Lung cancer is the biggest cause of cancer mortality worldwide and a major impediment to extending life expectancy. In comparison to other cancers, it has a relatively poor survival rate. In this paper, we have developed a mathematical model for lung cancer based on biological phenomena using nonlinear ordinary differential equations and analyzed it both analytically and numerically. According to the findings, CD8+ T cells and dendritic cells have a role in tumor cell variety. Surgery and chemotherapy have been used as treatment options, and we have observed that three doses of chemotherapy after surgery had the greatest results after examining several treatment options. During the treatment period, the cycle of each chemotherapy has been taken every 4 weeks, and the first dose has been taken after 28 days of surgery. Finally, we have evaluated the various starting dates for the best treatment choice and discovered that the patient who begins treatment sooner has a better probability of surviving.
{"title":"Mathematical Modeling and Analysis on the Effects of Surgery and Chemotherapy on Lung Cancer","authors":"Md. Ahsan Ullah, Uzzwal Kumar Mallick","doi":"10.1155/2023/4201373","DOIUrl":"https://doi.org/10.1155/2023/4201373","url":null,"abstract":"Lung cancer is the biggest cause of cancer mortality worldwide and a major impediment to extending life expectancy. In comparison to other cancers, it has a relatively poor survival rate. In this paper, we have developed a mathematical model for lung cancer based on biological phenomena using nonlinear ordinary differential equations and analyzed it both analytically and numerically. According to the findings, CD8+ T cells and dendritic cells have a role in tumor cell variety. Surgery and chemotherapy have been used as treatment options, and we have observed that three doses of chemotherapy after surgery had the greatest results after examining several treatment options. During the treatment period, the cycle of each chemotherapy has been taken every 4 weeks, and the first dose has been taken after 28 days of surgery. Finally, we have evaluated the various starting dates for the best treatment choice and discovered that the patient who begins treatment sooner has a better probability of surviving.","PeriodicalId":14766,"journal":{"name":"J. Appl. Math.","volume":"8 1","pages":"4201373:1-4201373:16"},"PeriodicalIF":0.0,"publicationDate":"2023-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80775153","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present a method of deriving analytical solutions for a two-dimensional Black-Scholes-Merton equation. The method consists of three changes of variables in order to reduce the original partial differential equation (PDE) to a normal form and then solve it. Analytical solutions for two cases of option pricing on the minimum and maximum of two assets are derived using our method and are shown to agree with previously published results. The advantage of our solution procedure is the ability of splitting the original problem into several components in order to demonstrate some solution properties. The solutions of the two cases have a total of five components; each is a particular solution of the PDE itself. Due to the linearity of the two-dimensional Black-Scholes-Merton equation, any linear combination of these components constitutes another solution. Some other possible solutions as well as the solution properties are discussed.
{"title":"An Alternative Method for Analytical Solutions of Two-Dimensional Black-Scholes-Merton Equation","authors":"Jun Yu, Michael J. Tomas","doi":"10.1155/2023/6725686","DOIUrl":"https://doi.org/10.1155/2023/6725686","url":null,"abstract":"We present a method of deriving analytical solutions for a two-dimensional Black-Scholes-Merton equation. The method consists of three changes of variables in order to reduce the original partial differential equation (PDE) to a normal form and then solve it. Analytical solutions for two cases of option pricing on the minimum and maximum of two assets are derived using our method and are shown to agree with previously published results. The advantage of our solution procedure is the ability of splitting the original problem into several components in order to demonstrate some solution properties. The solutions of the two cases have a total of five components; each is a particular solution of the PDE itself. Due to the linearity of the two-dimensional Black-Scholes-Merton equation, any linear combination of these components constitutes another solution. Some other possible solutions as well as the solution properties are discussed.","PeriodicalId":14766,"journal":{"name":"J. Appl. Math.","volume":"48 1","pages":"6725686:1-6725686:11"},"PeriodicalIF":0.0,"publicationDate":"2023-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75791775","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the two-sided market where the third-party platforms connect the providers and consumers, the online platforms become the significant distribution channel of digital products; therefore, the digital product firms face the hybrid channel pricing problem in the two-sided market in which the products are launched through the platform channel and the existing direct channel. Because the network externality effect is the significant economic characteristic of digital products and services, the current work presents the models of consumers’ utility obtained by adopting digital products from the direct and platform channels, and the utility models use the network effect in the direct and platform channels as the parameters. The optimization model on pricing is derived from the utility models and solved mathematically. The closed-form solutions show that the price in the direct channel is supposed to be lower than that in the platform channel, while the prices of digital products would be affected by the network effect only when the products are distributed through the direct channel. The comparative statics analysis on the network effect illustrates that the network effect in the direct channel and the platform channel would, respectively, have the positive and negative impact on the products’ prices and the firms’ profit. The current work explores the hybrid channel pricing problem and provides insights for the digital product firms on the optimal pricing decision in the context of the emerging platform economy.
