Modeling Friendship Dynamics in Social Networks: A Graph-Theoretic and Computational Approach

This paper explores the application of graph theory in analyzing social networks, focusing on the relationships between members represented by vertices (individuals) and edges (connections). It examines how members establish friendships, including mutual connections that arise when members share common connections, as well as the potential for information sharing within these networks. The study investigates methods for predicting mutual friends based on the number of individual friends, utilizing linear regression to analyze trends and residual analysis to assess model quality. Additionally, it delves into the probability of forming relationships using the binomial distribution and employs the McCulloch-Pitts neuron model to study about un-friendships between members. The findings highlight the trends between the number of friends and mutual friendships, with residuals used to evaluate the accuracy of the predictive model. Overall, this research contributes to the understanding of friendship dynamics within social networks by integrating statistical modeling and graph theory. Understanding friendship structures is important for analyzing how information spreads in social networks. People who are closely connected form groups that help share content. Over time, some users become more influential by gaining many followers. These influential users work as a key role in spreading information within the network. This study examines how user connections, especially those linked to influential people with many followers, affect the spread of information. It moves from studying friendships to analyzing information diffusion. The Susceptible-Infected-Recovered (SIR) model is used to study real Instagram data. The research focuses on 200 Instagram channels to understand how network structure, content, and user behavior shape information flow. These results show how content spreads and influences user interactions. This study will be helpful in insights for marketers, influencers, and policymakers.