GRAPH THEORY METHODS IN MATHEMATICAL MODELING OF COMPLEX NETWORK SYSTEMS

Source abstract

Abstract. The monograph investigates modern approaches to mathematical modeling, analysis, and optimization of complex systems based on graph theory, mathematical optimization, and artificial intelligence technologies. The feasibility of using graph models to formalize transport, information, logistics, and other network systems is substantiated, particularly for systems whose functional characteristics are determined by the structure of relationships between their constituent objects. The main mathematical optimization models are considered, including linear, nonlinear, and discrete programming, shortest-path problems, network flow optimization, and routing problems. The potential of heuristic and metaheuristic methods, including genetic algorithms and ant colony optimization, for solving complex combinatorial problems is analyzed. Particular attention is paid to multi-objective optimization and the use of Pareto optimality for developing compromise solutions. The prospects of integrating classical mathematical methods with machine learning and Graph Neural Networks (GNNs) are substantiated. A conceptual approach to intelligent routing is proposed, in which the prediction of network parameters using artificial intelligence is combined with mathematically justified optimization of the resulting solution. An approach to adaptive selection of an optimization algorithm depending on the structural and functional characteristics of the problem under investigation is developed. The practical significance of the results lies in the possibility of applying the proposed approaches to the modeling and optimization of transport, information, logistics, and communication networks, as well as to the development of decision-support software systems. The proposed approaches can serve as a basis for further research in mathematical modeling, graph theory, optimization, intelligent data analysis, and artificial intelligence.