+ Linear and dynamic models of the system of information security in social networks, taking into consideration the relationships between users, were studied and the resistance of the security system was analyzed. There is a practical interest in studying dependence of the behavior of the system of social network security on the parameters of users’ interaction. Dynamic systems of information security in social networks in the mathematical sense of this term were considered. A dynamic system refers to any object or process, for which the concept of state as a totality of certain magnitudes at a given time is unambiguously defined and the law that describes a change (evolution) of the initial state over time was assigned. The network of social interactions consists of a totality of social users and a totality of the relations between them. Individuals, social groups, organizations, cities, countries can act as users. Relations imply not only communication interactions between users but also relations of the exchange of various resources and activities, including conflict relations. As a result of research, it was found that the security systems of a social network are nonlinear. Theoretical study of the dynamic behavior of an actual object requires the creation of its mathematical model. The procedure for developing a model is to compile mathematical equations based on physical laws. These laws are stated in the language of differential equations. Phase portraits of the data security system in the MATLAB/Multisim program, which indicate the stability of a security system in the working range of parameters even at the maximum value of the impacts, were determined. Thus, the influence of users’ interaction parameters on the parameters of the system of social network security was explored. Such study is useful and important in terms of information security in the network, since the parameters of users’ interaction significantly affect, up to 100 %, the security indicator.
A mathematical model has been developed and a study of the model of personal data protection from network clustering coefficient and data transfer intensity in social networks has been carried out. Dependencies of protection of the system from the size of the system (and from the amount of personal data); information security threats from the network clustering factor. A system of linear equations is obtained, which consists of the equation: rate of change of information flow from social network security and coefficients that reflect the impact of security measures, amount of personal data, leakage rate, change of information protection from network clustering factor, its size, personal data protection. As a result of solving the system of differential equations, mathematical and graphical dependences of the indicator of personal data protection in the social network from different components are obtained. Considering three options for solving the equation near the steady state of the system, we can conclude that, based on the conditions of the ratio of dissipation and natural frequency, the attenuation of the latter to a certain value is carried out periodically, with decaying amplitude, or by exponentially decaying law. A more visual analysis of the system behavior is performed, moving from the differential form of equations to the discrete one and modeling some interval of the system existence. Mathematical and graphical dependences of the system natural frequency, oscillation period, attenuation coefficient are presented. Simulation modeling for values with deviation from the stationary position of the system is carried out. As a result of simulation, it is proved that the social network protection system is nonlinear.
In Ukraine, the right to protection of personal data is a constitutional guarantee, and the protection of personal data is one of the areas in which such a guarantee should be implemented. The subject of our research will not be objects in general, but dynamic systems of information protection in social networks in the mathematical sense of the term. The study developed a linear mathematical model and conducted a survey of the model of protection of personal data from a set of specific network parameters and the intensity of data transmission in social networks. Dependencies are considered: the amount of information flow in the social network from the components of information protection, personal data, and data flow rate; security of the system from the size of the system and from the amount of personal data; information security threats from a set of specific network parameters.A system of linear equations is obtained, which consists of the equation: rate of change of information flow from social network security and coefficients that reflect the impact of security measures, amount of personal data, leakage rate, changes in information protection from a set of specific network parameters, its size, personal data protection As a result of solving the system of differential equations, mathematical and graphical dependences of the indicator of personal data protection in the social network on various components are obtained. Considering three options for solving the equation near the steady-state of the system, we can conclude that, based on the conditions of the ratio of dissipation and natural frequency, the attenuation of the latter to a specific value is carried out periodically, with attenuation: amplitude, or exponentially fading law. A more visual analysis of the system behavior is performed, moving from the differential form of equations to the discrete one and modeling some interval of the system's existence. Mathematical and graphical dependences of the frequency of natural oscillations of the system, the period of oscillations, and the attenuation coefficient are presented. Simulation modeling for values with deviation from the stationary position of the system is performed. As a result of the simulation, it is proved that the social network protection system is nonlinear.