{"title":"Mathematical Optimization on Hybrid Channel Pricing Digital Products in Two-Sided Market with Network Effect","authors":"W. Li, Yan-peng He, Shuangbao Ma","doi":"10.1155/2023/4456931","DOIUrl":"https://doi.org/10.1155/2023/4456931","url":null,"abstract":"In the two-sided market where the third-party platforms connect the providers and consumers, the online platforms become the significant distribution channel of digital products; therefore, the digital product firms face the hybrid channel pricing problem in the two-sided market in which the products are launched through the platform channel and the existing direct channel. Because the network externality effect is the significant economic characteristic of digital products and services, the current work presents the models of consumers’ utility obtained by adopting digital products from the direct and platform channels, and the utility models use the network effect in the direct and platform channels as the parameters. The optimization model on pricing is derived from the utility models and solved mathematically. The closed-form solutions show that the price in the direct channel is supposed to be lower than that in the platform channel, while the prices of digital products would be affected by the network effect only when the products are distributed through the direct channel. The comparative statics analysis on the network effect illustrates that the network effect in the direct channel and the platform channel would, respectively, have the positive and negative impact on the products’ prices and the firms’ profit. The current work explores the hybrid channel pricing problem and provides insights for the digital product firms on the optimal pricing decision in the context of the emerging platform economy.","PeriodicalId":14766,"journal":{"name":"J. Appl. Math.","volume":"2023 1","pages":"4456931:1-4456931:8"},"PeriodicalIF":0.0,"publicationDate":"2023-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75443397","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Deep learning with specific network topologies has been successfully applied in many fields. However, what is primarily called into question by people is its lack of theoretical foundation investigations, especially for structured neural networks. This paper theoretically studies the multichannel deep convolutional neural networks equipped with the downsampling operator, which is frequently used in applications. The results show that the proposed networks have outstanding approximation and generalization ability of functions from ridge class and Sobolev space. Not only does it answer an open and crucial question of why multichannel deep convolutional neural networks are universal in learning theory, but it also reveals the convergence rates.
