Mathematical Modeling of Social Networks Using Graphical Theory and Network Centrality Measures

Abstract: Social Network Analysis (SNA) is an interdisciplinary approach for studying relationships, interactions, and information flows among individuals, groups, organizations, and other entities. A social network can be represented mathematically as a graph consisting of nodes (vertices) and links (edges), where nodes represent users or entities and edges represent social relationships such as friendship, collaboration, communication, or following. The rapid growth of social networking platforms such as Facebook, Twitter, Instagram, and other Web 2.0 applications has increased the importance of graph-based approaches for understanding online social interactions. Social graphs and labelled graphs provide useful representations of relationships and associated information. Graph-based community clustering can identify groups of users sharing common interests, behaviours, backgrounds, or preferences. Graph theory also helps analyse information diffusion by representing the movement of information from senders to receivers through different communication channels. Important network measures such as degree centrality help identify influential or highly connected nodes, while modularity measures the strength and quality of community structures by comparing actual connections with those expected by chance. These techniques have practical applications in business communication, recommendation systems, network optimization, and law enforcement. Thus, graph theory and SNA provide powerful mathematical and visual tools for understanding the structure, behaviour, and dynamics of modern social networks.

Keywords: SNA-Social Network Analysis, Communication, Nodes, links (edges), Diffusion, Optimization