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Differential Equations and Networks for Description of Natural and Social Systems: Methodology and Applications

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

Deadline for manuscript submissions: closed (30 September 2023) | Viewed by 23134

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Special Issue Editors

Prof. Dr. Nikolay Kolev Vitanov

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1. Institute of Mechanics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Block 4, 1113 Sofia, Bulgaria
2 Climate, Atmosphere and Water Research Institute, Bulgarian Academy of Sciences, Blvd. Tzarigradsko Chaussee 66, 1784 Sofia, Bulgaria
Interests: nonlinear dynamics; nonlinear time series analysis; fluid mechanics; nonlinear partial differential equations; application of the methods of statistics and probability theory to natural, social and economic systems
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Dr. Zlatinka I. Dimitrova

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Institute of Mechanics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Block 4, 1113 Sofia, Bulgaria
Interests: Solid state physics; Physics of complex systems; Nonlinear time series analysis and its application for extracting information from time series

Special Issue Information

Dear colleagues,

Networks are used as model of numerous complex natural and social systems. In order to study the characteristics of the processes in such networks, we often use differential equations and various quantities for network nodes or for larger segments of networks. This Special Issue intends to attract articles on the following topics:

Nonlinear differential equations and methods of obtaining exact and approximate solutions of these equations;
Quantities which can be used for characterization of processes in network nodes and segments containing many network nodes;
Models of flows and other processes in systems which have a network structure, with special attention on models based on differential and difference equations.

As we want to contribute to increase the application of mathematical methodologies for the study of processes in social systems, we especially encourage articles which are devoted to the application of network models for the study of complex social and economic systems.

Prof. Dr. Nikolay K. Vitanov
Dr. Zlatinka I. Dimitrova
Guest Editors

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Published Papers (8 papers)

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Research

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14 pages, 319 KiB

Open AccessArticle

Inferred Rate of Default as a Credit Risk Indicator in the Bulgarian Bank System

by Vilislav Boutchaktchiev

Entropy 2023, 25(12), 1608; https://doi.org/10.3390/e25121608 - 30 Nov 2023

Cited by 1 | Viewed by 1097

Abstract

The inferred rate of default (IRD) was first introduced as an indicator of default risk computable from information publicly reported by the Bulgarian National Bank. We have provided a more detailed justification for the suggested methodology for forecasting the IRD on the bank-group- and bank-system-level based on macroeconomic factors. Furthermore, we supply additional empirical evidence in the time-series analysis. Additionally, we demonstrate that IRD provides a new perspective for comparing credit risk across bank groups. The estimation methods and model assumptions agree with current Bulgarian regulations and the IFRS 9 accounting standard. The suggested models could be used by practitioners in monthly forecasting the point-in-time probability of default in the context of accounting reporting and in monitoring and managing credit risk. Full article

(This article belongs to the Special Issue Differential Equations and Networks for Description of Natural and Social Systems: Methodology and Applications)

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18 pages, 8885 KiB

Open AccessArticle

Physics-Based Differentiable Rendering for Efficient and Plausible Fluid Modeling from Monocular Video

by Yunchi Cen, Qifan Zhang and Xiaohui Liang

Entropy 2023, 25(9), 1348; https://doi.org/10.3390/e25091348 - 17 Sep 2023

Viewed by 2037

Abstract

Realistic fluid models play an important role in computer graphics applications. However, efficiently reconstructing volumetric fluid flows from monocular videos remains challenging. In this work, we present a novel approach for reconstructing 3D flows from monocular inputs through a physics-based differentiable renderer coupled with joint density and velocity estimation. Our primary contributions include the proposed efficient differentiable rendering framework and improved coupled density and velocity estimation strategy. Rather than relying on automatic differentiation, we derive the differential form of the radiance transfer equation under single scattering. This allows the direct computation of the radiance gradient with respect to density, yielding higher efficiency compared to prior works. To improve temporal coherence in the reconstructed flows, subsequent fluid densities are estimated via a coupled strategy that enables smooth and realistic fluid motions suitable for applications that require high efficiency. Experiments on synthetic and real-world data demonstrated our method’s capacity to reconstruct plausible volumetric flows with smooth dynamics efficiently. Comparisons to prior work on fluid motion reconstruction from monocular video revealed over 50–170x speedups across multiple resolutions. Full article

(This article belongs to the Special Issue Differential Equations and Networks for Description of Natural and Social Systems: Methodology and Applications)