{"title":"Error Bounds for Approximations Using Multichannel Deep Convolutional Neural Networks with Downsampling","authors":"Xinling Liu, Jingyao Hou","doi":"10.1155/2023/8208424","DOIUrl":"https://doi.org/10.1155/2023/8208424","url":null,"abstract":"Deep learning with specific network topologies has been successfully applied in many fields. However, what is primarily called into question by people is its lack of theoretical foundation investigations, especially for structured neural networks. This paper theoretically studies the multichannel deep convolutional neural networks equipped with the downsampling operator, which is frequently used in applications. The results show that the proposed networks have outstanding approximation and generalization ability of functions from ridge class and Sobolev space. Not only does it answer an open and crucial question of why multichannel deep convolutional neural networks are universal in learning theory, but it also reveals the convergence rates.","PeriodicalId":14766,"journal":{"name":"J. Appl. Math.","volume":"4 1","pages":"8208424:1-8208424:12"},"PeriodicalIF":0.0,"publicationDate":"2023-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72691187","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The fifth-order accurate Weighted Essentially Nonoscillatory space discretization developed by Jiang and Shu (WENO-JS) is studied theoretically. An exact Nonlinear Spectral Method (NSM) is developed based on an innovative yet simple methodology. The NSM explains the behaviour of nonsmooth solutions because it is valid for arbitrary modified wave numbers (MWN). The NSM clarifies the effects of the time integration methods and the Courant number. The mode isolation assumption, extensively used to analyse WENO-JS, is elucidated, and several novel findings are presented. The improved performance of the combination of WENO-JS with the forward Euler time integration method, compared to the Linear Fifth-Order Upwind discretization, is illustrated. The overdamping of the combination of WENO-JS with the popular third-order total variation diminishing Runge-Kutta method is discovered. Thus, the NSM covers several gaps in the current literature.
{"title":"Exact Nonlinear Spectral Analysis of Nonsmooth WENO-JS Solutions","authors":"T. Kasem","doi":"10.1155/2023/4941679","DOIUrl":"https://doi.org/10.1155/2023/4941679","url":null,"abstract":"The fifth-order accurate Weighted Essentially Nonoscillatory space discretization developed by Jiang and Shu (WENO-JS) is studied theoretically. An exact Nonlinear Spectral Method (NSM) is developed based on an innovative yet simple methodology. The NSM explains the behaviour of nonsmooth solutions because it is valid for arbitrary modified wave numbers (MWN). The NSM clarifies the effects of the time integration methods and the Courant number. The mode isolation assumption, extensively used to analyse WENO-JS, is elucidated, and several novel findings are presented. The improved performance of the combination of WENO-JS with the forward Euler time integration method, compared to the Linear Fifth-Order Upwind discretization, is illustrated. The overdamping of the combination of WENO-JS with the popular third-order total variation diminishing Runge-Kutta method is discovered. Thus, the NSM covers several gaps in the current literature.","PeriodicalId":14766,"journal":{"name":"J. Appl. Math.","volume":"9 1","pages":"4941679:1-4941679:21"},"PeriodicalIF":0.0,"publicationDate":"2023-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73071747","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
W. Iddrisu, Inusah D. Iddrisu, Abdul-Karim Iddrisu
In this study, we extend Codeço’s classical SI-B epidemic and endemic model from a deterministic framework into a stochastic framework. Then, we formulated it as a stochastic differential equation for the number of infectious individuals I t under the role of the aquatic environment. We also proved that this stochastic differential equation (SDE) exists and is unique. The reproduction number, R 0 , was derived for the deterministic model, and qualitative features such as the positivity and invariant region of the solution, the two equilibrium points (disease-free and endemic equilibrium), and stabilities were studied to ensure the biological meaningfulness of the model. Numerical simulations were also carried out for the stochastic differential equation (SDE) model by utilizing the Euler-Maruyama numerical method. The method was used to simulate the sample path of the SI-B stochastic differential equation for the number of infectious individuals I t , and the findings showed that the sample path or trajectory of the stochastic differential equation for the number of infectious individuals I t is continuous but not differentiable and that the SI-B stochastic differential equation model for the number of infectious individuals I t fluctuates inside the solution of the SI-B ordinary differential equation model. Another significant feature of our proposed SDE model is its simplicity.
