This paper examines the concept of small-world networks within the domain of graph theory. The paper provides a comprehensive introduction to the fundamental concepts of graph theory, explaining the history and basic properties of graphs to lay the foundations for further consideration. Following this, the paper delves in detail into various clustering methods. Techniques such as hierarchical clustering and density-based clustering are explored, and the significance of the clustering coefficient is investigated. Network theory is also explored in the paper, with a particular focus on complex and scale-free networks. Within this discussion, specific attention is given to small-world networks. The properties and models of small-world networks, including the Watts-Strogatz and Barabási–Albert models, are studied, and their practical applications and robustness are investigated. For a better understanding of these concepts, numerous simulations and analyses are conducted in the paper. These analyses reveal the key aspects and connectivity mechanisms inherent in these models. Additionally, a comparison between the Watts-Strogatz and Barabási–Albert models offers deeper insights into their differences and similarities. In conclusion, this paper summarizes the basic findings about small-world networks and their models, aiming for a better understanding of this fascinating area. As an additional resource, the paper provides the programming code used in the simulations, which can serve as a basis for further research and learning.