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Figure 1

Figure 1
Our framework consists of two primary algorithm components: the differentiable rendering component and the coupled density and velocity estimation component. In this diagram, yellow and blue arrows delineate the directional flow of data between external and internal modules, respectively. The differentiable rendering component is utilized to refine volumetric representations extracted from a temporal sequence of fluidic images. We note that the current density fields are associated with the temporal epoch t − 1. These outputs subsequently serve as inputs for the coupled density and velocity estimation component. This component initially estimates the velocity field based on the density at temporal epoch t − 1. It then advects this velocity field to the subsequent temporal epoch t while enforcing incompressibility constraints. Finally, the density field is advected in accordance with the updated velocity field. The advected density volumes, along with their corresponding input images, are inputted into the differentiable renderer for final corrections. Full article ">Figure 2
This figure provides a schematic diagram overviewing the relations between our proposed algorithms and the formulas presented in each subsection (Algorithm 2: Differentiable Renderer. Algorithm 3: Density Reconstruction. Algorithm 4: Coupled Density and Velocity Estimation. Algorithm 5: Velocity Reconstruction). Full article ">Figure 3
Derivative analysis of density optimization. The initial volume was configured in the shape of a rabbit, with the density field optimized to match a target smoke image. The derivative images show an increasing density in blue and decreasing density in red. This color mapping accurately captures the evolution of the derivatives throughout the entire density optimization process. Full article ">Figure 4
Validation of the physical constraint. (a) The input image sequence used to constrain the shape from orthogonal views. (b) The reconstructed results without the transport constraint. (c) The proposed constraint. We observed that adding the transport constraint significantly improved the reconstruction quality and temporal consistency. Full article ">Figure 5
Validating the velocity estimation. We compared center-slice velocities from our estimation to the ground truth at frames <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>70</mn> <mo>,</mo> <mn>110</mn> </mrow> </semantics></math>, and 147 along the front and side views. The results exhibited some bias near the base, as no specialized inflow treatment was implemented. However, overall, the estimated velocities reasonably matched the ground truth, suggesting that our method could effectively and reliably characterize the fluid flow dynamics. Full article ">Figure 6
Orthogonal view (angle 0<math display="inline"><semantics> <msup> <mrow/> <mo>∘</mo> </msup> </semantics></math> and angle 90<math display="inline"><semantics> <msup> <mrow/> <mo>∘</mo> </msup> </semantics></math>) reconstruction. The results demonstrate that our approach could produce highly realistic fluid motion that matched the ground truth closely. Our comparison with Franz et al. [<a href="#B5-entropy-25-01348" class="html-bibr">5</a>]’s method revealed that our method’s reconstructed ability was comparable to theirs. Full article ">Figure 7
Reconstruction from real-world data. We utilized a monocular video captured from real-world fluid flows as input into our method, reconstructing at a volumetric resolution of 64 × 96 × 64. As illustrated in (b), the reconstructed flows rendered via Mitsuba 3 demonstrated that our proposed approach could reconstruct physically plausible fluid motions from real-world video. Full article ">

22 pages, 820 KiB

Open AccessArticle

Unpredictable and Poisson Stable Oscillations of Inertial Neural Networks with Generalized Piecewise Constant Argument

by Marat Akhmet, Madina Tleubergenova and Zakhira Nugayeva

Entropy 2023, 25(4), 620; https://doi.org/10.3390/e25040620 - 6 Apr 2023

Cited by 3 | Viewed by 1364

Abstract

A new model of inertial neural networks with a generalized piecewise constant argument as well as unpredictable inputs is proposed. The model is inspired by unpredictable perturbations, which allow to study the distribution of chaotic signals in neural networks. The existence and exponential stability of unique unpredictable and Poisson stable motions of the neural networks are proved. Due to the generalized piecewise constant argument, solutions are continuous functions with discontinuous derivatives, and, accordingly, Poisson stability and unpredictability are studied by considering the characteristics of continuity intervals. That is, the piecewise constant argument requires a specific component, the Poisson triple. The B-topology is used for the analysis of Poisson stability for the discontinuous functions. The results are demonstrated by examples and simulations. Full article

(This article belongs to the Special Issue Differential Equations and Networks for Description of Natural and Social Systems: Methodology and Applications)

