ANALYSIS OF MATHEMATICAL MODELS OF RANDOM GRAPHS

Abstract

Relevance. In the modern world, the majority of social, biological, technological, and communication processes occur in the form of complex network structures. The analysis of such systems requires the construction of mathematical models capable of adequately describing their topology, stochastic nature, and dynamic properties. One of the most powerful tools for this is the theory of random graphs, which enables the modeling of a wide range of real-world phenomena – from the spread of viruses and information to the functioning of critical infrastructures. Classical models, particularly the Erdős-Rényi model, laid the foundation of modern graph theory; however, they have limitations in describing networks with high clustering or uneven degree distributions. Therefore, in recent years, modern approaches have gained special significance, including small-world models, which reflect the properties of real social and biological systems with short paths and a high level of clustering, and scale-free graphs, which model networks with heterogeneous degree distributions. The relevance of this topic is due to the need to select and analyze an appropriate mathematical model that can accurately represent the properties of real networks, enable justified prediction of their behavior, identify vulnerabilities, and optimize the functioning of complex systems. In this context, the analysis of mathematical models of random graphs is an important area of modern applied mathematics, computer science, and systems theory. The object of research: Random graphs as mathematical structures that model the topology and dynamics of complex networked systems. Purpose of the article: Research, systematization, and comparative analysis of mathematical models of random graphs to determine their suitability for modeling various types of complex networked systems. Research results. The article presents an analysis of mathematical models of random graphs that underpin modern network science. Beginning with the theoretical foundations of graphs, both classical and contemporary models for constructing random networks are examined, alongside methods of their analysis and practical areas of application. The methods of analysis are critically important for the practical use of random graphs, as they enable the models to be applied to forecasting, diagnostics, and the management of complex systems. These methods also provide a foundation for integrating mathematical models with real-world data, which is a key objective of contemporary network science. Classical models allow for the formalization of randomness in network connections; however, they have significant limitations in capturing the real topological properties of complex networks – particularly high clustering, degree heterogeneity, and growth mechanisms. In contrast, modern models correspond much more closely to the structure of real-world systems. They make it possible to model such essential features as the emergence of hubs, clustering, short average path lengths, and robustness to failures. Conclusions. Mathematical models of random graphs are a universal tool for the analysis and synthesis of various networked systems. Their application allows not only for the formal description of a complex system’s structure, but also for the discovery of its hidden patterns, the prediction of its behavior under different conditions, and the optimization of its functioning while accounting for real-world constraints.