{"title":"Modeling Cholera Epidemiology Using Stochastic Differential Equations","authors":"W. Iddrisu, Inusah D. Iddrisu, Abdul-Karim Iddrisu","doi":"10.1155/2023/7232395","DOIUrl":"https://doi.org/10.1155/2023/7232395","url":null,"abstract":"In this study, we extend Codeço’s classical SI-B epidemic and endemic model from a deterministic framework into a stochastic framework. Then, we formulated it as a stochastic differential equation for the number of infectious individuals \u0000 \u0000 I\u0000 \u0000 \u0000 t\u0000 \u0000 \u0000 \u0000 under the role of the aquatic environment. We also proved that this stochastic differential equation (SDE) exists and is unique. The reproduction number, \u0000 \u0000 \u0000 \u0000 R\u0000 \u0000 \u0000 0\u0000 \u0000 \u0000 \u0000 , was derived for the deterministic model, and qualitative features such as the positivity and invariant region of the solution, the two equilibrium points (disease-free and endemic equilibrium), and stabilities were studied to ensure the biological meaningfulness of the model. Numerical simulations were also carried out for the stochastic differential equation (SDE) model by utilizing the Euler-Maruyama numerical method. The method was used to simulate the sample path of the SI-B stochastic differential equation for the number of infectious individuals \u0000 \u0000 I\u0000 \u0000 \u0000 t\u0000 \u0000 \u0000 \u0000 , and the findings showed that the sample path or trajectory of the stochastic differential equation for the number of infectious individuals \u0000 \u0000 I\u0000 \u0000 \u0000 t\u0000 \u0000 \u0000 \u0000 is continuous but not differentiable and that the SI-B stochastic differential equation model for the number of infectious individuals \u0000 \u0000 I\u0000 \u0000 \u0000 t\u0000 \u0000 \u0000 \u0000 fluctuates inside the solution of the SI-B ordinary differential equation model. Another significant feature of our proposed SDE model is its simplicity.","PeriodicalId":14766,"journal":{"name":"J. Appl. Math.","volume":"146 1","pages":"7232395:1-7232395:17"},"PeriodicalIF":0.0,"publicationDate":"2023-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82896686","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The transient creeping motion of two rigid spheres oscillating in a boundless viscous fluid beneath the impact of the magnetic field is investigated. There is no slippage associated with a Stokes flow on the two rigid spherical surfaces with different sizes and radii. The solutions can be obtained using the boundary collocation scheme at low Reynolds numbers. The unsteady real and imaginary drag coefficients are estimated on the hard spherical particles. These coefficients are computed in tables for various parameters and illustrated graphically. The frequency parameter and Hartmann number also play a significant role in this study where the drag coefficients decrease or increase by 100 percent after the value of κ = 5.0 . Using available literature data, we tested the accuracy and reliability of our results.
{"title":"Interaction of Two Rigid Spheres Oscillating in an Infinite Liquid under the Control of a Magnetic Field","authors":"Shreen El-Sapa, W. Albalawi","doi":"10.1155/2023/1146872","DOIUrl":"https://doi.org/10.1155/2023/1146872","url":null,"abstract":"The transient creeping motion of two rigid spheres oscillating in a boundless viscous fluid beneath the impact of the magnetic field is investigated. There is no slippage associated with a Stokes flow on the two rigid spherical surfaces with different sizes and radii. The solutions can be obtained using the boundary collocation scheme at low Reynolds numbers. The unsteady real and imaginary drag coefficients are estimated on the hard spherical particles. These coefficients are computed in tables for various parameters and illustrated graphically. The frequency parameter and Hartmann number also play a significant role in this study where the drag coefficients decrease or increase by 100 percent after the value of \u0000 \u0000 κ\u0000 =\u0000 5.0\u0000 \u0000 . Using available literature data, we tested the accuracy and reliability of our results.","PeriodicalId":14766,"journal":{"name":"J. Appl. Math.","volume":"21 1","pages":"1146872:1-1146872:13"},"PeriodicalIF":0.0,"publicationDate":"2023-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90316076","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this study, the steady hydromagnetic flow of two immiscible couple stress fluids through a uniform porous medium in a cylindrical pipe with slip effect is investigated analytically. Essentially, the flow system is divided into two regions, region I and region II, which occupy the core and periphery of the system, respectively. The flow is driven by a constant pressure gradient applied in a direction parallel to the cylinder’s axis, and an external uniform magnetic field is