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29 pages, 2619 KiB

Open AccessArticle

Epidemic Waves and Exact Solutions of a Sequence of Nonlinear Differential Equations Connected to the SIR Model of Epidemics

by Nikolay K. Vitanov and Kaloyan N. Vitanov

Entropy 2023, 25(3), 438; https://doi.org/10.3390/e25030438 - 1 Mar 2023

Cited by 8 | Viewed by 5704

Abstract

The SIR model of epidemic spreading can be reduced to a nonlinear differential equation with an exponential nonlinearity. This differential equation can be approximated by a sequence of nonlinear differential equations with polynomial nonlinearities. The equations from the obtained sequence are treated by the Simple Equations Method (SEsM). This allows us to obtain exact solutions to some of these equations. We discuss several of these solutions. Some (but not all) of the obtained exact solutions can be used for the description of the evolution of epidemic waves. We discuss this connection. In addition, we use two of the obtained solutions to study the evolution of two of the COVID-19 epidemic waves in Bulgaria by a comparison of the solutions with the available data for the infected individuals. Full article

(This article belongs to the Special Issue Differential Equations and Networks for Description of Natural and Social Systems: Methodology and Applications)

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16 pages, 562 KiB

Open AccessArticle

Influence of Information Blocking on the Spread of Virus in Multilayer Networks

by Paulina Wątroba and Piotr Bródka

Entropy 2023, 25(2), 231; https://doi.org/10.3390/e25020231 - 27 Jan 2023

Cited by 3 | Viewed by 1492

Abstract

In this paper, we present the model of the interaction between the spread of disease and the spread of information about the disease in multilayer networks. Next, based on the characteristics of the SARS-CoV-2 virus pandemic, we evaluated the influence of information blocking on the virus spread. Our results show that blocking the spread of information affects the speed at which the epidemic peak appears in our society, and affects the number of infected individuals. Full article

(This article belongs to the Special Issue Differential Equations and Networks for Description of Natural and Social Systems: Methodology and Applications)

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Figure 1

Figure 1
An example of multilayer networks. Full article ">Figure 2
State changes in <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>I</mi> <mi>R</mi> </mrow> </semantics></math> model. Full article ">Figure 3
State changes for <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>I</mi> <mi>S</mi> </mrow> </semantics></math> model. Full article ">Figure 4
The results for different scenarios; our baseline is <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>I</mi> <mi>R</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>U</mi> <mi>A</mi> <mi>U</mi> </mrow> </semantics></math> to which we compare two other processes. Each bar represents how much faster the peak day was or how many more nodes became infected (until the peak day or until day 150) compared to <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>I</mi> <mi>R</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>U</mi> <mi>A</mi> <mi>U</mi> </mrow> </semantics></math>, that is, the scenario where both the virus and the information start to spread at the same time. Full article ">Figure 5
The results for different delay times (the baseline is <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>I</mi> <mi>R</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>U</mi> <mi>A</mi> <mi>U</mi> </mrow> </semantics></math> with 21 days delay) to which we compare two other delay periods. Each bar represents how much sooner or later (in case of negative values) the peak day was, or how many more or fewer (in the case of negative values) nodes became infected till peak day or till day 150. Full article ">

26 pages, 718 KiB

Open AccessFeature PaperArticle

Exact Travelling-Wave Solutions of the Extended Fifth-Order Korteweg-de Vries Equation via Simple Equations Method (SEsM): The Case of Two Simple Equations

by Elena V. Nikolova

Entropy 2022, 24(9), 1288; https://doi.org/10.3390/e24091288 - 13 Sep 2022

Cited by 4 | Viewed by 1519

Abstract

We apply the Simple Equations Method (SEsM) for obtaining exact travelling-wave solutions of the extended fifth-order Korteweg-de Vries (KdV) equation. We present the solution of this equation as a composite function of two functions of two independent variables. The two composing functions are constructed as finite series of the solutions of two simple equations. For our convenience, we express these solutions by special functions V, which are solutions of appropriate ordinary differential equations, containing polynomial non-linearity. Various specific cases of the use of the special functions V are presented depending on the highest degrees of the polynomials of the used simple equations. We choose the simple equations used for this study to be ordinary differential equations of first order. Based on this choice, we obtain various travelling-wave solutions of the studied equation based on the solutions of appropriate ordinary differential equations, such as the Bernoulli equation, the Abel equation of first kind, the Riccati equation, the extended tanh-function equation and the linear equation. Full article

