Theoretical foundations of soft topological granular structures for multi-criteria social network analysis.

Granular computing (GrC) models complex systems through hierarchically organized information granules, whereas soft set theory offers a flexible mathematical framework for parameter-dependent uncertainty. Motivated by their strong conceptual affinity, this paper develops a soft topological GrC framework grounded in Molodtsov's proximity-oriented soft topology, which is unexplored to a great extent until now by the researchers of soft computing and related areas. New granular operators, reachability relations and saturation concepts are introduced, which lead to a unified structure that integrates soft neighbourhoods with multi-level granularity. Several non-trivial theorems are established concerning granular closure, connectivity, convergence and structural equivalence with examples, which reveals the intrinsic multi-resolution nature of soft topological spaces. The theoretical framework is further interpreted in the context of multi-criteria social network analysis, where contextual interactions induce soft neighbourhoods, and granular communities represent stable influence formations. The present study is primarily theoretical in nature and lays a rigorous foundation for future computational and empirical investigations.