A mathematical model has been developed and a study of the model of personal data protection from network clustering coefficient and data transfer intensity in social networks has been carried out. Dependencies of protection of the system from the size of the system (and from the amount of personal data); information security threats from the network clustering factor. A system of linear equations is obtained, which consists of the equation: rate of change of information flow from social network security and coefficients that reflect the impact of security measures, amount of personal data, leakage rate, change of information protection from network clustering factor, its size, personal data protection. As a result of solving the system of differential equations, mathematical and graphical dependences of the indicator of personal data protection in the social network from different components are obtained. Considering three options for solving the equation near the steady state of the system, we can conclude that, based on the conditions of the ratio of dissipation and natural frequency, the attenuation of the latter to a certain value is carried out periodically, with decaying amplitude, or by exponentially decaying law. A more visual analysis of the system behavior is performed, moving from the differential form of equations to the discrete one and modeling some interval of the system existence.Mathematical and graphical dependences of the system natural frequency, oscillation period, attenuation coefficient are presented. Simulation modeling for values with deviation from the stationary position of the system is carried out. As a result of simulation, it is proved that the social network protection system is nonlinear.
A mathematical model (linear system of differential equations) was developed and a research of the model of personal data protection against the number of communities and the intensity of data transfer in social networks was conducted. The linear system of information protection in social networks in the mathematical sense of this term is considered. When described by linear models, the object should be linear, at least approximately. This approach makes it quite simple to consider mathematical models. If such a thing is not noticed, it is necessary to examine the security system for linearity. Such dependecies has been studied: the dependence of the amount of information flow in the social network on the components of information protection, the amount of personal data, and the speed of the data flow; the security of the system on the size of the system (as well as on the amount of personal data); information security threats on the number of communities, and also calculated: – coefficient representing the impact of information protection measures; – coefficient representing the impact of data leakage rate; – the coefficient representing the influence of the amount of data on its leakage; – the coefficient representing the influence of the system size on system security; – coefficient representing the impact of system security on data leakage; – the number of connections in the social networks; – number of vertices in the social networks; – the parameter can be used to configure the network partitioning algorithm. The solution has been obtained - the harmonic oscillator equation, which breaks down into three cases: pre-resonance zone, resonance zone and post-resonance zone. So, the impact of the parameters of the number of communities on the parameters of the social network security system was investigated. Such a study is useful and important from the point of view of information protection in the network, since the parameters of the number of communities has significant influence, up to 100%, per protection indicator. As the result of research, it was established that social network security systems are non-linear.
The object of research is the system of information protection of the social network. The article investigates the dynamic models of the information protection system in social networks taking into account the clustering coefficient, and also analyzes the stability of the protection system. In graph theory, the clustering factor is a measure of the degree to which nodes in a graph tend to group together. The available data suggest that in most real networks, and in particular in social networks, nodes tend to form closely related groups with a relatively high density of connections. It is probability is greater than the average probability of a random connection between two nodes. There are two variants of this term: global and local. The global version was created for a general idea of network clustering, while the local one describes the nesting of individual nodes. There is a practical interest in studying the behavior of the system of protection of social networks from the value of the clustering factor. Dynamic systems of information protection in social networks in the mathematical sense of this term are considered. A dynamic system is understood as any object or process for which the concept of state as a set of some quantities at a given moment of time is unambiguously defined and a given law is described that describes the change (evolution) of the initial state over time. This law allows the initial state to predict the future state of a dynamic system. It is called the law of evolution. The study is based on the nonlinearity of the social network protection system. To solve the system of nonlinear equations used: the method of exceptions, the joint solution of the corresponding homogeneous characteristic equation. Since the differential of the protection function has a positive value in some data domains (the requirement of Lyapunov's theorem for this domain is not fulfilled), an additional study of the stability of the protection system within the operating parameters is required. Phase portraits of the data protection system in MatLab/Multisim are determined, which indicate the stability of the protection system in the operating range of parameters even at the maximum value of influences.
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