(This article belongs to the Special Issue Differential Equations and Networks for Description of Natural and Social Systems: Methodology and Applications)

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Review

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55 pages, 714 KiB

Open AccessReview

Simple Equations Method (SEsM): An Effective Algorithm for Obtaining Exact Solutions of Nonlinear Differential Equations

by Nikolay K. Vitanov

Entropy 2022, 24(11), 1653; https://doi.org/10.3390/e24111653 - 14 Nov 2022

Cited by 17 | Viewed by 6625

Abstract

Exact solutions of nonlinear differential equations are of great importance to the theory and practice of complex systems. The main point of this review article is to discuss a specific methodology for obtaining such exact solutions. The methodology is called the SEsM, or the Simple Equations Method. The article begins with a short overview of the literature connected to the methodology for obtaining exact solutions of nonlinear differential equations. This overview includes research on nonlinear waves, research on the methodology of the Inverse Scattering Transform method, and the method of Hirota, as well as some of the nonlinear equations studied by these methods. The overview continues with articles devoted to the phenomena described by the exact solutions of the nonlinear differential equations and articles about mathematical results connected to the methodology for obtaining such exact solutions. Several articles devoted to the numerical study of nonlinear waves are mentioned. Then, the approach to the SEsM is described starting from the Hopf–Cole transformation, the research of Kudryashov on the Method of the Simplest Equation, the approach to the Modified Method of the Simplest Equation, and the development of this methodology towards the SEsM. The description of the algorithm of the SEsM begins with the transformations that convert the nonlinearity of the solved complicated equation into a treatable kind of nonlinearity. Next, we discuss the use of composite functions in the steps of the algorithms. Special attention is given to the role of the simple equation in the SEsM. The connection of the methodology with other methods for obtaining exact multisoliton solutions of nonlinear differential equations is discussed. These methods are the Inverse Scattering Transform method and the Hirota method. Numerous examples of the application of the SEsM for obtaining exact solutions of nonlinear differential equations are demonstrated. One of the examples is connected to the exact solution of an equation that occurs in the SIR model of epidemic spreading. The solution of this equation can be used for modeling epidemic waves, for example, COVID-19 epidemic waves. Other examples of the application of the SEsM methodology are connected to the use of the differential equation of Bernoulli and Riccati as simple equations for obtaining exact solutions of more complicated nonlinear differential equations. The SEsM leads to a definition of a specific special function through a simple equation containing polynomial nonlinearities. The special function contains specific cases of numerous well-known functions such as the trigonometric and hyperbolic functions and the elliptic functions of Jacobi, Weierstrass, etc. Among the examples are the solutions of the differential equations of Fisher, equation of Burgers–Huxley, generalized equation of Camassa–Holm, generalized equation of Swift–Hohenberg, generalized Rayleigh equation, etc. Finally, we discuss the connection between the SEsM and the other methods for obtaining exact solutions of nonintegrable nonlinear differential equations. We present a conjecture about the relationship of the SEsM with these methods. Full article

(This article belongs to the Special Issue Differential Equations and Networks for Description of Natural and Social Systems: Methodology and Applications)

30 pages, 491 KiB

Open AccessFeature PaperReview

Flows of Substances in Networks and Network Channels: Selected Results and Applications

by Zlatinka I. Dimitrova

Entropy 2022, 24(10), 1485; https://doi.org/10.3390/e24101485 - 18 Oct 2022

Cited by 5 | Viewed by 2525

Abstract

This review paper is devoted to a brief overview of results and models concerning flows in networks and channels of networks. First of all, we conduct a survey of the literature in several areas of research connected to these flows. Then, we mention certain basic mathematical models of flows in networks that are based on differential equations. We give special attention to several models for flows of substances in channels of networks. For stationary cases of these flows, we present probability distributions connected to the substance in the nodes of the channel for two basic models: the model of a channel with many arms modeled by differential equations and the model of a simple channel with flows of substances modeled by difference equations. The probability distributions obtained contain as specific cases any probability distribution of a discrete random variable that takes values of

0, 1, \dots

. We also mention applications of the considered models, such as applications for modeling migration flows. Special attention is given to the connection of the theory of stationary flows in channels of networks and the theory of the growth of random networks. Full article

(This article belongs to the Special Issue Differential Equations and Networks for Description of Natural and Social Systems: Methodology and Applications)

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