Asian Research Journal of Mathematics

Friendship networks provide a clear and intuitive framework for analyzing binary relations and understanding their structural properties. This paper presents a mathematical framework for representing social networks using binary relations on finite sets, with an emphasis on reflexivity, symmetry, transitivity and their role in community formation. In this framework, individuals are represented as elements of a finite set A, while the friendship relation is modeled as a subset of the Cartesian product A×A.  Its structural properties - reflexivity, symmetry, and transitivity are examined in detail. The study formulates fundamental propositions that connect relational properties with their graph-theoretic representations, showing that reflexivity corresponds to self-loops, symmetry to undirected edges, and transitivity to the formation of direct connections from indirect paths. When the friendship relation forms an equivalence relation, it naturally partitions the population into disjoint communities. Expanding the framework further, social connectivity is analyzed using measures such as degree, paths, reachability, and connected components to interpret influence, cohesion, and community formation. A comparative analysis of real-world social networks highlights the differences between directed and undirected networks and shows that, although strict global transitivity is rare, actual networks often exhibit high clustering, reflecting local or near transitivity.