applied in the direction perpendicular to the direction of fluid motion. Instead of the classical no-slip condition, the slip velocity along with vanishing couple stress boundary conditions is taken on the surface of the rigid cylinder, and continuity conditions of velocity, vorticity, shear stress, and couple stress are imposed at the fluid-fluid interface. The governing equations are modeled using the fully developed flow conditions. The resulting differential equations governing the flow in the two regions are converted to nondimensional forms using appropriate dimensionless variables. The nondimensional equations are solved analytically, and closed-form expressions for the flow velocity, flow rate, and stresses are derived in terms of the Bessel functions. The impacts of several parameters pertaining to the flow such as the magnetic number, couple stress parameters, Darcy number, viscosity ratio, Reynolds number, and slip parameter on the velocities in respective regions are examined and illustrated through graphs. The flow rate’s numerical values are also calculated for different fluid parameters and displayed in tabular form. It is found that increasing the magnetic number, viscosity ratio, Reynolds number, and slip parameters decreases the velocities of the fluids whereas increasing the couple stress parameter, Darcy number, and pressure gradient increases fluid velocities. The results obtained in this paper show an excellent agreement with the already existing results in the literature as limiting cases.
{"title":"Hydromagnetic Flow of Two Immiscible Couple Stress Fluids through Porous Medium in a Cylindrical Pipe with Slip Effect","authors":"Punnamchandar Bitla, Yitagesu Daba Kore","doi":"10.1155/2023/1902844","DOIUrl":"https://doi.org/10.1155/2023/1902844","url":null,"abstract":"In this study, the steady hydromagnetic flow of two immiscible couple stress fluids through a uniform porous medium in a cylindrical pipe with slip effect is investigated analytically. Essentially, the flow system is divided into two regions, region I and region II, which occupy the core and periphery of the system, respectively. The flow is driven by a constant pressure gradient applied in a direction parallel to the cylinder’s axis, and an external uniform magnetic field is applied in the direction perpendicular to the direction of fluid motion. Instead of the classical no-slip condition, the slip velocity along with vanishing couple stress boundary conditions is taken on the surface of the rigid cylinder, and continuity conditions of velocity, vorticity, shear stress, and couple stress are imposed at the fluid-fluid interface. The governing equations are modeled using the fully developed flow conditions. The resulting differential equations governing the flow in the two regions are converted to nondimensional forms using appropriate dimensionless variables. The nondimensional equations are solved analytically, and closed-form expressions for the flow velocity, flow rate, and stresses are derived in terms of the Bessel functions. The impacts of several parameters pertaining to the flow such as the magnetic number, couple stress parameters, Darcy number, viscosity ratio, Reynolds number, and slip parameter on the velocities in respective regions are examined and illustrated through graphs. The flow rate’s numerical values are also calculated for different fluid parameters and displayed in tabular form. It is found that increasing the magnetic number, viscosity ratio, Reynolds number, and slip parameters decreases the velocities of the fluids whereas increasing the couple stress parameter, Darcy number, and pressure gradient increases fluid velocities. The results obtained in this paper show an excellent agreement with the already existing results in the literature as limiting cases.","PeriodicalId":14766,"journal":{"name":"J. Appl. Math.","volume":"12 1","pages":"1902844:1-1902844:13"},"PeriodicalIF":0.0,"publicationDate":"2023-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87455139","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Issa Cherif Geraldo, Edoh Katchekpele, T. A. Kpanzou
In this paper, we consider a multivariate statistical model of accident frequencies having a variable number of parameters and whose parameters are dependent and subject to box constraints and linear equality constraints. We design a minorization-maximization (MM) algorithm and an accelerated MM algorithm to compute the maximum likelihood estimates of the parameters. We illustrate, through simulations, the performance of our proposed MM algorithm and its accelerated version by comparing them to Newton-Raphson (NR) and quasi-Newton algorithms. The results suggest that the MM algorithm and its accelerated version are better in terms of convergence proportion and, as the number of parameters increases, they are also better in terms of computation time.
