophysics II)

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21 pages, 737 KiB

Open AccessArticle

A Model for the Formation of Beliefs and Social Norms Based on the Satisfaction Problem (SAT)

by Bastien Chopard, Franck Raynaud and Julien Stalhandske

Entropy 2025, 27(4), 358; https://doi.org/10.3390/e27040358 - 28 Mar 2025

Viewed by 282

Abstract

We propose a numerical representation of beliefs in social systems based on the so-called SAT problem in computer science. The main idea is that a belief system is a set of true/false values associated with claims or propositions. Each individual assigns these values according to its cognitive system in order to minimize logical contradictions, thus trying to solve a satisfaction problem. Social interactions between agents that disagree on a proposition can be introduced in order to see how, in the long term, social norms and competing belief systems build up in a population. Among other metrics, entropy is used to characterize the diversity of belief systems. Full article

(This article belongs to the Special Issue Entropy-Based Applications in Sociophysics II)

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19 pages, 1138 KiB

Open AccessArticle

Democratic Thwarting of Majority Rule in Opinion Dynamics: 1. Unavowed Prejudices Versus Contrarians

by Serge Galam

Entropy 2025, 27(3), 306; https://doi.org/10.3390/e27030306 - 14 Mar 2025

Viewed by 459

Abstract

I study the conditions under which the democratic dynamics of a public debate drives a minority-to-majority transition. A landscape of the opinion dynamics is thus built using the Galam Majority Model (GMM) in a 3-dimensional parameter space for three different sizes, [...] Read more.

r = 2, 3, 4

, of local discussion groups. The related parameters are

(p_{0}, k, x)

, the respective proportions of initial agents supporting opinion A, unavowed tie prejudices breaking in favor of opinion A, and contrarians. Combining k and x yields unexpected and counterintuitive results. In most of the landscape the final outcome is predetermined, with a single-attractor dynamics, independent of the initial support for the competing opinions. Large domains of

(k, x)

values are found to lead an initial minority to turn into a majority democratically without any external influence. A new alternating regime is also unveiled in narrow ranges of extreme proportions of contrarians. The findings indicate that the expected democratic character of free opinion dynamics is indeed rarely satisfied. The actual values of

(k, x)

are found to be instrumental to predetermining the final winning opinion independently of

p_{0}

. Therefore, the conflicting challenge for the predetermined opinion to lose is to modify these values appropriately to become the winner. However, developing a model which could help in manipulating public opinion raises ethical questions. This issue is discussed in the Conclusions. Full article

(This article belongs to the Special Issue Entropy-Based Applications in Sociophysics II)

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13 pages, 447 KiB

Open AccessArticle

Biswas–Chatterjee–Sen Model Defined on Solomon Networks in (1 ≤ D ≤ 6)-Dimensional Lattices

by Gessineide Sousa Oliveira, David Santana Alencar, Tayroni Alencar Alves, José Ferreira da Silva Neto, Gladstone Alencar Alves, Antônio Macedo-Filho, Ronan S. Ferreira, Francisco Welington Lima and João Antônio Plascak

Entropy 2025, 27(3), 300; https://doi.org/10.3390/e27030300 - 14 Mar 2025

Viewed by 425

Abstract

The discrete version of the Biswas–Chatterjee–Sen model, defined on D-dimensional hypercubic Solomon networks, with

1 \leq D \leq 6

, has been studied by means of extensive Monte Carlo simulations. Thermodynamic-like variables have been computed as a function of the external noise [...] Read more.

The discrete version of the Biswas–Chatterjee–Sen model, defined on D-dimensional hypercubic Solomon networks, with

1 \leq D \leq 6

, has been studied by means of extensive Monte Carlo simulations. Thermodynamic-like variables have been computed as a function of the external noise probability. Finite-size scaling theory, applied to different network sizes, has been utilized in order to characterize the phase transition of the system in the thermodynamic limit. The results show that the model presents a phase transition of the second order for all considered dimensions. Despite the lower critical dimension being zero, this dynamical system seems not to have any upper critical dimension since the critical exponents change with D and go away from the expected mean-field values. Although larger networks could not be simulated because the number of sites drastically increases with the dimension D, the scaling regime has been achieved when computing the critical exponent ratios and the corresponding critical noise probability. Full article

(This article belongs to the Special Issue Entropy-Based Applications in Sociophysics II)

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35 pages, 7938 KiB

Open AccessArticle

Network Geometry of Borsa Istanbul: Analyzing Sectoral Dynamics with Forman–Ricci Curvature

by Ömer Akgüller, Mehmet Ali Balcı, Larissa Margareta Batrancea and Lucian Gaban

Entropy 2025, 27(3), 271; https://doi.org/10.3390/e27030271 - 5 Mar 2025

Viewed by 755

Abstract

This study investigates the dynamic interdependencies among key sectors of Borsa Istanbul—industrial, services, technology, banking, and electricity—using a novel network-geometric framework. Daily closure prices from 2022 to 2024 are transformed into logarithmic returns and analyzed via a sliding window approach. In each window, mutual information is computed to construct weighted networks that are filtered using Triangulated Maximally Filtered Graphs (TMFG) to isolate the most significant links. Forman–Ricci curvature is then calculated at the node level, and entropy measures over k-neighborhoods (

k = 1, 2, 3

) capture the complexity of both local and global network structures. Cross-correlation, Granger causality, and transfer entropy analyses reveal that sector responses to macroeconomic shocks—such as inflation surges, interest rate hikes, and currency depreciation—vary considerably. The services sector emerges as a critical intermediary, transmitting shocks between the banking and both the industrial and technology sectors, while the electricity sector displays robust, stable interconnections. These findings demonstrate that curvature-based metrics capture nuanced network characteristics beyond traditional measures. Future work could incorporate high-frequency data to capture finer interactions and empirically compare curvature metrics with conventional indicators. Full article

(This article belongs to the Special Issue Entropy-Based Applications in Sociophysics II)

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