The concept of network security is on the increase in the digital era and cyber threats have increasingly become more advanced and sophisticated. This study examines the research of finding critical nodes with the help of a graph theory and determining weaknesses in network structures. The analysis of the network based on centrality metrics which include degree, betweenness and closeness centrality names important node identities that are essential in ensuring the network remains connected and efficient. The effect of node failure is estimated using the MATLAB simulation, and it is concentrated on how the centrality of nodes is compromised to cause network fragmentation and performance deterioration. The findings demonstrate the importance of centrality metrics in identifying vulnerabilities in a network, and degree-centrality nodes are the most susceptible systems to failure. Betweenness and closeness centrality also underscore how important nodes are in regulating the movement of the network and removing delays. The results confirm the usefulness of the graph theory-related approaches to enhancing the security of the network through providing information on the identification of the key nodes and simulation of attacks. The paper also shows the capabilities of dynamic simulations versus the static methodology that offers real-time analysis of network vulnerabilities and security policies. It is a work in progress to expand these methods to larger networks (including real-time data to detect vulnerabilities) and improve recovery measures in the future.