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Optimization problems on many-weighted dynamic graphs

Keywords:

R.A. Kochkarov – Ph.D. (Econ.), Deputy IT Director, Financial University under the Government of the Russian Federation (Moscow). E-mail: rasul_kochkarov@mail.ru A.A. Kochkarov - Ph.D. (Phys.-Math.), Associate Professor of Applied Mathematics Department, Financial University under the Government of the Russian Federation (Moscow); Vice-Chief of R&D Centre, JSC «Academician A.L. Mints Radiotechnical Institute» (Moscow). E-mail: akochkar@gmail.com A.A. Baychorova - Associate Professor of Department of Informatics and Computational Mathematics, Karachay-Cherkessia State University named U.D. Aliyev (Cherkessk). E-mail: abaichora@gmail.com


Nowadays, the computational mathematics techniques applied for the analysis of such classical mathematical models as partial differential equations are rapidly developing. This is needed to improve the accurateness of computations using new generations of computers. The concept of dynamic networks is widely used for the study of complex networks with a variable structure of various natures. These are social networks, communication networks, cooperative networks, structures of stock markets, and structures of mutual obligations in the interbank system. Despite the accumulated empirical data about dynamic networks, the dynamic network analysis or network science has not yet been formed. To form such a branch of application science, a theoretical basis is required. The dynamic graph theory can provide such a basis. The main subject in this theory is the dynamic graph, which provides a model of a dynamic network. The principles and methods of dynamic graph theory are especially useful for the design of command and data interaction between mobile subscribers in network systems. Network systems are engineering systems based on networks. In this sense, network systems are in a greater degree an engineering concept than a rigorous mathematical one. The history of wireless network development shows that the field of application of this class of telecommunication technologies grows. Presently, wireless networks are superior to wired networks in terms of security, cost, reliability, functionality, and convenience of use. Furthermore, the range of tasks related to new applications of wireless technologies and networks still grows. There are two main fields of application of wireless networks – telecommunications and monitoring. In “large” systems, wireless networks can perform both data transfer and monitoring functions. The appearing dynamic graph theory can provide a theoretical basis for the design of algorithms of command and information interaction of mobile subscribers in network systems. The topology of a network consisting of mobile subscribers cannot be strictly fixed. Moreover, this topology must change due to a variety of reasons, e.g., due to the increase in the number of subscribers in the network. Since data transfer in a network depends on the length of the chain of subscribers, it is reasonable to control the network diameter as new subscrib-ers join the network. Dynamic introduction to graph theory is offered in the paper, some engineering applications are discussed. Description of the set of alternatives multicriterial problems on prefractal graphs is described. The general mathematical formulation of discrete multicriterial problems on many-weighted prefractal graph is proposed. The classification of multicriterial problems of many-weighted prefractal graphs is implemented, the computational complexity of the algorithms is defined. The approach to assessing the complexity of de-termining a set of alternatives is suggested. One of the main findings is the existence of hereditary properties of dynamic graphs.
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