{"title":"A Fast and Efficient Estimation of the Parameters of a Model of Accident Frequencies via an MM Algorithm","authors":"Issa Cherif Geraldo, Edoh Katchekpele, T. A. Kpanzou","doi":"10.1155/2023/3377201","DOIUrl":"https://doi.org/10.1155/2023/3377201","url":null,"abstract":"In this paper, we consider a multivariate statistical model of accident frequencies having a variable number of parameters and whose parameters are dependent and subject to box constraints and linear equality constraints. We design a minorization-maximization (MM) algorithm and an accelerated MM algorithm to compute the maximum likelihood estimates of the parameters. We illustrate, through simulations, the performance of our proposed MM algorithm and its accelerated version by comparing them to Newton-Raphson (NR) and quasi-Newton algorithms. The results suggest that the MM algorithm and its accelerated version are better in terms of convergence proportion and, as the number of parameters increases, they are also better in terms of computation time.","PeriodicalId":14766,"journal":{"name":"J. Appl. Math.","volume":"44 2 1","pages":"3377201:1-3377201:10"},"PeriodicalIF":0.0,"publicationDate":"2023-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79908592","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Networks are prevalent in real life, and the study of network evolution models is very important for understanding the nature and laws of real networks. The distribution of the initial degree of nodes in existing classical models is constant or uniform. The model we proposed shows binomial distribution, and it is consistent with real network data. The theoretical analysis shows that the proposed model is scale-free at different probability values and its clustering coefficients are adjustable, and the Barabasi-Albert model is a special case of p = 0 in our model. In addition, the analytical results of the clustering coefficients can be estimated using mean-field theory. The mean clustering coefficients calculated from the simulated data and the analytical results tend to be stable. The model also exhibits small-world characteristics and has good reproducibility for short distances of real networks. Our model combines three network characteristics, scale-free, high clustering coefficients, and small-world characteristics, which is a significant improvement over traditional models with only a single or two characteristics. The theoretical analysis procedure can be used as a theoretical reference for various network models to study the estimation of clustering coefficients. The existence of stable equilibrium points of the model explains the controversy of whether scale-free is universal or not, and this explanation provides a new way of thinking to understand the problem.
{"title":"Network Model with Scale-Free, High Clustering Coefficients, and Small-World Properties","authors":"Chuankui Yan","doi":"10.1155/2023/5533260","DOIUrl":"https://doi.org/10.1155/2023/5533260","url":null,"abstract":"Networks are prevalent in real life, and the study of network evolution models is very important for understanding the nature and laws of real networks. The distribution of the initial degree of nodes in existing classical models is constant or uniform. The model we proposed shows binomial distribution, and it is consistent with real network data. The theoretical analysis shows that the proposed model is scale-free at different probability values and its clustering coefficients are adjustable, and the Barabasi-Albert model is a special case of \u0000 \u0000 p\u0000 =\u0000 0\u0000 \u0000 in our model. In addition, the analytical results of the clustering coefficients can be estimated using mean-field theory. The mean clustering coefficients calculated from the simulated data and the analytical results tend to be stable. The model also exhibits small-world characteristics and has good reproducibility for short distances of real networks. Our model combines three network characteristics, scale-free, high clustering coefficients, and small-world characteristics, which is a significant improvement over traditional models with only a single or two characteristics. The theoretical analysis procedure can be used as a theoretical reference for various network models to study the estimation of clustering coefficients. The existence of stable equilibrium points of the model explains the controversy of whether scale-free is universal or not, and this explanation provides a new way of thinking to understand the problem.","PeriodicalId":14766,"journal":{"name":"J. Appl. Math.","volume":"74 1","pages":"5533260:1-5533260:11"},"PeriodicalIF":0.0,"publicationDate":"2023-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80968853","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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